Arnold Tongues
Explore frequency-locking and chaos in driven nonlinear oscillators. The tongue-shaped resonance zones emerge from the circle map as rational frequency ratios lock the oscillator — between them, quasiperiodic motion and chaos.
Viscous Fingering
Watch the Saffman-Taylor instability: a low-viscosity fluid invades a high-viscosity one in a Hele-Shaw cell. The flat interface is unstable — branching fingers grow, split, and compete in fractal patterns. The same math governs lightning, snowflakes, and DLA.
Pappus Chain
An infinite chain of circles packed inside the arbelos (shoemaker’s knife). The nth circle’s center sits exactly n diameters above the baseline — always, regardless of the split ratio. A theorem from 300 AD, proved in an instant by inversion.
Mercator Distortion
Visualize how the Mercator projection warps areas near the poles with Tissot indicatrices, and drag a reference circle to compare true size across latitudes.
Sorting Algorithm Race
Four sorting algorithms race side by side on identical data — watch bubble sort, insertion sort, merge sort, and quicksort reveal why O(n log n) wins.
Vernier Caliper
An interactive vernier caliper with draggable jaw, zoomed alignment view, and adjustable precision — learn how the vernier trick turns a ruler into a precision instrument.
Peg Solitaire
The classic marble-jumping puzzle — select a peg, jump over an adjacent peg into an empty hole to remove it, and try to clear the board down to a single peg.
Plasma Globe
A plasma globe simulation with fractal electric tendrils that dance from a bright core to the glass surface and reach toward your cursor when you move near the sphere.
Sand Timer
An hourglass with real particle physics — thousands of sand grains fall through a narrow neck, pile up in the bottom bulb, and drain from the top with emergent funnel flow.
Fresnel Lens
A cross-section simulator of a Fresnel lens — trace parallel or point-source rays refracting through sawtooth zones and converging at the focal point, with a conventional plano-convex overlay for comparison.
Pi from Billiards
The remarkable result that counting elastic collisions between two blocks and a wall gives digits of π — watch the phase-space circle emerge as collisions trace an arc that reveals why.
Epitrochoid Spirograph
An advanced spirograph drawing tool with full epitrochoid and hypotrochoid mathematics — animate the rolling circle mechanism, layer curves in multiple colors, and explore famous shapes from cardioids to astroids.
Acoustic Beating
Two tones of slightly different frequencies played together produce a pulsing beat at the difference frequency — adjust both frequencies, hear the interference, and see the envelope in real time.
Penrose Tiling Generator
Aperiodic Penrose tilings via iterative deflation — subdivide from level 0 to 7, toggle between P2 kite/dart and P3 rhombus tilings, and watch the golden ratio emerge in tile count ratios.
Telegraph Key
A working Morse code telegraph — tap the brass key to send dots and dashes decoded in real time, or type a message and hear it transmitted automatically at adjustable speed.
Riemann Sphere
Stereographic projection mapping the complex plane to a sphere — place points, draw circles, apply Möbius transformations, and watch how infinity maps to the north pole.
Combustion Engine
A four-stroke internal combustion engine animated from first principles — piston, crankshaft, valves, and spark plug in cross-section with a real-time PV diagram of the Otto cycle.
Elementary Cellular Automata
The full atlas of Wolfram’s 256 elementary 1D rules — select any rule number, paint custom initial conditions, and watch fractals, chaos, and computation emerge from three-cell neighborhoods.
Soap Film Catenoid
The minimal surface between two parallel rings — drag them apart until the critical separation where the soap film snaps. Rotate the 3D wireframe view and watch mean curvature stay exactly zero.
Langton’s Turmite
2D Turing machines on a grid — Langton’s ant, highway builders, spiral makers, and pure chaos. Choose presets or define custom rule tables and watch order emerge from nothing.
Catenary Arch
The ideal self-supporting masonry arch. Flip a hanging chain, place voussoir blocks, trace the thrust line, and distort the shape until it collapses.
The Equation of Time
The difference between sundial time and clock time — up to 16 minutes — decomposed into eccentricity and obliquity, with an animated analemma figure-eight.
Domino Toppling
Place dominos, tip the first, and watch energy transfer through the chain. A rigid-body cascade where wave speed, spacing, and friction shape the physics.
Voronoi Stippling
Lloyd’s relaxation turns random dots into pointillist art. Dark areas attract more points, light areas repel them — weighted Voronoi stippling, converging in real time.
Reaction-diffusion
Two chemicals diffuse and react. Complex patterns emerge from simple local rules.
Strange attractors
The Lorenz system never repeats, never escapes. Deterministic chaos made visible.
Mandelbrot set
Infinite boundary complexity from a two-line rule. Zoom in forever.
Conway’s Life
Four rules. Three states. Enough to simulate anything a computer can compute.
Voronoi diagrams
Every point belongs to its nearest neighbor. Cells of belonging.
Bifurcation diagram
A single parameter drives order into chaos — and back out again.
Gradient descent
How machine learning learns: a ball rolling downhill on a landscape you can shape.
Penrose tiling
Tiles that cover the plane without ever repeating. Almost periodic.
The missing fundamental
Your brain infers a pitch that isn’t in the signal. Inference, not illusion.
Network dynamics
How opinions spread through networks. Topology shapes consensus — or prevents it.
Slime mold
A brainless organism builds efficient transport networks. Thousands of agents, no plan, emergent optimization.
Bioluminescence
Disturb the water. Thousands of dinoflagellates flash in chain reactions — the ocean’s burglar alarm.
Sandpile
Drop grains of sand. Watch them topple. Fractals emerge from the simplest possible rule — self-organized criticality.
Sandpile model
BTW sandpile with continuous random grain addition. Avalanche sizes follow a power law — watch the log-log distribution build up as the system self-organizes to criticality.
Sandpile toppling
Watch avalanche wavefronts propagate in slow motion. Three view modes reveal grain height, accumulated toppling count (fractal geometry), and live activity.
Elementary cellular automata
256 one-dimensional rules. Some produce nothing. Some produce chaos. One is Turing complete. Time flows downward.
Percolation
Adjust a single probability. At the critical threshold, isolated clusters suddenly connect into a spanning network.
Pursuit curves
N agents at the vertices of a polygon, each chasing the next. Logarithmic spirals emerge from the simplest pursuit rule — Bouguer’s 1732 problem, animated.
Diffusion-limited aggregation
Random walkers stick on contact. Fractal dendrites emerge — snowflakes, lightning, and mineral veins from pure randomness.
L-systems
Rewriting rules become fractal geometry. Axiom, grammar, turtle — and plants that never existed grow from pure syntax.
Double pendulum
Two arms, deterministic equations, and wildly unpredictable motion. The simplest system where certainty dissolves into chaos.
Fourier transform
Any signal is just a sum of sine waves. Draw a waveform, watch it decompose, build one from harmonics, and see epicycles trace the shape.
Wave interference
Drop point sources into a ripple tank. Watch waves superpose, interfere, and diffract. The physics of superposition, visible.
Evolutionary game theory
Cooperators and defectors compete on a grid. Spatial structure enables cooperation that pure logic forbids. Watch strategies evolve.
Parrondo’s Paradox
Two individually losing games combine to produce a winning strategy. A counterintuitive result in game theory with implications for ratchet mechanisms, financial portfolios, and evolutionary biology.
Lotka-Volterra
Two species, four parameters. The predator-prey dance that never settles — orbits that close but never repeat exactly.
Ciphers
Type a message, encrypt it, then break it with frequency analysis — the technique Al-Kindi invented in 850 CE Baghdad.
Circle packing
Apollonian gaskets built circle by circle. Descartes’ Circle Theorem recursively fills every gap — a fractal connecting geometry, number theory, and hyperbolic space.
Schelling segregation
Mild individual preferences produce dramatic collective segregation. The gap between personal tolerance and social outcomes, visualized.
Galton board
Balls bounce randomly through pegs and pile into a bell curve. The central limit theorem — order from pure randomness.
Galois field arithmetic
Finite fields made visual. Explore GF(p) prime fields and GF(2⊃n) binary extensions through interactive multiplication tables, additive structure, and generator cycles — the algebra behind AES encryption and Reed-Solomon codes.
Chaos game
Pick a random vertex. Jump halfway. Repeat. Somehow, a fractal appears — order from pure randomness and a simple rule.
Ising model
A lattice of spins, each choosing up or down. Below a critical temperature, order emerges spontaneously — a phase transition from thermal chaos into collective magnetism.
Spin glass
Random ferromagnetic and antiferromagnetic couplings create frustrated magnets that can’t fully order. Watch competing interactions trap the system in complex metastable states — the physics of glassy dynamics, simulated annealing, and exponentially hard optimization.
Random walk
Walkers take random steps and trace unpredictable paths — yet from pure chaos, the bell curve emerges. Diffusion, Brownian motion, and the square-root law made visible.
Dead reckoning
Navigate by integrating noisy measurements from a known start. Watch error accumulate, experiment with corrections, and see why every integrating system drifts without ground truth.
Boids
Three simple rules — separation, alignment, cohesion — and complex flocking emerges with no leader and no global plan. Craig Reynolds’ 1987 algorithm, the science of self-organization made visible.
Brachistochrone
The curve of fastest descent. Race balls down a cycloid, straight line, circular arc, parabola, and catenary — and watch the cycloid always win. Johann Bernoulli’s 1696 challenge, animated.
Ant pheromone trails
No leader, no map, no plan. Ants leave chemical traces that recruit others, and efficient foraging trails emerge from pure local interaction — stigmergy, the coordination mechanism that scales from termites to Wikipedia.
N-body gravity
Every mass pulls every other mass. The two-body problem is solved; the three-body problem is chaos. Add bodies, launch them with velocity, and watch orbits form, destabilize, and eject.
Voting systems
Same voters, same preferences, different winners. Five voting methods applied to the same ballots — Arrow’s impossibility theorem made interactive.
Knot explorer
Click crossings to toggle over and under. Watch a trefoil become an unknot. Knot identity is irreducibly holistic — it lives in the whole, never in the parts.
Mycorrhizal networks
Trees and fungi trade nutrients for carbon underground. Slide the soil richness and watch mutualism become parasitism — the Wood Wide Web is more complicated than it sounds.
Siteswap juggling
Mathematical juggling notation, animated. Enter any siteswap sequence, watch balls fly between hands on parabolic arcs, and learn why 441 was discovered by math before any juggler threw it.
Steiner tree
Place points, find the shortest connecting network. Steiner points appear at 120° junctions — the geometry soap films discover naturally. NP-hard, yet physics solves it anyway.
Tuning systems
Hear the impossible compromise at the heart of Western music. Compare Equal Temperament, Just Intonation, and Pythagorean tuning — three solutions to the fact that log₂(3) is irrational.
Epidemic SIR model
Watch disease spread through a population of moving agents. Adjust infection and recovery rates, discover the herd immunity threshold, and vaccinate to contain outbreaks.
Aperiodic monotile
One shape that tiles the plane but never periodically. The hat polykite, discovered in 2023 after a 60-year search — the first aperiodic monotile.
Magnetic field visualization
Place magnetic dipoles and watch field lines emerge in real time. Superposition, dipole fields, and the invisible architecture of electromagnetism made visible.
Orbital mechanics
Place orbiting bodies around a central star, drag to set velocity, and watch Kepler’s laws emerge — ellipses, equal areas, and the T² ∝ a³ relationship that holds the solar system together.
Ray optics
Place lenses, mirrors, and prisms on a canvas and watch light rays trace through them — Snell’s law, focal points, chromatic dispersion, and total internal reflection, all rendered in real time.
Circuit simulator
Place batteries, resistors, LEDs, switches, and capacitors on a grid, wire them together, and watch animated electrons trace the current that Ohm’s law predicts and Kirchhoff’s laws constrain.
Cloth simulation
A grid of particles connected by distance constraints, integrated forward by Verlet’s method. Grab it, tear it, blow wind on it — position-based dynamics with no springs and no velocities, just geometry.
Genetic algorithm
A population of candidate routes evolves by natural selection. The shortest survive, reproduce, and mutate — no gradient, no formula, just variation, selection, and time solving the traveling salesman problem.
Maze generation & solving
Four algorithms carve passages through a grid — DFS winds deep corridors, Prim’s branches organically, Kruskal’s randomizes uniformly, recursive division builds rooms. Then four solvers race to find the path.
Neural network playground
Place points, choose their class, configure a network — layers, neurons, activation functions — and watch it learn. The decision boundary emerges as a heatmap: warm gold for one class, cool blue for the other.
Fluid dynamics
Draw obstacles in a wind tunnel and watch turbulence emerge. The Lattice Boltzmann Method — vortex streets, laminar-to-turbulent transitions, and the Navier-Stokes equations solved without solving them.
Non-Newtonian Fluid
Cornstarch suspensions shear-thicken under impact while ketchup shear-thins when shaken. Explore viscosity that depends on strain rate, with shear-thickening and viscoelastic fluids side by side.
Map projections
You can’t flatten a sphere without lying. Ten projections, Tissot circles, and a distortion overlay — see what every map hides.
Graph algorithms
Build a graph, pick an algorithm, watch it think. BFS, DFS, Dijkstra, A*, Kruskal, and Prim — step by step with animated data structures.
Sorting algorithms
Watch seven classic sorts race. Bubble, insertion, selection, merge, quick, heap, and radix — animated bar-by-bar with comparison and swap counters.
RSA cryptography
Public-key cryptography made visible. Pick two primes, watch the key generation math unfold, then encrypt and decrypt messages — every modular exponentiation on display.
Planetary system builder
Click to place stars and planets, drag to fling them into orbit. N-body gravity with Velocity Verlet integration, collisions, mergers, and orbital trails.
Diffraction patterns
Compute the 2D Fourier transform of point arrangements. See how Penrose tiling vertices produce the “forbidden” 10-fold symmetric diffraction that won Shechtman the Nobel Prize.
Auxetic lattice simulator
Stretch a lattice and watch what happens to its width. Re-entrant honeycombs get wider when pulled — negative Poisson’s ratio from geometry alone.
Moiré patterns
Overlay two periodic structures with a slight twist and watch large-scale interference patterns bloom — the spatial beat frequency, visible.
Hyperbolic geometry
The Poincaré disk model: an infinite plane of negative curvature inside a finite circle. Draw geodesics, explore tessellations, and see the geometry Euclid’s fifth postulate forbids.
Lissajous curves
Two perpendicular oscillations trace parametric curves whose shapes encode musical intervals — unison, octave, perfect fifth. The mathematics of harmonographs and oscilloscopes.
Mechanical linkages
Rigid bars and pivot joints convert one motion into another. From Watt’s steam engine to Jansen’s walking Strandbeests — the geometry of constrained motion, animated.
Langton’s Ant
An ant on a grid follows two rules: turn right on white, turn left on black. For ten thousand steps, chaos. Then an ordered highway emerges from the noise. Multi-color turmite extensions included.
Probability distributions
Every distribution is a different answer to “how is uncertainty structured?” Nine distributions with adjustable parameters, random sampling, and the law of large numbers made visible.
Chladni patterns
Sprinkle sand on a vibrating plate and it migrates to the nodal lines. Rectangular and circular plate modes, superposition mixing, and the geometry of resonance.
Harmonograph
Damped pendulums trace decaying parametric curves. Frequency ratios map to musical intervals — but unlike Lissajous figures, the pattern spirals inward as energy dissipates. Victorian drawing machines, simulated.
Heat equation
Paint heat sources onto a 2D surface and watch thermal diffusion unfold. Draw insulating walls, toggle boundary conditions, and see the finite difference method solve the heat equation in real time.
Huffman coding
David Huffman’s 1952 algorithm builds an optimal prefix-free binary tree from character frequencies. Watch the tree construct step by step and see how close it gets to Shannon entropy.
Dynamical billiards
A ball bouncing inside a closed boundary — no friction, perfect reflection. Circle: predictable. Stadium: chaos. The shape alone determines the dynamics. Phase space included.
Mathematical string art
Connect each point i to point (i × m) mod N on a circle. Only straight lines — but cardioids, nephroids, and epicycloids emerge from pure modular arithmetic.
Phyllotaxis
Sweep the divergence angle and watch spirals form. Only at the golden angle — 137.508° — do seeds pack without gaps, and the Fibonacci numbers appear in the spiral count.
Pendulum waves
A row of uncoupled pendulums with slightly different periods create stunning wave patterns — traveling waves, standing waves, and chaos before realigning.
Tessellations
Which polygons tile the plane? Explore the three regular and several semi-regular tilings — the symmetry groups behind crystallography and Islamic art.
Spirograph
Watch epitrochoid and hypotrochoid curves trace in real time as circles roll within circles — the parametric equations behind the classic toy.
Markov chains
Edit a transition matrix, watch probability flow through a state graph, and discover the steady-state distribution every ergodic chain converges to — the mathematics behind PageRank, weather models, and statistical physics.
Bayesian inference
Flip coins of unknown bias and watch beliefs update in real time. Prior times likelihood equals posterior — the optimal rule for learning from evidence, with the base rate fallacy that fools almost everyone.
Bézier curves
Drag control points and watch de Casteljau’s algorithm subdivide in real time. Recursive linear interpolation — the same mathematics behind every font you read and every smooth path in computer graphics.
Procedural noise
How simple gradients create complex worlds. Tweak octaves, persistence, and lacunarity to generate terrain, clouds, marble, and wood grain — Perlin noise and fractal Brownian motion, interactive.
Prime spiral
Arrange integers in a spiral and highlight the primes. Mysterious diagonal lines appear — artifacts of quadratic polynomials that nobody fully understands.
Verlet physics
Ropes, chains, curtains, and webs — all simulated with position-based Verlet integration. Grab, swing, cut, and watch the same constraint-based physics behind ragdoll animation and cloth simulation.
Wave Function Collapse
Procedural generation by constraint propagation. Each cell starts as a superposition of all possible tiles — observe, collapse, propagate, repeat. The algorithm behind procedural level design.
Catenary curves
The shape a hanging chain makes is not a parabola. Drag anchor points, adjust slack, toggle the parabola overlay, and see the curve Galileo got wrong and the Bernoullis solved.
Fourier epicycles
Draw any closed shape and watch rotating circles reconstruct it. The Discrete Fourier Transform as visible epicycles — from Ptolemy’s planetary models to modern signal processing.
Acoustic levitation
Standing wave pressure fields trap particles at pressure nodes, defying gravity through sound. Simulate the Gor’kov potential and watch objects float in an acoustic field.
Lenia
Conway’s Game of Life generalized to continuous space and time. Smooth creatures glide, pulse, and self-organize from pure mathematics — artificial life from a growth function.
Kinetic theory
Particles bounce in a box, trading momentum through elastic collisions. From this chaos, the Maxwell-Boltzmann speed distribution emerges — statistical mechanics made visible.
Kuramoto model
Coupled oscillators with different natural frequencies. Below critical coupling, disorder. Above it, spontaneous synchronization — the phase transition behind fireflies, neurons, and power grids.
Minimal surfaces
Surfaces that minimize area for given boundary constraints — the shapes soap films find naturally. Rotate catenoids, helicoids, and Enneper surfaces, and watch conjugate minimal surfaces continuously deform into each other.
Quorum sensing
Bacteria communicate by releasing signaling molecules. When enough accumulate, behavior switches collectively — a phase transition from individual to group identity, with cheaters and the public goods game.
Lennard-Jones potential
Atoms attract at a distance and repel up close. The 6-12 potential produces gas, liquid, and solid phases from nothing but pair interactions — molecular dynamics with real-time radial distribution functions.
Digital logic gates
Build digital circuits from logic gates. Place AND, OR, NOT, XOR, NAND, and NOR gates on a canvas, wire them together, toggle inputs, and watch signals propagate — Boolean algebra made visible.
Elastic collisions
Launch balls of different masses and watch momentum and kinetic energy conserve through elastic and inelastic collisions — from Newton’s cradle to particle physics.
Delaunay triangulation
Place points and watch the Delaunay triangulation form in real time. Toggle the dual Voronoi diagram, inspect circumcircles, and step through the Bowyer-Watson algorithm.
Color space explorer
Slice through RGB, HSL, HSV, CIELAB, and CMYK. See why equal steps in RGB look uneven, how CIELAB was designed to match human vision, and where sRGB runs out of gamut.
Soap films & foam physics
Voronoi foam with von Neumann coarsening — cells with fewer than six sides shrink, more than six grow, and hexagons stay stable. Plateau’s laws in action.
Traffic flow
The Nagel-Schreckenberg cellular automaton on a circular road. Phantom traffic jams emerge from nothing — backward-propagating waves with no cause, visible in the space-time diagram.
Cycloid curves
Rolling circles trace beautiful curves. Cycloids, epicycloids, hypocycloids — cardioids, nephroids, astroids — all from one mechanism: rolling without slipping.
Coupled spring-mass systems
A chain of masses connected by springs vibrates in normal modes. Energy sloshes between oscillators — the discrete precursor to every vibrating string and standing wave.
Brownian motion
A large particle buffeted by many small ones performs a random walk. The phenomenon Einstein explained in 1905, proving atoms exist — with MSD tracking and diffusion visualization.
Brownian ratchet
Feynman’s ratchet and pawl: a microscopic device that appears to extract work from heat, but at thermal equilibrium backward slips exactly cancel forward clicks. The second law, animated.
Riemann zeta function
Domain coloring of ζ(s) on the complex plane. Phase encodes hue, magnitude encodes brightness — the critical line, non-trivial zeros, and the deepest unsolved problem in mathematics.
Rope bridge physics
A rope hanging between anchor points, simulated with Verlet integration. Drag to disturb it, watch tension propagate as color — catenary curves, wind buffeting, and position-based dynamics.
3D strange attractors
Six strange attractors rendered in three dimensions with a rotating camera. Lorenz, Rössler, Chen, Halvorsen, Thomas, and Aizawa — chaos theory in full rotation.
Doppler effect
A moving source emits circular wavefronts. Compression ahead, expansion behind — the physics of ambulance sirens, redshift, and sonic booms at Mach 1.
Hydraulic erosion
Drop thousands of water particles onto a heightmap. Each droplet carries sediment downhill, carves channels, and deposits in valleys — the Grand Canyon in seconds.
Hydraulic jump
A thin supercritical jet of water suddenly thickens into a ring — the circle you see when water from a faucet hits a sink. Adjust flow rate and viscosity to shift the jump radius.
Hodograph
Trace velocity vectors in velocity space for a Keplerian orbit. Hamilton’s 1847 discovery: the hodograph is always a perfect circle, regardless of the orbit’s eccentricity.
Tippe top
A truncated sphere that flips upside-down when spun fast enough, raising its own center of mass. Friction drives gyroscopic precession into a paradox-resolving inversion.
Turing machine
An infinite tape, a read/write head, and a finite state table. Step through binary counters, busy beavers, and palindrome checkers — the foundation of all computation.
Kelvin-Helmholtz instability
Two fluid layers at different velocities shear into rolling vortices. The physics behind Jupiter’s bands and wind over water — beautiful wave patterns from velocity gradients.
Magnetic pendulum
A pendulum swings over magnets and settles unpredictably. Fractal basins of attraction emerge — chaos, sensitive dependence, and the impossibility of long-term prediction.
Space-filling curves
Hilbert, Peano, Z-order, and Dragon curves — one-dimensional lines that visit every point in a two-dimensional square. Step through iterations and watch the fractal unfold.
Monte Carlo methods
Estimate π by throwing random darts at a circle inscribed in a square. Watch convergence. Then estimate integrals by sampling — the brute-force method that solves the unsolvable.
Projectile motion
Launch a projectile and watch physics unfold. Angle, velocity, gravity, air resistance — adjust them all and see how drag transforms the clean parabola into something messier and more real.
Convex hull
Click to place points, then watch algorithms stretch a rubber band around the outermost ones. Gift wrapping, Graham scan, and quickhull — three strategies for the same boundary, animated step by step.
Quaternion rotations
Drag to rotate a 3D shape, then watch Euler angles seize up at the poles while quaternions glide through. SLERP interpolation, gimbal lock, and the geometry of 4D rotation.
Poisson disk sampling
Watch Bridson’s algorithm scatter points like seeds — never too close, never in rows. Compare blue noise to white noise and see why game engines prefer this elegant randomness.
Simple machines
Six machines that changed civilisation. Drag the fulcrum, add pulleys, tilt the ramp — watch force multiply and energy stay conserved. The physics of mechanical advantage.
PID controller
Adjust proportional, integral, and derivative gains and watch a system chase its setpoint — overshoot, oscillation, and steady-state error from three knobs.
Electromagnetic induction
Drag a magnet near a coil of wire and watch current appear from nothing but motion. Faraday’s law, Lenz’s law, and the birth of electromagnetism.
Origami crease patterns
Draw mountain and valley folds on a square, check flat-foldability with Kawasaki’s and Maekawa’s theorems, and explore the mathematics hidden in paper folding.
Cellular Potts model
A generalization of the Ising model to multi-cell biological simulation. Cells grow, shrink, move, and sort by differential adhesion — morphogenesis from statistical mechanics.
Lévy flight
A random walk with heavy-tailed step sizes. Local exploration punctuated by long jumps — the optimal foraging strategy found in albatrosses, financial markets, and search algorithms.
Trebuchet physics
A counterweight falls, an arm swings, a sling whips forward. Potential energy converts to kinetic energy through lever mechanics — medieval siege engineering, simulated.
Ant colony optimization
Virtual ants deposit pheromone on edges they traverse, converging on short tours. Nature’s heuristic for the Traveling Salesman Problem — Dorigo’s 1992 algorithm, animated.
Double spring
Two masses on springs in series, hanging under gravity. Normal modes, energy transfer between oscillators, and the phase portrait of coupled motion.
Langton’s ant variants
Extend Langton’s ant with multi-color rules. Highways, square spirals, space-filling diamonds, and symmetric patterns from simple turning instructions.
Hash function explorer
Type input, watch each character’s contribution to the hash. Change one bit and see the avalanche effect cascade through every output bit. Then hunt for collisions.
N-body simulation
Place gravitational bodies, drag to set velocity, scroll to adjust mass. Softened gravity, energy tracking, and presets from binary stars to the figure-8 three-body orbit.
Wave interference pool
Click to drop wave sources into a ripple tank. Circular waves spread, overlap, and interfere — constructive and destructive patterns from the 2D wave equation with damping.
Double-slit experiment
Wave interference from two slits — the experiment that reveals the wave nature of light and matter. Toggle particle mode and watch individual detections build the pattern quantum mechanics predicts.
Rubik’s Cube
A fully interactive 3D Rubik’s cube. Drag to rotate the view, click face buttons to twist layers. Scramble and solve — 43 quintillion states, one solved position.
Musical scales
Explore ten scales on a piano keyboard and chromatic circle. Hear each note, play ascending and descending, and see the interval patterns that give each scale its character.
Collatz conjecture
If even, halve it. If odd, triple and add one. Does every number reach 1? Nobody knows. Watch hailstone sequences animate and explore the tree of convergence.
Compass & magnetic declination
Magnetic north is not true north. Click a world map to place your compass and watch the needle respond — declination, dip angle, and field intensity vary by location and drift over decades.
Curve fitting
Click to place data points, choose a regression method, and watch the curve adapt in real time. Drag points, check residuals, and see why R² alone never tells the whole story.
Platonic solids
Five shapes — the only convex regular polyhedra that can exist. Rotate, zoom, toggle duals, and morph between them. V − E + F = 2, always.
Fourier drawing
Draw any shape and watch rotating epicycles reconstruct it. The Discrete Fourier Transform as visible spinning circles — add more frequencies, sharpen the fit.
Chaos game fractals
Jump toward random polygon vertices with restriction rules. Sierpinski triangles, carpets, Vicsek fractals, and unnamed structures from pure randomness and simple constraints.
Tides simulator
Watch tidal bulges form from Moon and Sun gravity. Toggle each body, see spring vs neap tides, track tide height at an observer — the physics of the ocean’s rhythm.
Brachistochrone racer
Five curves connect two points. Release the balls and watch them race under gravity — straight line, circular arc, parabola, catenary, and the cycloid that always wins. The calculus of variations, as a race.
Signal processing
Generate sine, square, sawtooth, and triangle waves, mix them together, add noise, and apply low-pass, high-pass, or band-pass filters. Time domain and FFT spectrum side by side — press play to hear the math.
Siphon
Fluid flows upward over a barrier and down the other side, driven by gravity and atmospheric pressure. Bernoulli’s equation, Torricelli’s law, and the ~10.3 m limit where the water column breaks.
Spirals in nature
Logarithmic spirals, the golden angle, Fibonacci rectangles, and sunflower seed packing. Why 137.5° is the most irrational angle, and how nature uses it to fill space without waste.
Wireworld
Four cell states, four rules, and enough expressive power to build clocks, diodes, and logic gates — a Turing-complete cellular automaton for simulating electronic circuits.
Wave interference
Two point sources emit circular waves. Where crests meet crests, amplitude doubles. Where crests meet troughs, they cancel. Superposition, nodal lines, and phase — the physics of interference.
Geodesics
Shortest paths on curved surfaces. Click two points on a sphere and watch the great circle bend with the curvature — the geometry that light and free-falling objects follow through spacetime.
Enigma machine
An interactive WWII Enigma cipher simulation. Configure rotors, ring settings, and plugboard, then type to watch the electrical signal path illuminate through each component.
Tower of Hanoi
The classic recursive puzzle. Drag disks between pegs or watch the optimal solution animate — 2n − 1 moves, exponential growth from the simplest recursion.
Spectrogram
Real-time audio frequency visualization. Microphone input or built-in tone generator, scrolling spectrogram, live spectrum, and configurable FFT — sound decomposed into frequencies.
Ideal gas law
PV = nRT made visible. Particles bounce inside a container with a draggable piston — adjust temperature, volume, and particle count, and watch pressure respond in real time with Boyle’s and Gay-Lussac’s laws.
Color vision
See through different eyes. Paint on a canvas and watch your art transform under protanopia, deuteranopia, tritanopia, and achromatopsia — trichromatic theory and cone response curves, interactive.
Steganography
Hide secret messages in the least significant bits of pixel data. Encode, decode, and visualize bit planes — the art of hiding information in plain sight.
Kaleidoscope
Draw with your mouse and watch dihedral symmetry replicate your strokes across mirror segments. The symmetry group Dn — rotations and reflections — turns gestures into mandalas.
Blackbody radiation
Planck’s law spectral curves at any temperature. Wien’s displacement, Stefan-Boltzmann total power, stellar presets, and the Rayleigh-Jeans ultraviolet catastrophe that launched quantum mechanics.
Topological surfaces
Rotate Möbius strips, Klein bottles, tori, and Boy’s surface in 3D. Wireframe, solid, and transparent rendering — with normal coloring and animated points that reveal non-orientability.
Voronoi fracture
Click a surface to shatter it. Cracks propagate along Voronoi cell boundaries — the grain-boundary fracture physics of glass, ceramics, and ice, with stress visualization and fragment separation.
Atmospheric scattering
Drag the sun across the sky and watch Rayleigh scattering paint the atmosphere. Blue at noon, red at sunset — because intensity scales as 1/λ&sup4; and longer paths scatter away the short wavelengths.
Spirograph advanced
A three-gear spirograph with visible mechanics. Watch nested gears rotate and trace hypotrochoid and epitrochoid curves — rose curves, cardioids, nephroids, and unexplored three-gear patterns.
Electric field
Place positive and negative charges and watch field lines and equipotential contours emerge in real time. Drag charges to rearrange — Coulomb’s law, superposition, and the invisible geometry of electrostatics.
Fluid dynamics (Stable Fluids)
Jos Stam’s stable fluids method in real time. Inject dye and velocity, place obstacles, and switch between five visualization modes — the Navier-Stokes equations solved without solving them.
Prisoner’s dilemma tournament
Ten strategies compete in iterated Prisoner’s Dilemma tournaments. Watch population dynamics shift over generations — Axelrod’s 1980 result that nice strategies win, made interactive.
Interference patterns
Point sources emit circular waves that spread, overlap, and interfere. Drag sources, adjust wavelength and frequency, and watch constructive and destructive patterns emerge from the superposition principle.
Rope physics
Ropes and chains simulated with Verlet integration and distance constraints. Grab, fling, pin, and cut them — elastic, rigid chain, and bungee types with tension visualization.
Number theory
Explore the hidden structure of integers. Animated sieve of Eratosthenes, Euler’s totient function, divisor graphs, modular multiplication circles, and factor trees — the queen of mathematics, interactive.
Benford’s law
The digit 1 appears as the leading digit ~30% of the time in real-world data. Toggle between powers of 2, Fibonacci, factorials, and uniform random to see why — and why fraud detectors use it.
Rayleigh-Taylor instability
Heavy fluid on top of light fluid under gravity. Small perturbations grow into finger-like structures, mushroom caps, and turbulent mixing — the instability behind supernovae and inertial confinement fusion.
Hohmann transfer orbit
The most fuel-efficient way to move between two circular orbits. Two burns, one ellipse, and the fundamental trade-off of orbital mechanics — with delta-v calculations and presets for LEO-GEO and Earth-Mars.
Gabriel’s Horn
Torricelli’s Trumpet: a 3D surface with finite volume (π) but infinite surface area. Fill it with paint, but never paint its surface — the painter’s paradox, visualized and rotating.
Catmull-Rom splines
Place control points and draw smooth curves that pass through every one. Drag to reshape, adjust tension, toggle tangent vectors — the interpolating spline behind animation paths and camera tracks.
Toroidal Life
Conway’s Game of Life on a 3D torus. Every Life grid wraps toroidally — here you see it. Gliders trace paths across a rotating donut surface, never teleporting, always continuous.
Dragon curve
Take a strip of paper, fold it in half repeatedly, unfold to 90°. The Heighway dragon emerges — a space-filling fractal from nothing but right-angle folds, colored by iteration depth.
Spring pendulum
A mass on a spring that swings and bounces. Energy flows chaotically between radial and angular modes — the simplest system where two oscillations couple into chaos.
Pigeonhole principle
Birthday paradox, sock drawer, and hash collisions. The most elementary counting argument produces the most counterintuitive results — most people guess 183 when the answer is 23.
Inverted pendulum
Balance a pole on a moving cart — manually with arrow keys or with a tunable PID controller. The classic control theory problem where three gains separate stability from collapse.
Nyquist sampling
Sample a signal below twice its frequency and phantom aliases appear. The sampling theorem made visible — with sinc reconstruction, multiple signal types, and the red line where information is lost.
Central limit theorem
Average samples from any distribution — uniform, exponential, bimodal, or one you draw yourself — and watch the histogram converge to a bell curve. The most important theorem in statistics, animated.
Lorenz waterwheel
Water pours into leaky cups on a spinning wheel. At low flow it sits still, at medium flow it rotates steadily, at high flow — chaos. The physical system behind the Lorenz attractor.
Monty Hall problem
Pick a door, the host reveals a goat, then decide: switch or stay? Play the game yourself, then auto-run a thousand rounds and watch the counterintuitive 2/3 emerge.
String vibration
Pluck a string fixed at both ends and watch standing waves form. Toggle individual harmonics, mix them together, and see the frequency spectrum — the physics of every stringed instrument.
Gravity assist
A spacecraft steals kinetic energy from a moving planet — the slingshot maneuver. Toggle reference frames to see why speed changes in the sun frame but not the planet frame.
Gosper curve
The Gosper curve (flowsnake) is a space-filling fractal generated by a hexagonal L-system. Watch it grow through recursion depths 1–6, zoom into self-similar detail, and reveal the Gosper island that tiles the plane.
Wolfram elementary automata
256 rules. One row of cells. One time step. Wolfram’s elementary cellular automata show how simple local rules produce fractals (Rule 90), chaos (Rule 30), and even Turing-complete computation (Rule 110).
Kapitza pendulum
A pendulum with a rapidly oscillating pivot can stand stably upright — defeating gravity through parametric stabilization. Tune drive frequency and amplitude to find the stability threshold, and watch the phase portrait trace the attractor.
Barnsley fern
Four affine transformations, chosen at random, draw an entire fern point by point. Edit the IFS coefficients to grow new plants from pure mathematics.
Parabolic reflector
Why satellite dishes, headlights, and telescopes are all parabolas. Parallel rays converge to a focus; a point source at the focus becomes a parallel beam. Compare with circular reflectors to see spherical aberration.
Photoelectric effect
Photons hit a metal surface and eject electrons — but only above a threshold frequency. Einstein’s 1905 equation E = hf − φ, the birth of quantum mechanics, made interactive.
Van der Pol oscillator
Nonlinear damping creates a limit cycle — every trajectory converges to the same closed orbit. Increase μ and watch smooth sinusoids sharpen into relaxation oscillations.
Planetary gears
Sun, planets, ring, and carrier — lock any component and drive another. The epicyclic gear train behind automatic transmissions, with live gear ratio calculations.
Coriolis effect
Launch a ball across a spinning turntable and toggle between inertial and rotating frames. The same straight-line motion appears curved — the fictitious force that steers hurricanes.
Predator-prey flocking
Reynolds’ three flocking rules plus a predator. Watch the flock split, scatter, and reform as flee overrides cohesion — the dynamics of starling murmurations under hawk attack.
Recamán sequence
Subtract n or add n — whichever lands on a new positive integer. Semicircular arcs alternate above and below a number line, building an unexpectedly beautiful pattern from a trivial rule.
DLA 3D
Diffusion-limited aggregation in three dimensions. Particles random-walk through space and stick on contact, building fractal dendrites you can rotate with your mouse — snowflakes in full spatial glory.
Ulam spiral
Write the integers in a spiral, highlight the primes. Mysterious diagonal lines appear — artifacts of prime-rich quadratic polynomials that nobody fully understands. Hover for factorizations, zoom to explore.
Ant colony bridge-building
Army ants form living bridges across gaps. Individual ants lock their bodies in place while others walk across. The bridge self-optimizes — ants join when traffic is high, leave when it drops. No leader, no plan.
Snowflake crystal growth
Hexagonal ice crystal growth using the Reiter cellular automaton. Vapor diffuses on a triangular grid and freezes at crystal boundaries — six-fold symmetry from simple local rules.
Soliton waves
The Korteweg–de Vries equation produces solitary waves that maintain their shape and pass through each other unchanged. Create solitons of different amplitudes and watch them collide.
Marangoni effect
Surface tension gradients from temperature differences drive fluid flow. Place heat sources and watch particles stream from hot to cold — the physics behind wine tears and welding pool convection.
3D Spirograph
Hypotrochoid curves traced by nested rotating circles, lifted into three dimensions with perspective projection. The classic toy elevated into a space you can orbit around.
Continued fractions
Every real number unfolds into a nested tower of integers. Watch convergents approach from alternating sides — the best rational approximations, the golden ratio’s stubbornness, and why 355/113 is miraculous.
Strange attractors gallery
Six famous attractors beyond Lorenz — Rössler, Thomas, Aizawa, Dadras, Chen, and Halvorsen — rendered as 3D point clouds with adjustable parameters and mouse-drag rotation.
Weaire-Phelan structure
The best known solution to the Kelvin problem: tiling 3D space with equal-volume cells of minimum surface area. Two cell types, 0.3% better than Kelvin’s truncated octahedra, and still unbeaten since 1993.
Estimating π
Four methods for computing the same transcendental number. Monte Carlo darts, Buffon’s needles, the Leibniz series, and Archimedes’ polygon squeeze — all converging to 3.14159…
2D wave equation
Click to disturb a vibrating membrane. Waves propagate outward, reflect off boundaries, and interfere — the full two-dimensional wave equation on a grid, with adjustable damping and boundary conditions.
Binary tree
Interactive binary search tree with insert, delete, search, AVL balancing, animated traversals, and heap mode. Educational annotations explain time complexity at each step.
Combustion
Particle-based fire simulation with temperature-driven color gradients, smoke, wind, and gravity. Candle, campfire, and explosion modes — from slow burn to rapid ignition.
Electromagnetic wave
3D visualization of EM wave propagation showing perpendicular E and B fields, polarization modes, and a rotatable filter. Maxwell’s equations in motion.
Catapult physics
A counterweight falls, an arm pivots, a projectile launches. Adjust mass, arm geometry, and release angle — watch energy transfer from potential to kinetic and trajectories bend under drag.
Voronoi crystal growth
Nucleation seeds grow outward into crystals colored by orientation. Grain boundaries form where growth fronts collide — the physics behind metals, ice, and ceramics.
Topology puzzle
Are these two shapes topologically equivalent? Classify pairs of surfaces — sphere, torus, Möbius strip, Klein bottle, double torus — and watch animated deformations prove it.
Riemann sums
The definite integral as a limit of sums. Pick a function, choose left, right, midpoint, or trapezoidal method, and watch rectangles converge to the true area as n increases.
Magnetic domains
Ferromagnetic domain simulation. Tiny arrow-magnets align with neighbors, form domains, and respond to external fields — with hysteresis loops, temperature control, and Barkhausen jumps.
Anamorphic art
Draw on a flat canvas and watch it stretch into a distortion that looks correct only from a specific viewing angle or reflected in a cylindrical mirror — Renaissance optical illusions, interactive.
Pascal’s triangle
Every entry is the sum of the two above it. Color by divisibility and fractals appear — Sierpinski triangles, Fibonacci diagonals, and binomial coefficients made visible.
Shive wave machine
A row of torsion-coupled rods. Flick one end and watch the wave propagate, reflect, and superpose. Fixed and free boundary conditions change everything.
Cryptography playground
Four classic ciphers side by side — Caesar, Vigenère, substitution, rail fence. Live encryption, frequency analysis, and a crack mode for breaking codes.
Fermat spiral explorer
Fermat’s spiral creates sunflower seed patterns. Adjust the divergence angle and watch optimal packing emerge at the golden angle — Fibonacci parastichies from pure geometry.
Telegraph simulator
An interactive Morse code telegraph. Click the key or press spacebar to tap out dots and dashes, decoded in real time. Switch to receive mode and decode incoming messages.
Probability distribution explorer
Seven distribution families with PDF and CDF side by side. Drag parameters, sample and overlay histograms, and watch key statistics update live — density and cumulative views of uncertainty.
Knot theory explorer
Visualize mathematical knots as 2D projections with proper over/under crossings. Trefoil, figure-eight, cinquefoil, torus knots — gradient-colored strands, crossing numbers, and an unknot untangling animation.
Sundial simulator
A horizontal sundial viewed from above. Set latitude, date, and time — watch the shadow trace hour lines at the correct speed. Equation of time, midnight sun, and polar night behavior.
Langton’s ant 3D
Langton’s ant walks on the surface of a cube. Six face grids, correct edge-wrapping orientation, and the classic highway behavior routing across 3D geometry.
Practical rope knots
An interactive step-by-step guide to five essential knots — bowline, clove hitch, figure-eight, sheet bend, and square knot. Animated rope with over/under crossing indicators.
Tidal forces
How the Moon and Sun create two tidal bulges on Earth. Drag the Moon, toggle the Sun, and watch spring tides align and neap tides cancel — the far-side bulge explained.
Cellular automata music
Elementary cellular automata generate music. Each column maps to a note in a scale — alive cells play. Rule 30 on pentatonic sounds surprisingly musical.
Marching squares
Paint scalar values on a grid and watch contour lines form in real time. The marching squares algorithm — 16 lookup cases, linear interpolation, and the technique behind topographic maps and medical imaging.
The cardioid
One heart-shaped curve, three constructions: a rolling circle’s trace, the envelope of a modular multiplication circle, and the main bulb of the Mandelbrot set. The same curve keeps appearing in unrelated mathematics.
Dijkstra’s algorithm
Build a weighted graph, pick start and end nodes, and watch the algorithm explore outward — always expanding the closest unvisited node. The greedy strategy behind GPS navigation and network routing.
Kaprekar’s routine
Sort descending, sort ascending, subtract, repeat. Every 4-digit number converges to 6174 in at most 7 steps. Watch the routine animate and explore the step-count histogram for all starting numbers.
Seismic waves
Click to trigger an earthquake and watch P-waves and S-waves propagate through Earth’s layered interior. Snell’s law refracts rays at each boundary — the physics behind shadow zones and seismology.
Nim
Play against a perfect AI that uses XOR strategy. Multiple heaps, binary Nim-sum visualization, and the Sprague–Grundy theorem — the mathematical backbone of combinatorial game theory.
Stereographic projection
Map points between a sphere and a plane through the north pole. Circles map to circles, angles are perfectly preserved — conformal geometry from Riemann spheres to astrolabes, interactive.
Thermal erosion
Paint mountains onto a heightmap and watch gravity sculpt them. Material slides downhill when slope exceeds the angle of repose — thermal weathering without water, shaping scree slopes and talus cones.
Euclidean algorithm
The oldest known algorithm, animated. Watch rectangle-cutting find the GCD, step through division tables, and compute Bézout coefficients — twenty-three centuries of mathematics in one visualization.
Lorenz system
The butterfly effect made visible. Two trajectories a hair’s breadth apart diverge exponentially on the Lorenz attractor — adjustable σ, ρ, β parameters and 3D phase space projections.
Buffon’s needle
Drop needles onto parallel lines and estimate π from crossings. The 1777 experiment that founded geometric probability — watch the convergence in real time.
Chaos pendulum
A single driven damped pendulum on the route to chaos. Phase portrait, Poincaré section, and bifurcation diagram reveal the period-doubling cascade as drive amplitude increases.
Bloch sphere
Interactive 3D visualization of a qubit state on the Bloch sphere. Apply quantum gates and watch the state vector rotate in real time.
Maxwell-Boltzmann distribution
Simulate ideal gas molecules and visualize the Maxwell-Boltzmann speed distribution. Adjust temperature and see how molecular speeds change in real time.
Crystal lattice viewer
Interactive 3D crystal structure viewer. Explore SC, BCC, FCC, HCP, and Diamond Cubic lattices with rotation, zoom, and bond visualization.
Shannon entropy
Interactive visualization of Shannon entropy and information content. Type text and watch entropy, character frequencies, and compression ratios update in real time.
Light polarization
Simulate polarized light passing through a chain of polarizing filters. Visualize Malus’s law, electric field oscillations, and the famous crossed-polarizer paradox.
Hertzsprung-Russell diagram
Interactive HR diagram plotting stellar luminosity vs temperature. Explore the main sequence, red giants, white dwarfs, and watch stars evolve across the diagram.
Benham’s disk
A spinning black-and-white disk that tricks your brain into seeing color. Explore Fechner colors, temporal color mixing, and the surprising limits of human cone response.
Phase diagram explorer
An interactive pressure-temperature phase diagram. Drag a point across the boundaries between solid, liquid, and gas to watch molecules rearrange in real time.
Rigid body dynamics
Push and spin 2D rigid bodies. Every click applies a force that creates both translation and rotation — see the interplay of linear and angular momentum.
Spiraling vortex
Particles spiral inward through vortex velocity fields. Click to place vortex centers with positive or negative rotation — complex swirling patterns from simple superposition of irrotational flow.
Theremin
An interactive theremin instrument. Mouse X controls pitch, mouse Y controls volume — sine wave synthesis with real-time oscilloscope visualization, waveform selection, and reverb.
Apollonian gasket
Three mutually tangent circles define a fourth. Recurse into every gap and a fractal emerges — Descartes’ Circle Theorem, Hausdorff dimension 1.3057, and the geometry of hyperbolic space.
Hilbert curve
A space-filling curve that maps a 1D line onto a 2D plane. Animated construction at levels 1–8 with gradient coloring that reveals locality preservation — the property behind spatial indexing.
Standing waves
A string fixed at both ends vibrates in harmonics 1–12. Nodes stay still, antinodes oscillate at maximum amplitude — toggle the two traveling waves that compose the standing pattern.
Vernier caliper
An interactive vernier caliper with sub-millimeter precision. Drag the sliding jaw to measure randomly-sized objects and learn how a 0.1 mm difference between scale divisions achieves ten times the resolution.
Penrose stairs
The impossible staircase — a figure walks endlessly up or down, never reaching a higher or lower level. Toggle the reveal to expose the geometric trick.
Hooke’s law
Drag a mass on a spring, release it, and watch F = −kx in action. Real-time graphs of displacement, velocity, and force. Compare multiple springs side by side.
Koch snowflake
Infinite perimeter, finite area. Watch the fractal construction level by level as the coastline paradox unfolds — dimension log4/log3 ≈ 1.2619.
Gyroscope
A spinning disk on a tilted axis. Gravity creates torque, torque drives precession. Toggle gravity off and watch the magic stop.
Lemniscate
The figure-eight curve where the product of distances to two foci is constant. Bernoulli (1694), Cassini ovals, and the infinity symbol.
Roman mosaic
Generative pattern builder with meanders, guilloche, waves, chevrons, and rosettes. Click tiles, apply symmetry, export your design.
Doppler ultrasound
Simulate a medical Doppler display — blood flowing through a vessel, an angled ultrasound beam, and the spectral waveform clinicians read to diagnose stenosis and flow abnormalities.
Abacus
An interactive soroban with clickable beads, decimal display, and step-by-step tutorials for addition, subtraction, and multiplication. Switch to the Chinese suanpan for the older 2/5 configuration.
Fresnel zones
Concentric ellipsoidal zones between transmitter and receiver. Drag obstacles into the path and watch signal strength drop — the physics behind why cell towers need clearance above terrain.
Archimedes’ Principle
Buoyant force equals the weight of displaced fluid. Adjust object and fluid densities, watch force vectors balance, and see objects float, sink, or bob to equilibrium.
Light caustics
Parallel rays refract through a wavy surface and converge into bright caustic curves. Snell’s law, envelope theory, and the shimmering patterns on a swimming pool floor.
Starling murmuration
Thousands of birds following their nearest neighbors — topological, not metric. Compression waves, predator avoidance, and scale-free correlations in a simulated dusk flock.
Ant bridge
Army ants form living bridges with their bodies to create shortcuts across gaps. The bridge self-optimizes — growing when traffic is heavy, shrinking when it drops. No leader, no plan, just local rules.
Structural color
Thin-film interference creates vivid color without pigment. Adjust film thickness, refractive index, and viewing angle to see how soap bubbles, oil slicks, and butterfly wings get their shimmer.
Theremin 3D
A spatial musical instrument: mouse X controls pitch, mouse Y controls volume. Sound waves radiate in 3D perspective, color shifting with frequency — Léon Theremin’s 1920 invention, playable.
Faraday cage
A conducting shell shields its interior from external electric fields. Watch charges redistribute, field lines bend around the cage, and the interior stay field-free — no matter how strong the external field.
Fresnel diffraction
Near-field diffraction from slits, circular apertures, and straight edges. Slide the screen distance and watch Fresnel patterns transition smoothly into Fraunhofer far-field — with the Cornu spiral that connects them.
Rutherford scattering
Alpha particles scatter off a gold nucleus via Coulomb repulsion. Hyperbolic trajectories, the forbidden zone of closest approach, and the experiment that discovered the atomic nucleus in 1911.
Stern-Gerlach experiment
A beam of spin-1/2 particles passes through an inhomogeneous magnetic field and splits into exactly two discrete beams. The first direct evidence of spatial quantization — classical physics predicted a continuous smear.
Wheatstone bridge
Four resistors in a diamond, a galvanometer across the middle. When R₁/R₂ = R₃/R₄, no current flows — the balance condition behind precision resistance measurement, strain gauges, and sensor interfaces.
Torsion pendulum
A disk on a wire oscillates rotationally with restoring torque τ = −κθ. Add damping and it decays. Drive it at the natural frequency and watch resonance amplify the motion.
Resonance tube
Standing waves in an air column — Kundt’s tube. A speaker drives a tube with a movable piston; particles pile up at pressure nodes, revealing the harmonic series for open and closed tubes.
Stokes drag & terminal velocity
Drop spheres into viscous fluid and watch them reach terminal velocity. Stokes drag at low Reynolds number, quadratic drag at high Re — with real-time velocity graphs and force diagrams.
Van de Graaff generator
A moving belt carries charge to a hollow metal sphere, building enormous static potential. Corona discharge, branching lightning sparks to a grounding sphere, and hair that stands on end.
Carnot engine
The most efficient heat engine possible between two temperatures. Four reversible stages on a PV diagram with an animated piston — the second law of thermodynamics in action.
Chemical equilibrium
Reactants and products interconvert at rates that depend on concentration. Add species, change temperature, adjust volume — and watch Le Chatelier’s principle restore balance.
DNA transcription
RNA polymerase unwinds the double helix and assembles mRNA one nucleotide at a time. A–U, T–A, G–C, C–G — the central dogma of molecular biology, animated.
Error-correcting codes
Hamming(7,4) encodes 4 data bits into 7. Flip any bit to introduce an error, then watch the syndrome detect and correct it — with Venn diagram and parity check matrix.
Quantum tunneling
A wave packet hits a potential barrier higher than its energy — and probability leaks through exponentially. The physics behind nuclear fusion and flash memory, solved with the split-operator method.
Special relativity
Interactive Minkowski spacetime diagram with light cones, Lorentz boosts, and the twin paradox. Drag the velocity slider to see time dilation and length contraction emerge from geometry.
Nuclear decay chains
Unstable atoms decay randomly through alpha, beta, and gamma processes. The exponential decay law emerges from randomness — watch daughter isotopes accumulate along the U-238 chain.
Plate tectonics
Drag tectonic plates and watch boundaries respond. Divergent ridges create new crust, convergent zones build mountains, and transform faults produce earthquakes.
Protein folding (HP model)
A chain of hydrophobic and polar residues folds on a 2D lattice to minimize energy. Monte Carlo search, contact maps, and the NP-hard optimization behind the hydrophobic collapse.
Room acoustics
Ray-traced 2D sound simulation. Emit rays from a source, watch them bounce off walls, and measure the impulse response and RT60 reverberation time.
Bridge stress simulator
Build truss bridges from nodes and beams. Apply loads and solve for internal forces — blue for tension, red for compression. Can your bridge hold?
Stellar lifecycle
Watch a star evolve on the HR diagram from molecular cloud to main sequence to its fate — white dwarf, neutron star, or black hole, depending on mass.
Data compression
Watch LZ77 find repeated patterns in text, building a dictionary step by step. See the sliding window, match highlights, and compression ratio in real time.
Ecological food web
Multiple species connected through predation. Remove one and watch trophic cascades ripple through the ecosystem.
Formal logic
Build propositional logic expressions and generate truth tables. Spot tautologies, contradictions, and explore expression trees.
Thin-film interference
Soap bubbles and oil slicks: white light interferes in thin films, producing iridescent color that shifts with thickness.
Vortex shedding
A cylinder in a flow sheds alternating vortices — the von Karman street. Lattice Boltzmann simulation with adjustable Reynolds number.
Weather system
Temperature, pressure, wind, and moisture on a 2D grid. Place heat sources and oceans, watch weather patterns and precipitation emerge.
Buoyancy
Archimedes’ principle: drop objects into fluid and watch them float, sink, or hover as you adjust densities.
Cell division
Mitosis animated stage by stage — chromosome condensation, spindle fibers, separation, and cytokinesis.
Pendulum clock
An escapement mechanism simulation. The pendulum, anchor, and escape wheel tick in concert to keep time.
Phase space
Phase portrait explorer for classic dynamical systems. Click to place trajectories and watch them flow.
Schwarzschild lensing
Gravitational lensing around a black hole. Starlight bends near the event horizon, forming an Einstein ring.
Sudoku solver
Constraint propagation and backtracking, visualized step by step as the solver deduces and guesses its way to a solution.
Chua’s circuit
The simplest electronic circuit that exhibits chaos. Two capacitors, one inductor, and a nonlinear resistor produce the famous double-scroll attractor — adjust parameters and watch order dissolve.
Sand dune self-organization
Wind picks up sand through saltation and deposits it in wind shadows. Flat terrain with random perturbations evolves into barchan crescents and transverse ridges — self-organization from nothing but wind and gravity.
Tensegrity
Rigid compression struts floating in a web of tension cables. No strut touches another — the structure holds its shape through pre-tension alone. Rotate, perturb, and watch it oscillate back to equilibrium.
Zoetrope
A spinning slitted cylinder turns discrete frames into fluid motion. Persistence of vision — the 19th-century illusion behind cinema, television, and every screen you own.
Lava lamp
Metaball blobs rise, merge, split, and sink in a lamp-shaped container. Rayleigh–Bénard convection with implicit surface rendering — the physics of buoyancy made hypnotic.
Möbius strip
A surface with one side and one edge. Drag to rotate in 3D, watch an ant prove single-sidedness, then cut it down the middle and discover the result is not two loops but one.
Ferrofluid
A magnetic fluid under a movable field source. Drag the magnet across the surface and watch spikes form through the Rosensweig instability — the same phenomenon that gives real ferrofluid its alien, spiky beauty.
Bragg diffraction
X-rays scattering from a crystal lattice. When waves reflected from successive lattice planes arrive in phase, they constructively interfere — producing the sharp diffraction peaks that reveal atomic structure.
Brusselator
A two-component autocatalytic reaction with diffusion. When inhibitor diffuses faster than activator, Turing instability spontaneously breaks symmetry — producing spots, stripes, and spiral waves from a uniform chemical soup. The 1968 model behind biological morphogenesis.
Beat frequency
Two sine waves with slightly different frequencies produce a pulsing amplitude pattern — beats. Hear and see the interference with Web Audio, waveform visualization, and frequency spectrum.
Sonic boom
A point source moves through a medium emitting spherical wavefronts. Watch Doppler compression go subsonic, pile into a shock front at Mach 1, and form a Mach cone as the source goes supersonic.
Optical illusions
Five classic visual illusions — Müller-Lyer, Ebbinghaus, Café Wall, Hermann Grid, and Ponzo — expose the shortcuts your visual cortex uses to construct reality. Toggle the reveal to see the truth.
K-means clustering
Place points, pick k, and watch centroids converge. Assign to nearest, move to mean, repeat — the unsupervised learning algorithm that finds structure in unlabeled data, with inertia tracking.
Leidenfrost effect
Water droplets on a hot surface. Below the Leidenfrost point they boil away fast; above it, a vapor cushion forms and droplets skitter like tiny hovercrafts. The characteristic lifetime dip and recovery, animated.
Magnus effect
A spinning ball curves through the air. Topspin dips, backspin floats, sidespin bends. Adjust spin rate, launch angle, and speed — then compare against a non-spinning reference. The physics behind every curveball.
Kaleidocycle
A ring of linked tetrahedra that turns inside-out continuously without disconnecting. Drag to evert the cycle, toggle wireframe or filled faces, and choose your color scheme. Topology in motion.
Reaction time
Measure your visual and auditory reaction time. Wait for the signal, click as fast as you can, and track your mean, median, and standard deviation over 20 trials. Compare against typical human performance.
TSP solver
Place cities and watch algorithms hunt for the shortest tour. Nearest neighbor, 2-opt improvement, and simulated annealing compete in real time. The classic NP-hard problem, visualized step by step.
Vernier engine
A pressure-fed bipropellant rocket engine, animated from first principles. Oxidizer and fuel flow through valves, atomize in the combustion chamber, and expand through a de Laval nozzle. Throttle control, temperature gradients, and the Tsiolkovsky rocket equation.
Anamorphic mirror
Draw on a canvas and see the radially distorted anamorphic version that looks correct when reflected in a cylindrical mirror. Polar inversion, geometric optics, and Renaissance visual puzzles — interactive.
3D Harmonograph
Three damped pendulums controlling x, y, z trace intricate curves through 3D space. Frequency ratios create knotted Lissajous-like patterns that spiral inward as energy dissipates. Drag to rotate, tune parameters, watch the curve emerge.
Faraday Waves
Shake a fluid layer and its surface erupts into standing wave patterns — stripes, squares, hexagons, and quasicrystalline order. Faraday instability drives parametric resonance at half the driving frequency. Tune frequency, amplitude, and surface tension to reshape the geometry.
Hopfield Network
A recurrent neural network as associative memory. Draw patterns and store them via Hebb’s rule, then corrupt one and watch asynchronous neuron updates descend an energy landscape to recall the original. The physics of memory, made visual.
Granular Segregation
The Brazil nut effect: shake a container of mixed particles and the largest ones rise to the top. Real rigid-body collision physics with void-filling, convection rolls, and size-driven segregation. Switch to inverse mode to make large, dense particles sink instead.
Nematic Liquid Crystal
A nematic liquid crystal sits between solid crystal and isotropic liquid: rod-like molecules share a common orientation called the director. Watch the director field relax via Frank elastic free energy, half-integer topological defects nucleate and annihilate, and the isotropic-nematic phase transition unfold as you tune temperature past the clearing point.
Bjerknes Vortex Rings
Two coaxial vortex rings of equal circulation perform the leapfrog: the rear ring accelerates into the front, which slows and expands, until they pass through each other — then roles reverse indefinitely. Try head-on collisions, three-ring cascades, and offset rings. V. F. K. Bjerknes studied these as a mechanical analogy for electromagnetic atom-atom forces.
Taylor–Couette Flow
Viscous fluid between rotating concentric cylinders. Below the critical Taylor number the flow is purely azimuthal (Couette). Spin up the inner cylinder and watch centrifugal instability break the flow into toroidal Taylor vortices, wavy vortex flow, and turbulence. G. I. Taylor’s 1923 experiment was one of the first quantitative tests of the Navier-Stokes equations.
Muscle Fiber
Interactive simulation of the sliding filament theory of muscle contraction. Watch actin and myosin filaments, cross-bridge cycling, and the force-velocity relationship in real time.
Zeeman Effect
Watch spectral lines split as magnetic field strength increases. Explore normal and anomalous Zeeman splitting, polarization selection rules, and the quantum mechanics of angular momentum in a field.
Action Potential
Hodgkin-Huxley neuron simulator. Visualize how sodium and potassium ion channels open and close to fire an action potential, and watch it propagate along an axon.
Hopf Fibration
Visualize the Hopf fibration — a stunning map from the 3-sphere to the 2-sphere where every point on S² corresponds to a circle in S³, projected into 3D as interlocked tori.
Aurora Borealis
Simulate the aurora borealis — charged solar wind particles spiraling along magnetic field lines, exciting atmospheric atoms to glow in curtains of light. Adjust solar wind speed, magnetic field, and particle energy.
Cherenkov Radiation
The electromagnetic shockwave emitted when a charged particle travels faster than light in a medium. See the Cherenkov cone, Frank-Tamm theory, and the eerie blue glow of nuclear reactors.
A* Pathfinding — Iris
Interactive A* pathfinding visualizer. Draw walls, place start and end points, and watch A* find the shortest path with f/g/h scores displayed.
Abacus Arithmetic — Iris
Learn to calculate on a soroban — the Japanese abacus that
Acoustic Doppler — Iris
Visualize and hear the Doppler effect with a moving sound source. Watch wavefronts compress and stretch, and listen as pitch shifts in real time.
Airfoil — Iris
Visualize airflow around a wing. Streamlines, pressure distribution, lift and drag — and what happens when angle of attack gets too high.
Airy Disk — Iris
Visualize diffraction through a circular aperture. See the Airy pattern form, adjust wavelength and aperture size, and discover the Rayleigh criterion for resolving two point sources.
Amoeba Simulation — Iris
Simulate a soft-body amoeba that moves via pseudopod extension and cytoplasmic flow. Place food and watch chemotaxis behavior unfold.
Anamorphic Cylinder
Draw on a radial grid and see how a cylindrical mirror placed at the center would reflect the distorted image into a correct one — cylindrical anamorphosis in action.
Anamorphic Text — Iris
Create text that only looks correct when viewed from a specific angle, using perspective distortion and anamorphic stretching.
Ant Brood Sorting — Iris
Interactive simulation of ant brood sorting — a self-organization algorithm where simple agents cluster objects without centralized control. Watch two types of brood spontaneously sort into clusters through purely local rules.
Antenna Radiation Pattern — Iris
Visualize electromagnetic radiation patterns from different antenna types. Explore dipole, monopole, Yagi, and phased array patterns with interactive controls for element spacing, phase steering, and animated field visualization.
Apollonius Circles
Given three circles, find all circles tangent to all three simultaneously. Eight solutions, circle inversion, and coaxial pencils.
Apollonius Gasket — Iris
The Apollonian gasket fractal built from Descartes
Archimedes Screw — Iris
One of humanity
Archimedes' Trammel
An interactive ellipsograph: watch two sliders glide along perpendicular tracks as a pencil traces a perfect ellipse. Drag to reshape, adjust rod lengths to change eccentricity.
Astrolabe — Iris
A working model of a medieval astrolabe — the analog astronomical computer. Rotate the rete to find star positions, tell time, and explore stereographic projection.
Balancing Bird — Iris
A toy bird that defies intuition by balancing on your fingertip. Explore how shifting the center of mass below the pivot point creates stable equilibrium.
Bayesian Network — Iris
Interactive Bayesian network with draggable nodes. Set evidence by clicking nodes and watch probabilities update via belief propagation. Explore conditional independence in the classic Sprinkler/Rain/Wet Grass network.
Bead on a Rotating Hoop — Iris
A bead on a rotating hoop demonstrates pitchfork bifurcation and spontaneous symmetry breaking. Above a critical angular velocity, the bottom becomes unstable.
Belousov-Zhabotinsky Reaction — Iris
Simulate the famous BZ reaction — a chemical oscillator that spontaneously produces spiral waves. Watch non-equilibrium chemistry create structure from nothing.
Bénard Convection — Iris
Interactive simulation of Rayleigh-Bénard convection: watch heated fluid form rolling convection cells, transition from conduction to turbulence, and explore how temperature and viscosity shape fluid flow.
Benham’s Disc — Iris
A spinning black-and-white disc that tricks your brain into seeing phantom colors. Explore Fechner colors, temporal retinal response, and the surprising limits of human color perception.
Bernoulli’s Principle — Iris
Visualize Bernoulli
Bifurcation Diagram — Iris
Interactive logistic map bifurcation diagram. Explore period doubling, chaos, and the Feigenbaum constant with a synchronized cobweb diagram.
Birthday Paradox — Iris
Interactive simulation of the birthday problem. Watch collisions emerge as people enter the room and see why 23 people gives a 50% chance of a shared birthday.
Bloom Filter — Iris
Interactive Bloom filter visualization. Add strings, test membership, and watch false positives emerge as the bit array fills up.
Bode Plot — Iris
Interactive Bode plot for control systems and signal processing. Build transfer functions by placing poles and zeros on the complex plane. See magnitude and phase response in real-time.
Boolean Logic Gates — Iris
Interactive logic gate circuit builder. Drag and drop AND, OR, NOT, NAND, XOR, NOR gates, wire them together, and watch signals propagate in real time.
Boundary layer — Iris
Interactive boundary layer simulator. Watch fluid particles slow near a surface, see the Blasius velocity profile develop, and observe the laminar-to-turbulent transition over a flat plate.
Brachistochrone Comparison — Iris
Race balls down six different curves simultaneously — straight line, parabola, circular arc, cycloid, catenary, and ellipse. The cycloid always wins despite not being the shortest path. Clear timing display and percentage comparisons.
Braess's Paradox | Iris Lab
Bubble Chamber — Iris
Simulate a particle physics bubble chamber. Click to create particle events and watch charged particles spiral through a magnetic field, leaving tracks like the detectors that revealed the subatomic world.
Bubble Raft — Iris
Simulate a 2D bubble raft: soap bubbles with Lennard-Jones forces self-organize into hexagonal crystal packing. Poke the raft to create dislocations, grain boundaries, and vacancies.
Crystal Dislocations
Edge and screw dislocations glide through a crystal lattice under applied stress. Watch how line defects enable plastic deformation and interact to produce work hardening.
Hexagonal Sphere Packing
Kepler’s conjecture proved: hexagonal close packing achieves π/(3√2) ≈ 74.05% density — the densest possible arrangement. Explore why bees, soap bubbles, and ice all independently discover the same geometry.
Burning Ship Fractal — Iris
Interactive Burning Ship fractal explorer. The asymmetric cousin of the Mandelbrot set, producing dramatic ship-like structures from absolute-value iteration.
Camera Obscura — Iris
Simulate pinhole camera physics. Watch light rays trace through an adjustable aperture to form an inverted image, exploring the optics behind photography.
Cantor Set — Iris
Visualize the Cantor set: iteratively remove middle thirds from [0,1] to build an uncountable set of measure zero. Watch the Devil
Carnot Cycle — Iris
Interactive Carnot cycle engine with animated piston, PV diagram, and TS diagram. See the four thermodynamic stages in real time and why the Carnot engine sets the theoretical efficiency limit.
Cassini Ovals — Iris
Interactive exploration of Cassini ovals: the locus of points where the product of distances to two foci equals a constant. Drag foci, adjust parameters, compare with ellipses.
Catalan Numbers — Iris
Visualize Catalan numbers through Dyck paths, binary trees, and polygon triangulations.
Centripetal Force — Iris
Explore centripetal force and circular motion. Spin objects on strings, release them to see tangential flight, and compare scenarios from banked curves to satellite orbits.
Chebyshev Polynomials — Iris
Visualize Chebyshev polynomials, their roots on the unit circle, equioscillation property, and how they eliminate Runge
Chemical Clock
A simulation of the Belousov-Zhabotinsky oscillating chemical reaction using the Oregonator model, with real-time concentration plots and a color-changing beaker.
Verlet Cloth Tearing — Iris
Interactive cloth tearing simulation using Verlet integration. Click and drag to rip fabric apart — constraints break when stretched beyond their threshold, and tears propagate naturally through the mesh.
Cloud Chamber — Iris
Simulate a Wilson cloud chamber showing particle tracks. Alpha, beta, muon, and cosmic ray particles leave characteristic vapor trails curved by a magnetic field.
Coandă Effect — Iris
Visualize the Coanda effect: a fluid jet curving to follow a nearby convex surface. Adjust speed, curvature, and distance to see flow attachment and detachment.
Color Mixing — Iris
Interactive demonstration of additive vs subtractive color mixing. Drag RGB light circles and CMY pigment circles to see how colors combine.
Conformal Mapping — Iris
Visualize how complex functions transform the plane — preserving angles but warping shapes. Watch grids flow through z-squared, inversion, the exponential, and the Joukowski airfoil transform.
Connect Four — Iris
Classic Connect Four game with an AI opponent using minimax and alpha-beta pruning. Choose your difficulty and try to beat the machine.
Couette Flow & Taylor Vortices — Iris
Interactive Couette flow simulation. Watch laminar flow between concentric cylinders give way to Taylor vortices, wavy vortex flow, and turbulence as rotation speed increases.
Crookes Radiometer — Iris
Simulate a Crookes radiometer (light mill) and explore the radiometer paradox. Drag a light source, adjust gas pressure, and see why black sides are pushed away from light through thermal transpiration, not radiation pressure.
Frequency Analysis — Iris
Break a substitution cipher the way the Arab mathematician al-Kindi did 1200 years ago — by counting letter frequencies.
Crystal Growth — Iris
Simulate crystal nucleation and growth in real time. Watch faceted and dendritic crystals emerge from a lattice model with tunable symmetry, supersaturation, and anisotropy.
Cymatics — Iris
Visualize vibration patterns on a 2D plate. Watch nodal lines and resonance patterns emerge from the wave equation in real time.
Devil's Curve — Iris
Visualize the devil
Dielectric Breakdown — Iris
Simulate Lichtenberg figures and dielectric breakdown patterns. Watch fractal lightning trees grow via diffusion-limited aggregation with adjustable branching and growth probability.
3D Diffusion-Limited Aggregation
Watch a fractal cluster grow in three dimensions as random-walking particles stick on contact, with live fractal dimension estimation.
Double Pendulum 3D
The double pendulum extended into three dimensions: chaotic motion sweeps through a hemisphere, tracing incandescent webs of trajectory. Rotate the view, launch multiple shadow pendulums to see sensitive dependence on initial conditions in full 3D.
Vibrating Drum Membrane — Iris
Visualize the 2D vibration modes of a circular drum membrane. Explore Bessel function solutions, nodal lines, and Chladni-like patterns in real-time animation.
Duffing Oscillator — Iris
Explore the Duffing oscillator: a driven nonlinear spring that exhibits chaos, period-doubling bifurcations, strange attractors, and sensitive dependence on initial conditions.
Echolocation — Iris
Navigate a dark room using sound pulses. Click to emit wavefronts that reflect off walls and objects, revealing the space through echoes alone.
Eddy Currents — Iris
Visualize electromagnetic eddy currents induced in a conductor by a moving magnet, demonstrating Faraday
2D Elastic Collision Simulator — Iris
Pool-table-like arena with multiple discs of varying mass and radius. Full elastic collision physics with momentum and energy conservation.
Elliptic Curves — Iris
Interactive elliptic curve visualization. Adjust curve parameters, explore point addition geometry, and see how the group law works.
Entropy & the Second Law — Iris
Watch entropy increase in real-time as colored particles mix irreversibly. Features Loschmidt
Euler’s Disk — Iris
Simulate Euler
Euler Line
Drag the vertices of a triangle to explore its Euler line — the remarkable collinearity of the circumcenter, centroid, orthocenter, and nine-point center.
Fiber Optic — Iris
Visualize total internal reflection in an optical fiber. Watch light rays bounce inside a curved waveguide, explore Snell
Fireworks Physics — Iris
Lab #400 — A spectacular particle physics simulation of fireworks. Launch mortar shells that burst into hundreds of sparks governed by gravity, air drag, and burst velocity. Five shell types: peony, ring, willow, palm, crossette.
Ford Circles — Iris
Explore Ford circles — tangent circles sitting on the number line at each rational p/q. Discover the Stern-Brocot tree and Farey sequence mediant relationships.
Foucault Pendulum — Iris
Simulate Foucault
Four-Bar Linkage
Animate a four-bar linkage mechanism and trace the coupler curve. Explore crank-rocker, double-crank, and double-rocker configurations.
Four Color Theorem — Iris
Can you color any map with just four colors so no adjacent regions share a color? Try it yourself — and discover why this simple problem took 124 years and a computer to prove.
Electrochemical Cell — Iris
Simulate a galvanic (voltaic) cell. Watch zinc dissolve, electrons flow through the wire, and copper deposit. Select different metal pairs and observe voltage changes.
D’Arsonval Galvanometer — Iris
Simulate a D
Gambler's Ruin — Iris
Interactive simulation of the Gambler
Gaussian Cannon — Magnetic Accelerator | Iris Lab
Gaussian Beam — Iris
A laser beam isn
Gears & Gear Trains — Iris
Interactive gear train simulator. Spin meshing gears with proper tooth profiles, see real-time gear ratio calculations, and build custom gear trains.
Geiger Counter — Iris
A radioactive decay simulator with Poisson statistics. Watch random particles strike a Geiger-Müller tube, hear the clicks, and see order emerge from randomness as the histogram converges to an exponential distribution.
Geodesic Dome — Iris
Interactive 3D geodesic dome built from icosahedral subdivision. Explore Buckminster Fuller
Geodesic Paths on Curved Surfaces — Iris
Explore geodesics — shortest paths — on sphere, torus, hyperboloid, and ellipsoid. Drag endpoints, reveal tangent vectors, and see how curvature shapes geometry.
The Golden Ratio — Iris
Explore the golden ratio phi = (1+sqrt(5))/2 through golden rectangles, Fibonacci spirals, phyllotaxis patterns, and the convergence of Fibonacci ratios.
Gravitational Lensing — Iris
Interactive gravitational lensing simulation. Drag a massive object to bend light from background sources — watch Einstein rings form, images multiply, and spacetime curve in your browser.
Gravity Train — Iris
Drill a straight tunnel between any two cities through the Earth and let a train fall through it — powered only by gravity. Remarkably, the trip always takes exactly 42 minutes.
Gravity Well Fabric — Iris
The rubber-sheet analogy for general relativity. Place masses on a 3D wireframe grid and watch spacetime deform. Fire test particles to trace geodesics around gravity wells.
Guilloche
Generate intricate geometric guilloche patterns — the fine engraved curves found on banknotes, certificates, and watches.
Halftone Rendering — Iris
Interactive halftone printing simulation. Explore ordered dithering, Floyd-Steinberg error diffusion, and CMYK dot halftoning with adjustable parameters.
Hall Effect — Iris
Simulate the Hall effect: watch charge carriers deflect in a magnetic field and build up a transverse voltage across a conductor.
Harmonograph with Decay — Iris
A 2-pendulum harmonograph simulator. Decaying pendulums trace spiraling Lissajous figures. Adjust frequency ratios, phase offsets, and decay rates to create beautiful spirograph patterns.
Hele-Shaw Cell — Iris
Simulate viscous fingering in a Hele-Shaw cell. Watch the Saffman-Taylor instability create fractal branching patterns as low-viscosity fluid displaces high-viscosity fluid.
Helmholtz Resonator — Iris
Simulate a Helmholtz resonator: adjust cavity volume, neck length, and diameter to change the resonant frequency. Hear the tone and see air pressure oscillations.
Hénon Map
The Hénon map's strange attractor — a fractal fold repeated at every scale — emerges from just two quadratic equations. Vary a and b to transition between order, periodicity, and chaos.
Hodgkin-Huxley Model — Iris
The Nobel Prize-winning mathematical model of the neuronal action potential. Watch membrane voltage, gating variables, and ion channel dynamics in real time.
Hourglass — Iris
Sand particle simulation flowing through a narrow opening. Particles fall under gravity, collide, and pile up realistically. Adjust grain count, size, and gravity strength. Flip the hourglass to restart the flow.
Hydraulic Press — Iris
Pascal
Magnetic Hysteresis — Iris
Interactive visualization of ferromagnetic hysteresis: drag the applied field H and watch the B-H curve trace its classic S-shaped loop. Explore domain wall motion, coercivity, remanence, and soft vs hard ferromagnets.
Image Convolution — Iris
Visualize how convolution kernels transform images. Apply blur, sharpen, edge detection, and custom kernels to see pixel-level operations in real time.
Impossible Objects — Iris
Interactive impossible geometric objects: Penrose triangle, Penrose staircase, impossible cube, and blivet. Rotate to reveal how the illusions exploit 2D projection ambiguity.
Impossible Triangle — Iris
The Penrose triangle — an object that can exist in 2D but never in 3D. Rotate it to discover the illusion, or explore other impossible figures.
Inverse Kinematics — Iris
Interactive robot arm simulation using the FABRIK (Forward And Backward Reaching Inverse Kinematics) algorithm. Drag the end effector or enable mouse tracking to watch a chain of joints solve in real time.
Isogonal Conjugates — Iris
Drag a point P inside triangle ABC and watch its isogonal conjugate P
Josephson Junction
A superconducting tunnel junction that converts DC current into AC oscillations at microwave frequencies. Apply a bias current, watch Shapiro steps emerge under microwave drive, and see quantum phase locking made visible.
Julia Set — Iris
Explore Julia sets interactively. Click on the Mandelbrot set to choose c and watch the corresponding Julia set render in real time with smooth escape-time coloring.
Kalman Filter — Iris
Track a moving object through noisy measurements. Watch the Kalman filter fuse prediction and observation in real time, converging on truth despite uncertainty.
Kármán Vortex Street — Iris
Watch alternating vortices shed behind a cylinder in fluid flow. Explore the Strouhal number, Reynolds number, and the beautiful asymmetric patterns of vortex streets.
Kelvin Water Dropper
Simulate Lord Kelvin
Kelvin Wake Pattern | Iris Lab
Kelvin Water Dropper — Iris
Simulate Lord Kelvin
Knight's Tour — Iris
Navigate a chess knight to visit every square on the board exactly once. Solve it manually or watch Warnsdorff
Laser Cavity — Iris
Simulate stimulated emission inside a laser cavity. Watch photons multiply as they bounce between mirrors, building a coherent beam from quantum noise.
Lenz's Law — Iris
Drop a magnet through a copper tube and watch eddy currents brake its fall. Compare to free-falling objects and see Faraday
Lenz's Law Demo — Iris
Drop a magnet through a conducting tube and watch eddy currents slow its fall. Side-by-side comparison with free fall demonstrates electromagnetic braking in real time.
Lichtenberg Figures — Iris
Simulate electrical dielectric breakdown creating fractal Lichtenberg figures. Watch branching lightning patterns grow via diffusion-limited aggregation with electric field bias.
Logistic Map
The logistic map r·x·(1−x) transitions from steady states through period-doubling cascades into chaos, with Feigenbaum's constant governing each bifurcation.
Magnetic Bottle
A charged particle spirals along converging magnetic field lines, its pitch angle increasing until it reflects — demonstrating the magnetic mirror effect.
Magnetic Domains — Iris
Simulate ferromagnetic domain dynamics with a real-time hysteresis loop. Apply external fields, heat past the Curie temperature, and watch magnetic domains form, grow, and collapse.
Magnetic Hysteresis — Iris
Visualize the B-H hysteresis loop of a ferromagnetic material. Watch magnetic domains align and flip as you sweep an external field through soft iron, hard steel, and rare earth magnets.
Magnetic Reconnection
Visualize magnetic reconnection: opposing field lines merge, snap, and release energy as high-speed particle jets — a fundamental process in space and fusion plasma physics.
Map Coloring — Iris
Interactive demonstration of the four-color theorem. Color random planar maps using only four colors so no adjacent regions share a color. Auto-solve with a greedy algorithm or solve it yourself.
Match Stick Combustion — Iris
Simulate a match being struck and burning with realistic flame rendering, combustion chemistry, and temperature curves.
Maurer Rose — Iris
Interactive Maurer rose pattern generator. A rose curve r=sin(n*theta) with points connected at regular degree intervals creates stunning geometric line patterns. Adjust petals and step angle, animate parameter sweeps.
Maxwell-Boltzmann Distribution — Iris
Watch gas molecules bounce in a box, color-coded by speed. See the Maxwell-Boltzmann distribution emerge from the chaos of molecular collisions.
Maxwell’s Demon — Iris
Simulate Maxwell
Maxwell Speed Distribution
Watch gas molecules bounce in a box while their speed histogram converges to the Maxwell-Boltzmann distribution. Adjust temperature, mass, and particle count.
Mechanical Clock — Iris Lab
Meissner Effect — Iris
Visualize superconductor levitation and the Meissner effect. Watch magnetic field lines get expelled as temperature drops below the critical threshold.
Menger Sponge — Iris
A 3D fractal constructed by recursively removing cubic sections. Rotate the sponge and adjust the iteration level to watch complexity explode exponentially.
Metaballs
Classic blobby implicit surface visualization. Charged blobs merge and split as they drift across the canvas.
Metronome Synchronization — Iris
Watch Huygens
Michelson Interferometer — Iris
Simulate a Michelson interferometer. A beam splitter divides a laser into two paths that recombine to produce interference fringes. Drag the mirror to shift fringes in real-time.
Mohr’s Circle — Iris
Interactive stress transformation visualization. Drag to rotate the stress element and watch the corresponding point trace Mohr
Morphogenesis
Interactive Turing morphogenesis simulation. Watch biological patterns — stripes, spots, labyrinths — self-organize from a homogeneous field via activator-inhibitor dynamics. Based on Alan Turing
Navier-Stokes 2D — Iris
2D Navier-Stokes fluid simulation on a grid. Stable fluid with advection, diffusion, and pressure solve. Add obstacles by clicking. Visualize velocity, vorticity, or streamlines.
Newton Fractal — Iris
Interactive Newton fractal explorer. Newton
Newton’s Cradle — Iris
A physics simulation of Newton
Newton’s Method — Iris
Interactive Newton
Nim Strategy — Iris
Play Nim against an optimal AI. Visualize the XOR-based winning strategy with binary heap representations and animated Nim-sum calculations.
Nomogram — Iris
Interactive nomogram — the beautiful graphical calculators used before electronic computers. Click scales to multiply numbers or compute BMI.
Number Spiral — Iris
Explore the Ulam spiral and variants. Plot integers on spirals and discover how primes cluster on diagonal lines, revealing deep patterns in number theory.
Olbers’ Paradox — Iris
Why is the night sky dark? If the universe were infinite, static, and uniformly filled with stars, every line of sight would end on a star. Explore the paradox and its resolution through redshift, finite age, and star lifetimes.
Mechanical Orrery — Iris
A brass-styled mechanical orrery showing the inner solar system. Watch gear-driven planets orbit the sun with accurate orbital periods.
Pantograph — Iris
A working mechanical pantograph that copies and scales drawings through parallelogram linkage geometry. Draw on one side and watch a scaled copy appear on the other.
Particle in a Box — Iris
Visualize quantum wavefunctions and probability densities for a particle confined to a 1D infinite potential well. Toggle energy levels, create superpositions, and watch probability oscillate in time.
2D Particle in a Box — Iris
Quantum mechanics in two dimensions — watch probability density patterns emerge as standing waves in a rectangular well. Adjust quantum numbers to see nodal patterns.
Pascal's Triangle Modular Sieve — Iris
Render Pascal
Prisoner’s Dilemma Tournament — Iris
Axelrod-style iterated Prisoner
Pedal Curve Explorer — Iris
Interactive pedal curve explorer. Drop a perpendicular from a pedal point to every tangent line of a base curve and trace the locus of feet. Drag the pedal point to morph the pedal curve in real time.
Peltier Effect — Iris
Simulate thermoelectric cooling and heating at a Peltier junction. Watch charge carriers move through N-type and P-type semiconductors as heat flows from cold to hot.
Penrose Unilluminable Room
Explore the Penrose unilluminable room — a reflective chamber shaped so no single point light source can illuminate every point. Drag the light and watch rays bounce.
Piezoelectric effect — Iris
Interactive piezoelectric effect simulator. Squeeze a crystal lattice to generate voltage, or apply voltage to deform it. Explore direct and inverse piezoelectric effects with quartz, PZT, and barium titanate.
Pixel Sort — Iris
Artistic glitch art tool that sorts pixel rows and columns by brightness, hue, or saturation. Watch sorting happen in real time to create beautiful glitch art.
Planimeter — Iris Lab
Poincaré Section Visualizer — Iris
Poincare section and return map visualizer for a driven pendulum. Watch phase space trajectories, see strange attractors emerge from Poincare sections as driving force increases.
Electric Field Visualization — Iris
Visualize electric fields from point charges. Place positive and negative charges, see field lines, equipotential contours, and vector fields in real time.
Pursuit & Evasion — Iris
Simulate pursuit curves where a predator chases prey along mathematical pursuit paths, exploring pure pursuit, proportional navigation, and constant bearing strategies.
QR Code Generator — Iris
Generate QR codes from scratch with no libraries. Watch the encoding process, explore anatomy mode to see finder patterns, timing, data, and error correction regions.
Quantum Eraser
Visualize the quantum eraser and delayed-choice experiment. See interference appear, collapse with which-path detection, and restore with an eraser. Wave-particle duality made visible.
Quasicrystal Diffraction
Interactive quasicrystal simulation. Generate aperiodic tilings and their diffraction patterns. Explore 5-fold, 8-fold, and 12-fold symmetry that is impossible in ordinary crystals.
Crookes Radiometer — Iris
The spinning vanes in a Crookes radiometer seem to prove light carries momentum — but the real explanation is far more subtle. Explore three competing theories.
Radon Transform / CT Scanner — Iris
See how CT scanning works: watch X-ray beams rotate around a phantom, build a sinogram, and reconstruct the image using filtered back-projection.
Mechanical Railroad Switch — Iris
Interactive mechanical simulation of a railroad switch (points/turnout). Click the lever to move the switch rails and send a train through. Realistic track geometry, sleepers, ballast, signal, and mechanical linkage.
Rattleback — Iris
Simulate the rattleback (Celtic stone): a semi-ellipsoidal top that spontaneously reverses its spin direction due to chirality and energy transfer between modes.
Ray Marching — Iris
Interactive ray marching renderer: watch rays step through space using signed distance functions. See the 2D cross-section of marching circles and the full 3D rendered scene.
Rayleigh Scattering
Simulate white light passing through an atmosphere to see why the sky is blue and sunsets are red.
RC Circuit — Iris
Simulate charging and discharging a capacitor through a resistor. Watch animated charges flow through the circuit while real-time voltage and current graphs update.
Reuleaux Triangle — Iris
A curve of constant width that isn
River Meander Formation — Iris
Simulate how rivers form meanders through differential erosion and deposition. Watch a straight channel evolve into dramatic curves that eventually pinch off into oxbow lakes.
Rope Computer — Iris
Ancient Egyptians used ropes with evenly spaced knots to create perfect right angles. Explore how 3-4-5 triangles and other Pythagorean triples emerge from a simple knotted rope.
Rube Goldberg Machine — Iris
A 2D physics chain-reaction machine with balls, dominoes, seesaws, ramps, buckets, and pulleys. Watch 8 stages trigger in sequence with adjustable gravity and time scale. Energy conservation tracked in real time.
Rubens' Tube — Iris
Simulate a Rubens
Ruled Surface Explorer — Iris
Interactive 3D ruled surface explorer. Visualize how straight lines sweep out curved surfaces: hyperboloid, helicoid, Mobius strip, conoid, and Whitney umbrella.
Runge's Phenomenon — Iris
Watch polynomial interpolation fail at the edges with equally-spaced points. Compare with Chebyshev nodes to see how the right point placement tames oscillation.
Abelian Sandpile — Iris
Interactive Abelian sandpile automaton. Drop grains, watch cascading avalanches, and discover the fractal identity element of the BTW sandpile model.
Sand Pendulum — Iris
A pendulum that releases sand as it swings, tracing Lissajous-like figures in a sand tray. Adjust frequency ratios, amplitude, and damping to create beautiful patterns.
Sand Pendulum Patterns — Iris
A pendulum dripping sand traces Lissajous-like patterns as it swings and decays. Adjust the initial conditions to create spirographs written by gravity and friction.
Schlieren Photography
Simulate schlieren photography to visualize density gradients, convection currents, and thermal plumes in transparent media.
Schlieren Optics — Iris
Simulate schlieren photography to visualize refractive index gradients. Place heat sources and watch convection plumes appear as beautiful density patterns in the schlieren image.
Seifert Surface — Iris
Visualize Seifert surfaces: the orientable surfaces bounded by mathematical knots. Rotate 3D wireframe knots and explore the topology of trefoils, figure-eights, and torus knots.
Seismograph — Iris
Interactive seismograph simulator. Watch a mechanical seismograph respond to ground motion. Explore natural frequency, damping, frequency response, and trace seismograms in real time.
Analemma & Shadow Clock — Iris
Simulate the shadow of a gnomon throughout the day and year. Watch the analemma emerge as Earth
Sieve of Eratosthenes | Iris Lab
Simpson's Paradox — Iris
A trend that appears in combined data reverses when you split by group. Interactive scatter plot demonstrating Simpson
Skip List — Iris
Visualize the skip list probabilistic data structure. Watch search, insertion, and deletion operations animate step by step through multi-level express lanes.
Slide Rule — Iris
A working interactive slide rule — the analog computer that got us to the moon. Drag the slide and cursor to multiply, divide, find squares and cubes.
Smith Chart — Iris
Interactive Smith chart for RF/microwave engineering. Plot impedance and admittance, transform along transmission lines, and design matching networks.
Smoke Ring Physics — Iris
Simulate vortex rings (smoke rings) in cross-section. Watch toroidal vortices propagate, interact, and leapfrog through viscous air. Fire rings and adjust physics parameters.
Snell's Law Refraction — Iris
Interactive demonstration of Snell
Soap Bubble Physics — Iris
Simulate soap bubble thin-film interference, merging, and drainage. Watch iridescent rainbow colors shift as film thickness changes.
Soap Bubble Thin-Film Interference
See how thin-film interference creates rainbow colors in soap bubbles, with real optical path difference calculations.
Prism Spectrometer — Iris
Simulate a prism spectrometer that splits white light into its component wavelengths via dispersion. Adjust prism angle, material, and incident angle to explore Cauchy
Spherical Harmonics — Iris
Visualize the real spherical harmonics Y_l^m used in quantum mechanics, acoustics, and computer graphics. Rotate, explore, and understand the shapes that solve Laplace
Spinning Top | Iris Lab
Star Chart — Iris
An interactive celestial map showing real constellations with proper star positions. Drag to rotate the sky view, toggle constellation lines, and explore the night sky.
Stereogram Generator — Iris
Generate random-dot autostereograms (Magic Eye images). Type text or draw shapes, then view the hidden 3D image by diverging your eyes. Controls for depth, dot density, and pattern width.
Stern-Brocot Tree — Iris
Explore the Stern-Brocot tree, a binary tree containing every positive rational number exactly once. Navigate to any fraction via mediant construction and continued fractions.
Stigmergy
Simulate stigmergic construction: termite-like agents build organized mound structures from purely local rules, with no central plan.
Stirling Engine — Iris
Watch a Stirling engine cycle through four thermodynamic strokes. See the PV diagram traced in real time as heat converts to work.
Strange Attractors — Iris
Interactive 3D strange attractors: Lorenz, Rössler, Chen, Aizawa, and Thomas. Rotate with mouse drag, color particles by speed or age, adjust parameters to reshape chaos in real time.
Superfluid Helium
Interactive simulation of superfluid helium-4. Watch quantum vortices form, interact, and annihilate. Explore the Gross-Pitaevskii equation and Bose-Einstein condensation.
Sympathetic Resonance
Coupled oscillators and sympathetic resonance: drive one tuning fork and watch energy transfer selectively to forks of matching frequency. Visualize the resonance curve in real time.
Tangram — Iris Lab
Tautochrone & Brachistochrone — Iris
The tautochrone is the curve where every ball reaches the bottom in exactly the same time — it
Taylor Series — Iris
Visualize Taylor and Maclaurin series approximations. Watch polynomials converge to sin, cos, exp, and more as you add terms.
Morse Code Telegraph — Iris
Interactive Morse code telegraph. Send dots and dashes with a telegraph key, decode incoming Morse, and encode text into audible Morse code patterns.
Telegraph Network — Iris
Build a telegraph network between cities and watch messages propagate across wires in real time. Experience the first electrical communication network.
Telegraph Sounder — Iris
A detailed mechanical simulation of a telegraph sounder. Type messages to hear them in Morse code, or tap manually with the spacebar. Real click sounds via Web Audio API.
Tesseract
Interactive 4D hypercube visualization. Rotate a tesseract across all six planes of four-dimensional space.
Theremin Waves — Iris
A visual theremin where mouse position controls pitch and volume, with real-time waveform oscilloscope and frequency spectrum visualization using Web Audio API.
Thomson Problem — Iris
N charged particles on a sphere seeking electrostatic equilibrium. Watch Coulomb repulsion and damping find minimum-energy configurations — tetrahedra, octahedra, icosahedra.
Three-Body Problem — Iris
Simulate the gravitational three-body problem. Three massive bodies interact via Newtonian gravity in a famously chaotic system. Explore periodic solutions like the figure-eight orbit, drag bodies, and watch chaos unfold.
Coulomb's Torsion Balance
Interactive simulation of Coulomb
Turing Reaction-Diffusion — Iris
Gray-Scott reaction-diffusion system producing Turing patterns: spots, stripes, coral, maze, worms. Two chemicals, two parameters, infinite biological complexity.
Venturi Tube
Interactive Venturi effect demonstration showing how fluid velocity increases and pressure drops through a constriction, visualized with particles and pressure gauges.
Vernier Effect — Iris
Interactive demonstration of the Vernier measurement principle and moiré patterns. See how slightly different spacings create a magnification effect that enables sub-division measurement.
Fractal Generator — Iris
Interactive fractal generator: Vicsek cross, Sierpinski carpet, Koch island, T-square, and Cantor dust. Animate construction step by step and explore self-similarity.
Fortune's Sweep Line
Fortune's algorithm constructs a Voronoi diagram in O(n log n) by sweeping a line through space, maintaining a parabolic beach line that traces the growing cell boundaries.
Wada Basins
Three interleaved attraction basins where every boundary point simultaneously borders all three regions — fractal chaos made visible.
Wallpaper Groups — Iris
Visualize all 17 wallpaper symmetry groups — the only 17 ways to tile a plane with a repeating pattern. Draw a motif and see it replicated by translations, rotations, reflections, and glide reflections.
Clepsydra — Water Clock Simulation | Iris Lab
Water Hammer — Iris
Interactive water hammer simulation. Close a valve and watch the pressure wave propagate through a pipe — the Joukowsky equation in action. See why pipes bang and why slow-closing valves matter.
Water Drop Lens — Iris
Simulate how a water droplet acts as a lens. Adjust curvature, refractive index, and ray angle to see Snell
Watt Governor — Iris
The centrifugal governor — a mechanical feedback controller that keeps steam engines from running away. Watch the spinning balls rise and fall to regulate speed.
2D Wave Table — Iris
A ripple tank simulation — drop point sources onto a 2D water surface and watch waves interfere, diffract through slits, and reflect off barriers.
Wave Packet — Iris
Visualize wave packet propagation, group vs phase velocity, and dispersive spreading. Watch a Gaussian wave packet evolve under different dispersion relations.
Wave Superposition — Iris
Interactive demonstration of wave superposition — add multiple sinusoidal waves and see constructive and destructive interference.
Wavelet Transform — Iris
Interactive wavelet transform visualization. Compose signals from sine waves, view the scalogram heatmap, and compare wavelet vs FFT analysis.
Weierstrass Function
Continuous everywhere, differentiable nowhere — the 1872 pathological curve that shattered intuitions about smoothness.
Bathtub Vortex — Iris
Simulate a whirlpool draining from a circular basin. Tracer particles reveal the spiral flow governed by conservation of angular momentum. Click and drag to add rotation.
Wilberforce Pendulum
A mass on a helical spring that couples vertical bouncing to axial rotation. When the frequencies are nearly equal, energy pours completely from one mode into the other and back — a slow, hypnotic oscillation between two kinds of motion.
Betz Limit Wind Turbine — Iris
Interactive Betz limit demonstration. Adjust turbine extraction to find the optimal 59.3% power coefficient. Visualize wind streamlines, stream tube expansion, and the real-time power curve.
Zener Diode — Iris
Interactive IV characteristic curve of a Zener diode. Explore forward bias, reverse bias, and Zener breakdown. See how a Zener diode regulates voltage in a simple circuit with animated current flow.
Zipf's Law — Iris
Paste any text and watch Zipf
Ant Colony TSP
Ant Colony Optimization for the Traveling Salesman Problem. Pheromone trails guide ants toward shorter routes through animated visualization.
Bézier Curves
Interactive Bézier curve construction — drag control points, watch de Casteljau's algorithm animate the curve construction.
Boids 3D
Boids flocking simulation in projected 3D using Reynolds' separation, alignment, and cohesion rules.
Collatz Tree
Collatz conjecture tree visualization — every tested number eventually reaches 1 following the 3n+1 rule.
Delaunay Triangulation
Interactive Delaunay triangulation — click to add points, watch the triangulation update. The dual graph of the Voronoi diagram.
Double Pendulum Chaos
Two double pendulums started with nearly identical conditions diverge exponentially — a vivid demonstration of chaos and sensitive dependence on initial conditions.
Elastic Collision
2D elastic collision physics with momentum and kinetic energy conservation. Watch balls bounce with realistic physics.
SIR Epidemic Model
Interactive SIR epidemic spreading model on a grid. Watch infection waves propagate, tune transmission and recovery rates, and observe herd immunity thresholds emerge from simple local rules.
Farey Sequence
Farey sequence and Stern-Brocot tree — all fractions with denominators up to n, in order, with beautiful geometric properties.
FitzHugh-Nagumo Model
FitzHugh-Nagumo excitable system simulation with spiral waves on a 2D grid.
Flocking with Obstacles
Boids flocking simulation with obstacle avoidance. Agents navigate around circular obstacles using steering behaviors.
Forest Fire Model
Drossel-Schwabl forest fire cellular automaton. Trees grow, lightning strikes, fire spreads. A self-organized critical system with power-law fire size distributions.
Granular Gas
Inelastic ball collisions spontaneously cluster. Unlike molecular gases, a granular gas loses energy at each collision and collapses into dense clumps.
Ising Model
2D Ising model Monte Carlo simulation using the Metropolis algorithm. Watch the ferromagnetic phase transition near T=2.27.
Kuramoto Oscillators
Coupled phase oscillators synchronizing via the Kuramoto model. Watch spontaneous synchronization emerge as coupling strength crosses a critical threshold.
L-System Plants
L-system plant generator — rewriting rules create fractal plants, ferns, trees and snowflakes from simple string grammars.
Langton's Ant
Langton's ant cellular automaton — simple rules produce complex emergent behavior including a mysterious 'highway' pattern.
Modular Arithmetic
Modular multiplication circle patterns — connect n to (n×k) mod m with lines on a circle. Beautiful patterns emerge from simple number theory.
Neural Network XOR
A 2-layer neural network learning the XOR function with live training visualization, weight display, and loss curve.
Percolation Threshold
Site percolation on a square lattice. Near the critical threshold p≈0.5927, a spanning cluster emerges from local randomness.
Potts Model
q-state Potts model, a colorful generalization of the Ising model with q spin states and rich domain pattern dynamics.
Prime Gaps
Distribution of gaps between consecutive primes, twin prime visualization, and the mysterious patterns in the prime number sequence.
2D Random Walk
2D random walk simulation with mean squared displacement statistics, showing diffusive behavior and the square-root law.
Rule 110
Rule 110 elementary cellular automaton — proven Turing complete, producing complex patterns from a single bit and three-cell neighborhoods.
Schelling Segregation
Schelling's segregation model shows how mild individual preferences for similar neighbors lead to strong macro-level segregation.
Self-Organized Criticality
Bak-Tang-Wiesenfeld sandpile model: add grains one at a time and watch avalanches follow a power-law distribution — the signature of self-organized criticality.
Sugarscape
Epstein & Axtell's Sugarscape: agents with varying vision and metabolism harvest sugar, generating wealth inequality from simple rules.
Tinkerbell Map
The Tinkerbell map is a 2D discrete dynamical system producing a butterfly-shaped strange attractor through simple iterative equations.
Traveling Salesman Problem
TSP nearest-neighbor heuristic and 2-opt local search optimization — watch the route improve in real time.
Turing Diffusion
Turing reaction-diffusion pattern formation — spots, stripes, and labyrinthine patterns emerge from two interacting chemicals.
Voronoi Diagram
Interactive Voronoi diagram with Fortune's algorithm and Lloyd relaxation — click to add sites, watch regions form.
Barkhausen Effect
Magnetic domain visualization — discrete domain-wall jumps as a field sweeps, producing Barkhausen noise in ferromagnets.
Neural Cellular Automata
Brain-like excitable media with firing, refractory, and resting states — spiral waves and propagating action potentials.
Laplacian Growth / DLA
Diffusion-limited aggregation — random-walking particles build fractal dendrites with dimension ~1.71.
Newton's Rings
Thin film interference between a curved lens and flat glass — concentric rainbow rings with wavelength and refractive index controls.
Osmosis
Particle simulation of osmotic pressure — water molecules crossing a semipermeable membrane driven by concentration gradients.
Photoelectric Effect
Einstein's quantum photoelectric effect — photon energy, work function, stopping potential, and KE vs. frequency plots.
Lissajous Curves
x = A sin(aτ+δ), y = B sin(bτ) — the geometry of coupled oscillators, frequency ratios, and phase sweeps.
Lorenz Waterwheel
Malkus waterwheel — a mechanical realization of the Lorenz attractor that unpredictably reverses direction.
Tsunami Propagation
Shallow-water wave simulation — bathymetry, shoaling, and speed v=√(gh). Trigger earthquakes and watch waves reach shore.
Totalistic Cellular Automata
Beyond Conway's Life — explore all B/S birth and survival rules in 2D outer totalistic automata, from gliders to chaos.
Voronoi Growth
Crystal nucleation — competing growth fronts carve territory. Models grain growth, cell territories, and foam coarsening.
Spin Precession
Bloch sphere visualization — spin-½ Larmor precession, Rabi oscillations, and the FID signal that underlies MRI.
Fourier Heat Equation
∂u/∂t = α∇²u — temperature diffusion with Fourier mode decomposition showing how higher harmonics decay fastest.
Langton's Ant Extended
Multi-color turmites with arbitrary turn sequences — discover highways, triangles, spirals, and chaotic patterns.
Gravitational Waves
Binary black hole inspiral and chirp signal — GW150914-style merger with h₊ polarization and phase portrait.
Prime Sieve
Sieve of Eratosthenes in real time — watch composites get eliminated, explore twin primes and the prime number theorem.
Double-Slit Experiment
Quantum interference built one photon at a time — toggle which-way information to destroy the fringes.
Magnetic Field Lines
Interactive field line tracer — place poles and current wires, watch streamlines trace the B field in real time.
Phase Transitions
2D Ising model — ferromagnetic order emerging below Tc ≈ 2.27. Order parameter, critical exponents, Onsager exact solution.
Geodesic Flow
Shortest paths on curved surfaces — great circles on a sphere, wound helices on a torus, diverging lines on a saddle.
Voronoi Lloyd Relaxation
Lloyd's algorithm iteratively moves each point to its Voronoi centroid, converging to a beautifully uniform distribution.
BTW Abelian Sandpile
Bak-Tang-Wiesenfeld self-organized criticality: drop grains at center, watch toppling avalanches build fractal patterns.
Double-Slit Interference
Huygens wavelets from two slits superpose into interference fringes — the wave-particle duality experiment.
Rule 110 Space-Time
1D cellular automaton Rule 110, proven Turing-complete — complex, irregular patterns from a 3-cell neighborhood rule.
Vicsek Flocking Model
Particles align with neighbors, perturbed by noise — tune η to watch the order-disorder flocking phase transition.
Thomas Cyclical Attractor
René Thomas's 3D strange attractor with cyclic symmetry — parameter b controls the dissipation and fractal structure.
Harmonic Phase Portrait
Damped and driven oscillator trajectories in phase space — click to set initial conditions, explore fixed points and limit cycles.
Langton's Ant Multi-Color
Turing-complete 2D automaton that builds chaotic patterns then suddenly constructs a periodic highway — multi-rule variants.
Gray-Scott Patterns
Two-chemical reaction-diffusion PDE generates spots, stripes, and worms — same equations, different feed/kill rates.
Logistic Cobweb Diagram
Cobweb diagram for xₙ₊₁=rx(1−x) — watch period-doubling cascade to chaos as r increases past 3.57.
Lissajous Figures
Parametric curves x=sin(at+δ), y=sin(bt) — rational frequency ratios yield closed knots, irrational ratios fill the plane.
Turing Morphogenesis
Turing's 1952 activator-inhibitor system spontaneously breaks symmetry, generating animal-coat spots, stripes, and labyrinths.
Percolation Phase Transition
Watch the giant cluster emerge as site occupancy crosses the critical threshold p_c ≈ 0.593.
Lattice Boltzmann Flow
D2Q9 lattice Boltzmann fluid simulation — watch vortex shedding behind obstacles. Click to add or remove barriers.
Truchet Tiling
Random tiling with quarter-circle arcs connecting into labyrinthine paths. Click any tile to flip it.
Coupled Pendulums
Two pendulums linked by a spring — energy sloshes back and forth, producing beats when coupling is weak.
Differential Growth
A 1D curve grows by inserting nodes and repelling itself, producing organic coral and brain-like forms.
Erdős–Rényi Graph
G(n,p) random graph: watch the giant component suddenly emerge as edge probability crosses p=1/(n−1).
Julia Set Morphing
Hover over the Mandelbrot set to choose c — the corresponding Julia set updates in real time.
Lyapunov Fractal
Color 2D parameter space (a,b) by the Lyapunov exponent of the logistic map driven by a repeating AB sequence.
Random Matrix Eigenvalues
Eigenvalue density of Gaussian random matrices converges to Wigner's semicircle law as matrix size grows.
Rep-Tiles
Shapes that tile themselves at a larger scale — subdivide the L-tromino, sphinx, and pinwheel triangle recursively.
Winding Number
Draw a closed curve, place a point, and watch the integer winding number update as you drag the point around.
Reaction-Diffusion
Gray-Scott model grows Turing patterns as two chemicals diffuse and react — click to seed new structures.
Fourier Drawing
Draw any shape and watch a stack of rotating epicycles reconstruct it via the Discrete Fourier Transform.
Bifurcation Diagram
The logistic map's period-doubling route to chaos, with infinite self-similar structure near the Feigenbaum constant.
Abelian Sandpile
Add grains to a grid; cells with four or more topple into their neighbors, producing fractal avalanches at criticality.
Apollonian Gasket
Infinitely nested tangent circles generated by Descartes' Circle Theorem — a fractal built from curvatures.
Hilbert Curve
A space-filling curve that winds through every point in a grid; watch it animate order by order toward a 2D bijection.
Halvorsen Attractor
A symmetric strange attractor in 3D — drag to rotate and explore its fractal basin of attraction.
Parametric Curves
Lissajous figures, rose curves, spirographs, epitrochoids, and butterfly curves — tune the parameters live.
Voronoi Relaxation
Lloyd's algorithm moves each Voronoi site to its cell's centroid — watch random points converge to a perfectly even tiling.
Lorenz Attractor
The classic butterfly orbit of chaos theory — two nearly-identical trajectories diverge exponentially; drag to rotate in 3D.
Brownian Tree
Diffusion-Limited Aggregation: random-walking particles stick on contact, growing a fractal crystal with dimension ~1.71.
Maze Generator
Recursive backtracker carves a perfect maze cell by cell; then A* finds the shortest path from start to finish.
Penrose Tiling
Aperiodic kite-and-dart tiling via deflation — five-fold symmetric yet never repeating; kite-to-dart ratio converges to phi.
Torus Knot
A (p,q) torus knot winds p times around the longitude and q times around the meridian — drag to rotate; tune p and q live.
Chaos Billiards
A point particle reflecting inside a stadium table is fully chaotic; the circle is integrable — compare their trajectories.
Poincare Disk
The Poincare disk model of hyperbolic geometry: geodesics are circular arcs, all triangles have angle sum under 180 degrees.
Snowflake Growth
Reiter's cellular automaton for ice crystal growth on a hexagonal grid — three parameters produce wildly different flake shapes.
Cellular Noise
Worley noise: distance to nearest feature points generates organic textures; F2-F1 cracks, F1 blobs, animated or static.
Fourier Square Wave
Add odd harmonics to build a square wave; the Gibbs phenomenon produces a permanent ~9% overshoot at every discontinuity, no matter how many terms you add.
Spring Lattice
A 2D grid of masses connected by springs — click to pluck nodes and watch transverse waves, reflections, and interference propagate through the mesh.
Snell's Law
Drag the light source across a refractive interface; the beam bends by n₁sinθ₁ = n₂sinθ₂ and vanishes above the critical angle via total internal reflection.
Pixel Fire
Classic "doom fire" algorithm: hot pixels propagate upward with random drift and cooling, producing a convincing flame from a single bottom row of heat.
Epicycloid Generator
A rolling circle traces cardioids, nephroids, deltoids, and hypotrochoids — tune the fixed radius, rolling radius, and pen arm length to explore the family.
Galton Board
Balls deflect randomly left or right at each peg level; the binomial distribution emerges in the bins below — the Central Limit Theorem made physical.
Arnold's Cat Map
The torus map (x,y) → (2x+y, x+y) mod 1 scrambles any image into noise — but is exactly periodic, returning the original after a finite number of steps.
Drunkard's Walk
Multiple 2D random walks grow in RMS distance as √N — the signature of diffusion — with colored trails and a live theory comparison ring.
Circle Map & Arnold Tongues
The circle map models coupled oscillators; its Arnold tongue diagram reveals locking regions where the winding number snaps to a rational value.
Flocking (Boids)
Craig Reynolds' three-rule model — separation, alignment, cohesion — produces lifelike flocking from simple local interactions; click to scatter the flock.
Gravity Simulator
N-body gravitational simulation with softening; drag to place bodies with custom velocity, or load preset solar system, binary star, and random cluster scenarios.
Sorting Visualizer
Watch bubble, selection, insertion, merge, quick, and heap sort animate step-by-step with comparison and swap counts — see why O(n log n) matters.
Fibonacci Spiral
The golden angle (137.5°) used to pack seeds in sunflowers, pinecones, and daisy heads — watch how irrational rotation creates optimal density with no overlaps.
Harmonics Visualizer
Build any periodic waveform by layering sine waves. Adjust individual harmonic amplitudes to construct sawtooth, square, or triangle waves — Fourier synthesis in your hands.
Plasma Simulation
Charged particles in electromagnetic fields: positive ions and electrons interact via Coulomb forces while a tunable magnetic field causes helical orbits. Click to inject energy.
Noise Terrain
Procedural worlds from fractal Brownian motion — layered Perlin noise octaves sculpt mountains, valleys, and coastlines. Tune roughness, sea level, and scale in real time.
Epicycle Art
Any curve is a sum of rotating circles. Choose a preset (heart, star, flower) or draw your own path — the DFT decomposes it into epicycles that perfectly reconstruct it.
Crystal Growth Automata
A stochastic cellular automaton of crystal nucleation and growth — dendritic snowflake-like patterns emerge from local freezing probabilities on square or hexagonal grids.
3D Phase Portrait
Lorenz, Rössler, and Halvorsen strange attractors drawn as 3D trajectory bundles. Drag to rotate; tune parameters to move through bifurcations and watch the attractor reshape.
Gravity Well 3D
Spacetime curvature from Einstein's field equations visualized as a deformed 3D grid. Place multiple masses and watch the fabric warp around them; drag to rotate the view.
Wave Refraction
Huygen's wavelets bend at the boundary between two media. Adjust refractive indices to see total internal reflection, the critical angle, and Snell's law come to life.
Torus Flow
Geodesics on a torus as p/q-frequency orbits: rational ratios close into torus knots; irrational ratios never close, densely covering the surface — ergodic flow made visible.
Prime Factorization Art
Every integer's unique prime decomposition drawn as nested circles — the Fundamental Theorem of Arithmetic as generative art. Hover to reveal any number's factor tree.
Springs Network
A 2D grid of mass-spring oscillators: drag nodes to disturb the lattice and watch waves, standing modes, and resonance patterns propagate through the elastic medium.
Clifford Attractor
Strange attractors from trigonometric recurrences: x'=sin(ay)+c·cos(ax), y'=sin(bx)+d·cos(by). Tune four parameters to morph between galaxies, butterflies, and storms — all chaos, no repetition.
Quadtree Decomposition
Adaptive spatial subdivision: regions are recursively split until each quadrant is uniform enough. Adjust the error threshold to trade detail for abstraction — low threshold gives photorealism, high gives cubist painting.
Conway's Game of Life
Four rules, infinite complexity. Watch the Gosper glider gun perpetually fire gliders across the grid. Load classic patterns — pulsars, spaceships, pentominoes — or draw your own and see what survives.
Wang Tiles
Square tiles with colored edges that must match their neighbors. Certain small sets of Wang tiles can tile the plane only aperiodically — never repeating — a fact that proved tiling undecidable (Berger, 1966).
Particle Life
Four particle types governed by a matrix of attraction and repulsion forces. Remarkably simple rules produce cell-like clusters, predator-prey dynamics, and self-organizing chains — artificial chemistry in motion.
Procedural Terrain
Diamond-square fractal algorithm generates realistic heightmaps. Tune roughness from alpine jagged to rolling hills, adjust sea level, and switch between biome rendering, raw heightmap, and topographic contour views.
Prime Constellations
Five geometric mappings of prime numbers: polar Archimedean spiral, Ulam spiral (diagonal alignments), Sacks spiral (parabolic arcs), gap histogram, and modular grid — each revealing a different facet of prime structure.
RC / RL / LC Circuit Simulator
Visualize voltage and current responses in RC, RL, LC, and RLC circuits. Sliders control component values in real time, with a schematic diagram and time-domain graph showing charge, decay, and damped oscillations.
Hyperbolic Tilings
Poincaré disk model of hyperbolic geometry: {p,q} tessellations pack p-gons with q at every vertex — impossible in Euclidean space. All tiles are geometrically equal despite shrinking toward the boundary.
Spring Network
Verlet-integrated mass-spring systems: grids, cloth, rings, spider webs, and jelly blobs. Drag nodes to perturb the system; springs color by strain. Toggle gravity and wind to explore elastic wave propagation.
Superellipse / Lamé Curves
|x/a|ⁿ + |y/b|ⁿ = 1 — the family between diamonds and rectangles. Slide the exponent from astroid (n=½) through ellipse (n=2) to the squircle (n=4) and near-rectangle (n=8), with live multi-curve overlay and animation.
Trochoidal Ocean Waves
Gerstner waves: the only exact nonlinear solution to the Euler equations. Each water particle traces a perfect circle — radius decays exponentially with depth. Watch particles spiral as waves pass, with adjustable wavelength and amplitude.
Newton's Cannon
Fire a cannonball from a mountain and watch it orbit. Adjust launch speed from sub-orbital arc through circular orbit, elliptical orbit, and escape trajectory — Newton's thought experiment made interactive.
VSEPR Molecular Geometry
Valence Shell Electron Pair Repulsion: explore 9 classic molecules in 3D — from linear BeCl₂ to octahedral SF₆. Drag to rotate; see how lone pairs distort ideal geometries (tetrahedral → bent, trigonal → pyramidal).
Rule 30 — Wolfram's Chaos
Stephen Wolfram's Rule 30 is a 1D cellular automaton that generates provably unpredictable output — used in Mathematica's random number generator. Explore all 256 elementary rules, including Sierpiński's Rule 90 and universal Rule 110.
Möbius Transformation Explorer
f(z) = (az+b)/(cz+d) maps circles and lines to circles and lines. Drag sliders to deform a grid of the complex plane in real time — identity, inversion, rotation, translation, and the Cayley map all appear as special cases.
Aeolian Sand Ripples
Wind-driven saltation self-organizes flat sand into periodic ripple patterns. Grains bounce forward and dislodge others; small irregularities grow into stable wavelike ridges. Adjust wind speed and grain size to control ripple spacing.
Torus & Knot Geometry
Drag to rotate a ring, horn, or spindle torus — or switch to a torus knot T(p,q) that winds p times longitudinally and q times meridionally. T(2,3) is the trefoil; T(3,5) the cinquefoil. Continuously colored by parameter.
Acoustic Room Modes
Standing pressure waves in a rectangular room: f(m,n) = (c/2)√((m/Lx)²+(n/Ly)²). Animated pressure field shows nodes (quiet) and antinodes (loud). Click to place a microphone and read relative SPL at that position.
Peter de Jong Attractor
xₙ₊₁ = sin(ay) − cos(bx), yₙ₊₁ = sin(cx) − cos(dy): four parameters, infinite variety. Millions of iterations accumulate into intricate filaments — density-mapped with fire, ice, plasma, or emerald color schemes.
Chord Diagram
Interactive circular chord diagram showing connections between groups. Hover over arcs to highlight relationships and outflows. Adjust group count and connection density — a go-to tool for visualizing network flows and inter-group relationships.
Hyperbolic Tessellation
The Poincaré disk model of hyperbolic geometry: tiles that look smaller near the boundary are all the same size in hyperbolic space. Choose from {4,5}, {3,7}, {7,3} and other Schläfli types. Drag to translate through the infinite hyperbolic plane.
Dendritic Crystal Growth
Phase-field simulation of dendritic solidification — the branching crystal patterns that form when a melt cools rapidly. Anisotropy controls branch symmetry; undercooling drives tip speed. The same math explains snowflakes, metallic dendrites, and frost.
Strange Attractors Gallery
Six 3D chaotic attractors — Chen, Dadras, Halvorsen, Thomas, Finance, and Aizawa — traced via RK4 integration and projected in real time. Adjust rotation speed and trail fade; watch orbits fill intricate fractal structures never repeating yet forever bounded.
3D Cellular Automata
Conway's Game of Life extended into three dimensions. Choose from Crystal (4/4/5), Amoeba, Stable, Cloud, and Explode rulesets — each produces radically different emergent structures. Drag to rotate the voxel grid and watch generations evolve.
Entropy & Time's Arrow
Gas particles released from one corner spread to fill a box — entropy rises. The microscopic equations are time-reversible, but the macroscopic arrow of time emerges statistically. Hit "Reverse Time" to watch entropy try (and fail) to decrease.
Adiabatic Compression
Drag a piston to compress or expand a gas with no heat exchange. Watch temperature spike on compression (PVᵞ = const), see the P-V diagram trace the adiabat in real time. Switch between diatomic air, monatomic noble gas, and polyatomic CO₂.
Fourier Optics
A lens performs an optical Fourier transform: the far-field diffraction pattern of an aperture IS its 2D Fourier magnitude spectrum. Draw any aperture or choose presets — circle, slit, double slit, grating, annular — and see the diffraction pattern instantly.
p-adic Numbers
The p-adic numbers redefine "closeness" via divisibility: two integers are close if their difference is divisible by a large power of p. The p-adic tree visualizes this ultrametric space. Includes a valuation calculator and the famous result that −1 = …11111 in ℚ₂.
Desargues' Theorem
Two triangles in perspective from a point are in perspective from a line — always. Drag any vertex to deform the configuration; the three intersection points of corresponding sides remain collinear. A cornerstone of projective geometry since 1648.
Kuramoto Synchronization
N oscillators with random natural frequencies couple and spontaneously synchronize above a critical coupling K_c. Watch the order parameter r jump from near-zero to near-one — a continuous phase transition from disorder to collective rhythm, modeling fireflies, neurons, and power grids.
Taxicab Geometry
In taxicab geometry, distance is measured along grid lines instead of straight lines — making "circles" into diamonds. Explore how this non-Euclidean metric changes circles, bisectors, and distance ratios. Compare taxicab vs Euclidean distances interactively.
Cyclic Cellular Automata
Rock-paper-scissors dynamics on a grid: each cell advances to the next state when enough neighbors already have it. With low thresholds, stunning rotating spiral waves emerge — a simple local rule generating global order. Adjust states, thresholds, and neighborhoods.
Arithmetic Billiards
A billiard ball fired at 45° in an a×b rectangle bounces until it reaches a corner. The path encodes number theory: bounce count = a+b−2, path segments = LCM/GCD, and GCD appears as symmetry. Watch animated trajectories reveal hidden arithmetic structure.
Polyomino Tiling
Place pentominoes, tetrominoes, and other polyomino shapes to fill a grid. The 12 pentominoes famously tile 6×10, 5×12, and other rectangles — one of the great combinatorial puzzles. Rotate, flip, and place pieces; or watch the auto-solver at work.
Spatial Predator-Prey
Lotka-Volterra dynamics in a 2D cellular world: grass grows, prey eats grass, predators eat prey. Local interactions generate emergent population oscillations — spiraling waves of boom and bust visible in real time, with a population history chart.
Stochastic Resonance
Counter-intuitively, adding noise to a sub-threshold signal can enhance its detection. Too little noise: no crossings. Too much: all noise. At the sweet spot, the signal "rides" the noise over the threshold — a phenomenon found in neurons, sensory biology, and climate science.
Prime Walk
Encode the prime numbers as directions in a 2D walk and reveal their hidden structure. Prime digits control turns, residues mod 4 encode direction (reflecting Dirichlet's theorem), and prime gaps leave visual signatures. Four walk types with color modes.
Fourier Epicycle Drawing
Draw any closed shape with your mouse, then watch rotating circles (epicycles) reconstruct it perfectly using the Discrete Fourier Transform. Each circle corresponds to one frequency component. Adjust the number of epicycles to see how accuracy builds with more terms.
Quantum Wave Functions
Visualize particle-in-a-box wave functions ψₙ(x), their probability densities |ψ|², and quantum superpositions. Watch animated time evolution, see how energy levels scale as n², and explore interference between states — the foundation of quantum mechanics.
Penney's Game
A non-transitive coin paradox: pick any sequence of H/T flips, and the computer picks a sequence that beats yours. Like rock-paper-scissors, no sequence is unbeatable. Conway's algorithm always finds the optimal counter — simulate thousands of games to verify.
Feigenbaum Universality
The logistic map x→r·x·(1−x) undergoes period-doubling cascades as r increases. The ratio of successive bifurcation intervals converges to δ≈4.6692 — universal across all smooth unimodal maps. Zoom into bifurcation diagram, mark period-doubling points, and discover Feigenbaum's constant.
Fibonacci Word Fractal
The Fibonacci word is generated by substitution: 1→12, 2→1. Drawing a path with turns at each character produces a self-similar fractal whose structure mirrors the Fibonacci sequence. Adjust iteration depth, turn angle, and color scheme to explore this elegant substitution geometry.
Bunimovich Stadium Billiards
A rectangle capped with semicircles creates a chaotic billiard: almost every trajectory is ergodic and explores the entire stadium. Nearby trajectories diverge exponentially (positive Lyapunov exponent). Click to place balls and watch them fill the table — a landmark result in mathematical physics from 1974.
Peano Curve
Giuseppe Peano (1890) shocked mathematics by constructing a continuous curve that passes through every point in a square. At order n, it visits all 3ⁿ×3ⁿ grid cells exactly once. Watch the curve animate cell by cell, or jump to any order to see the full space-filling structure.
Cantor Dust
The 2D Cantor set removes the middle third in both x and y directions, leaving a fractal dust with Hausdorff dimension log(4)/log(3)≈1.261. Explore three views: the 2D dust, the 1D Cantor set generation, and the Devil's Staircase — a continuous monotone function with derivative zero almost everywhere.
Hopalong Attractor
Barry Martin's "Hopalong" map — xₙ₊₁ = y−sign(x)·√|bx−c|, yₙ₊₁ = a−x — creates intricate feathery attractors from three simple parameters. Despite being a deterministic 2D map with no differential equations, it generates structures of astonishing complexity. Explore dozens of preset parameter combinations.
Zaslavsky Web Map
The Zaslavsky map models a kicked oscillator and creates stunning web-like stochastic structures at resonance. For integer r, KAM tori form symmetric webs with r-fold symmetry. Increasing the kick strength K progressively breaks the tori, revealing the transition from order to chaos in Hamiltonian systems.
Gingerbread Man Map
The gingerbread map xₙ₊₁ = 1−y+|x|, yₙ₊₁ = x is area-preserving and reversible, yet its orbits trace out a figure resembling a gingerbread man. Discovered by R. Devaney (1984). Explore single orbits, density plots of repeated trajectories, and basin boundary maps revealing fractal structure.
Ikeda Map Attractor
Kensuke Ikeda (1979) derived this map from light circulating in a nonlinear optical cavity. As the coupling parameter u increases, the fixed point undergoes period-doubling bifurcations and eventually becomes a strange attractor. The resulting spiral structure is one of the most beautiful chaotic attractors in 2D.
Rössler Attractor
Otto Rössler (1976) designed the simplest possible chaotic flow: ẋ=−y−z, ẏ=x+ay, ż=b+z(x−c). The strange attractor resembles a folded ribbon — one-dimensional folding creates sensitivity to initial conditions. View projections in XY, XZ, YZ planes or rotate a live 3D projection by dragging.
Nagel-Schreckenberg Traffic
A cellular automaton model of highway traffic: cars accelerate, brake to avoid collisions, and randomly slow down. Phantom traffic jams emerge spontaneously from local interactions — no bottleneck required. Watch the space-time diagram reveal the characteristic backward-propagating jam waves.
Galaxy Merger Simulation
Two galaxies collide under gravity using the Toomre & Toomre (1972) restricted 3-body approximation: massive cores interact via full N-body gravity while thousands of test-particle stars orbit both cores. Watch tidal tails, bridges, and shells form as the galaxies merge over cosmic time.
HP Protein Folding Model
The HP (Hydrophobic-Polar) lattice model reduces protein folding to a 2D optimization: H residues minimize free energy by maximizing non-bonded hydrophobic contacts. Monte Carlo with pivot and crankshaft moves searches conformational space. A playground for understanding NP-hard optimization in biology.
Reynolds Flow Transition
Drag tracer particles through flow around a cylinder as the Reynolds number Re = ρvL/μ spans five decades. Watch the transition from symmetric creeping flow (Re<5) through steady vortex pairs (Re~50) to the von Kármán vortex street (Re~100) and turbulent wake (Re>1000).
Quantum Walk vs Classical
A quantum particle in superposition spreads quadratically faster than its classical counterpart: σ∝t vs σ∝√t. The Hadamard coin creates interference between left- and right-moving amplitudes, producing a characteristic double-peaked distribution — the core mechanism behind quantum speedup algorithms.
Circadian Oscillator Network
The Goodwin 3-variable feedback loop (mRNA→Protein→Clock product→inhibits mRNA) produces the ~24-hour biological rhythm found in nearly all life. Tune coupling strength between cells, Hill coefficient, and light zeitgeber input to see synchronization, phase locking, and circadian disruption.
Water Wave Dispersion
Deep-water waves are dispersive: ω²=gk, so long waves travel faster. Click to launch a wave packet and watch it spread as its components separate. The dispersion relation ω²=gk·tanh(kh) smoothly connects deep-water (vg=½vp) to shallow-water (non-dispersive) limits. The group velocity tangent is shown live.
Topological Charge & Skyrmions
Skyrmions are localized spin textures in 2D magnetic materials with integer topological charge Q — stable against smooth deformations. Visualize the spin field, topological charge density, and watch Landau-Lifshitz relaxation smooth the configuration while preserving Q. Switch between skyrmion, antiskyrmion, meron, and vortex states.
Differential Gear Mechanism
The automotive differential obeys one elegant equation: ω_drive = (ω_left + ω_right)/2. Planet gears on a rotating spider allow the output shafts to rotate at different speeds during cornering. Watch animated gears, wheel rotations, and explore what happens when one wheel loses traction (locks).
Sonoluminescence
A bubble trapped by ultrasound oscillates, expands to ~10× its rest size, then violently collapses — compressing the interior to plasma temperatures (~50,000 K) and emitting a picosecond flash of light. The Rayleigh-Plesset equation governs the dynamics. Tune drive pressure and frequency to find the sonoluminescence regime.
Fourier Mode Decomposition
Decompose square waves, sawtooth, triangle, and custom drawn waveforms into individual frequency harmonics. Watch the spectrum light up, cycle through each mode, and observe the Gibbs phenomenon: a square wave's Fourier series overshoots by ~9% near discontinuities no matter how many terms you add.
Brownian Bridge
A random walk conditioned to return to its start — or any target value. Watch multiple bridge paths with empirical mean ± std envelope versus the theoretical variance σ²t(T−t)/T. Drag on canvas to set the endpoint, adjust volatility and path count. The Gaussian achieves minimum uncertainty.
Hénon-Heiles System
A 2D nonlinear Hamiltonian H=½(ẋ²+ẏ²+x²+y²)+x²y−y³/3 with a critical energy Ec=1/6. Below Ec, orbits are regular KAM tori; above it, chaos spreads. The right panel shows the Poincaré section (y=0 crossings) — smooth curves vs scattered points reveal the order-chaos transition in real time.
Hexagonal Life
Conway's Game of Life reimagined on a hexagonal grid where every cell has 6 neighbors. Explore multiple rule sets: Hex Life (B2/S3,4), glider rules, maze-filling, and annealing. Draw cells, step manually, or run continuously. Hex topology changes which patterns are stable, oscillating, or chaotic.
Small-World Networks
The Watts-Strogatz model starts with a ring lattice and rewires edges with probability β. Even a tiny β creates shortcuts that collapse average path length while preserving high clustering — the hallmark of social networks and neural circuits. Click nodes to explore BFS layers; watch L and C statistics shift.
Fourier Uncertainty Principle
The Heisenberg-Gabor bound: σ_t·σ_f ≥ 1/(4π). Squeeze a Gaussian pulse in time and its frequency spectrum broadens inversely — and vice versa. Explore Gaussian, rectangular, triangular, and chirp windows. The STFT spectrogram shows the time-frequency trade-off visually; the Gaussian achieves the lower bound exactly.
2D Wave Tank
Interactive finite-difference simulation of the 2D wave equation. Draw walls to create reflections, add oscillating sources, and set up a double-slit to see interference fringes. Toggle wave speed and damping. Walls act as fixed (Dirichlet) boundaries; the absorbing outer edge prevents artificial reflections.
Ptolemy's Theorem
For any cyclic quadrilateral: AC·BD = AB·CD + AD·BC. Drag the vertices around the circle to verify the identity dynamically. Explore how Ptolemy used this to build the first trigonometric table (chord table), and see the pentagon tab reveal the golden ratio φ hidden in diagonal-to-side ratios.
Tiling Explorer
Explore periodic and aperiodic tilings: square, hexagonal, triangular (all 3 regular), Cairo pentagonal (type 4, found in Islamic architecture), octagon-square, chair fractal (self-similar substitution), and the pinwheel tiling (infinitely many orientations). Adjust scale, rotation, and color mode live.
Continued Fraction Tree
The Stern-Brocot and Calkin-Wilf trees each contain every positive rational exactly once. Navigate the tree to find any number — the path encodes its continued fraction [a₀;a₁,a₂,...]. Watch convergents oscillate above and below the target, closing in exponentially. The golden ratio φ=[1;1,1,1...] takes the slowest path.
Elastic Waves
P-waves (longitudinal compression) and S-waves (transverse shear) in an elastic medium. Particle grid shows displacement vectors. Layered Earth mode traces ray paths refracting at density boundaries via Snell's law. Dispersion mode shows how frequency-dependent wave speed spreads a wavepacket into a chirp.
Hexaflexagon
A strip of paper folded into a hexagon with hidden faces — click to flex it and cycle through all faces. Trihexaflexagon (3 faces) or hexahexaflexagon (6 faces), invented by Arthur Stone in 1939.
Markov Music
A first-order Markov chain generates melodies from a transition matrix you edit. Choose scale, adjust the probability of jumping between notes, then press Play to hear stochastic music via the Web Audio API.
Josephus Problem
37 soldiers in a circle, every k-th eliminated — where do you stand to survive last? Watch the animated elimination and verify the recurrence J(n,k) = (J(n−1,k)+k−1) mod n + 1.
Euler Path Finder
Draw a graph by placing nodes and connecting edges, then find an Eulerian path — a trail crossing every edge exactly once. Includes the Königsberg bridge preset that proves no such path exists.
Weaving Draft
Design loom weave patterns using the traditional draft notation: threading, tie-up, and treadling grids update the drawdown fabric instantly. Includes plain weave, twill, basket, satin, and herringbone presets.
Moon Phases
Calculate and visualize the Moon's phase for any date using Julian Day arithmetic and the 29.53-day synodic period. Monthly calendar shows mini moon icons for every day.
Retrograde Motion
Watch a planet appear to reverse direction against background stars as Earth overtakes it. Heliocentric view shows the geometry; sky view traces the apparent looping path that puzzled ancient astronomers.
Ballistic Range
Fire a projectile and compare vacuum trajectory (parabolic, optimal at 45°) to the drag trajectory computed by RK4 integration. Adjust angle, speed, drag coefficient, mass, and gravity.
Ramsey Graph
Color every edge of a complete graph red or blue — Ramsey theory guarantees a monochromatic triangle for K₆ and beyond. Try to beat the theorem, or watch it find the inevitable triangle.
Solenoid Magnetic Field
Visualize the magnetic field of a solenoid computed via Biot-Savart summation over discrete coil loops. Adjust turns, length, radius, and current direction — field lines are uniform inside and fringe at the ends.
SIR Epidemic Model
The classic compartmental model — Susceptible, Infected, Recovered — shows how R₀ = β/γ determines whether an outbreak dies out or explodes. Drag sliders to explore the epidemic threshold.
Morse Code
Encode and decode Morse code in real time, visualize the dot-dash timeline, and hear it played back via Web Audio at adjustable speed — with a full reference table.
Musical Overtones
Sculpt a waveform by mixing the first eight harmonics of a fundamental frequency. Hear the result and see how the Fourier spectrum defines timbre — why violin and flute differ.
Petri Net Simulator
Model concurrent processes with Petri nets: click enabled transitions to fire them, or auto-fire at intervals. Includes producer-consumer, mutual exclusion, and dining philosophers presets.
Roman Numeral Calculator
Convert between Arabic and Roman numerals and perform full arithmetic — addition, subtraction, multiplication, division — in the ancient notation system up to MMMCMXCIX.
Shadow Puppets
Drag shapes and a point light source to cast mathematically projected shadows on a wall. Shadow size depends on distance from light — geometry made tangible.
Penrose Tiling (P3)
Generate a Penrose P3 aperiodic tiling via iterated deflation of fat and thin golden rhombs. The pattern has 5-fold symmetry yet never repeats — the geometry behind quasicrystals.
Soma Cube
The Soma cube: 7 irregular polycubes (all but the straight tetromino) that can be assembled into a 3×3×3 cube in 240 distinct ways. Manipulate pieces in 3D and find a solution.
Spherical Harmonics
Visualize real spherical harmonics Y(l,m) in 3D — the eigenfunctions of the angular part of the Laplacian. These describe electron orbital shapes, acoustic resonances, and CMB anisotropy.
Origami Fold Simulator
Draw mountain and valley crease lines on a square sheet and animate the fold. Load classic bases (crane, waterbomb, blintz) and explore Kawasaki and Maekawa's flat-fold theorems.
Braid Group Visualizer
Compose generators σ₁, σ₂, σ₃ and their inverses to build braid words — the algebraic objects behind knot theory. Close the braid to see the resulting link or knot.
Polymer Chain Explorer
Self-avoiding random walks model polymer conformations. Watch thermal fluctuations via the pivot Monte Carlo algorithm and measure end-to-end distance versus chain length.
Spectral Color Mixer
Add Gaussian spectral peaks across 380–700 nm and see the perceived color computed via CIE 1931 XYZ color-matching functions — explore metamerism and trichromacy.
Hodgkin-Huxley Neuron
The Nobel Prize model of nerve impulse generation. Inject current to trigger action potentials and watch voltage-gated Na⁺ and K⁺ channels produce the classic spike waveform.
Gear Clock
A mechanical clock driven by gear trains — every tooth counts. Gear ratios produce the 60:1 and 720:1 reductions from fast escapement wheel to minute and hour hands.
pH Indicator & Titration
Mix strong or weak acids with NaOH and watch indicator dyes change color in real time. Trace the full titration curve and locate the equivalence point using Henderson-Hasselbalch.
Michelson Interferometer
Move a mirror by nanometres and watch circular interference fringes shift. The 1887 experiment that disproved the aether — adjust tilt and coherence to explore fringe patterns.
IFS Fractal Explorer
Any fractal is the unique attractor of a set of contractive affine maps. Edit the transformation matrices or load presets (fern, Sierpinski, dragon) and render millions of points.
Chladni Patterns
Sand on a vibrating plate collects at nodal lines — zero-amplitude regions of 2D standing waves. Choose mode numbers m and n to reveal the intricate Chladni figure.
3D Cellular Automata
Conway's Game of Life extended into three dimensions with multiple rule sets — amoeba, coral, cloud, pyroclastic. Rendered as a rotatable voxel cloud with 26-neighbor rules.
Galton Board
Balls falling through a triangular peg array produce the binomial distribution, which converges to a Gaussian by the Central Limit Theorem. Tune bias and watch the normal curve emerge.
Tensor Product Visualizer
The outer product u⊗v forms a rank-1 matrix. Explore how vectors combine into tensors, visualize SVD decomposition, and see separable quantum states as product tensors.
Karplus-Strong String Synthesis
A burst of noise fed through a delay loop with low-pass filtering simulates a plucked string. Play a full piano octave in your browser — all synthesis done in real time via the Web Audio API with no external libraries.
Buddhabrot
Unlike the Mandelbrot set, the Buddhabrot plots the trajectories of escaped orbits — each pixel counts how many paths passed through it. The result is a haunting Buddha-like figure hidden within the Mandelbrot set, discovered by Melinda Green (1993).
Hydraulic Erosion
Simulated raindrops flow downhill, erode terrain, and deposit sediment elsewhere. Thousands of virtual drops carve realistic river valleys, ridges, and drainage networks from fractal noise using the Beyer (2006) hydraulic erosion algorithm.
Riemann Zeta Zeros
The Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re(s)=½. Visualize the zeta function on the critical line as a spiraling curve that crosses the origin at each zero — γ₁≈14.13, γ₂≈21.02, … computed via the alternating Dirichlet eta series.
Collatz Conjecture
If even, halve it; if odd, triple and add 1. Every number eventually reaches 1 — but nobody has proved it. Trace any sequence, compare stopping times across thousands of integers, and explore the inverse Collatz tree rooted at 1.
3D Lissajous Figures
x=sin(at+δ), y=sin(bt), z=sin(ct+φ) — three coupled sinusoids trace space curves that close into knots for integer ratios. Drag to rotate, animate the phase offset to watch the curve morph, and color by time, speed, or depth.
Rose Curves (Rhodonea)
r=cos(kθ) traces a k-petaled rose for odd k, or 2k petals for even k. Rational k=n/d creates intricate multi-lobed flowers that close after d revolutions. Explore presets from trefoil to 16-petal and animate the phase for hypnotic morphing.
SIR Epidemic Model
Kermack-McKendrick (1927): Susceptible → Infected → Recovered. Adjust β (transmission), γ (recovery), and spatial mixing to watch epidemic waves sweep a grid. R₀=β/γ determines whether disease spreads or dies — herd immunity threshold at 1−1/R₀.
Fourier Heat Equation
∂u/∂t = α∂²u/∂x² — draw any initial temperature profile and watch it diffuse. Fourier's solution decomposes the profile into sinusoidal modes that each decay at rate e^(−αn²π²t). High-frequency components vanish first, smoothing sharp edges.
Indo-European Language Tree
All ~3 billion Indo-European speakers share Proto-Indo-European ancestry (~5000 BCE). Hover the phylogenetic tree to see cognate words — mother/māter/mātṛ/мать — transformed by millennia of systematic sound change following Grimm's law and others.
Rayleigh–Bénard Convection
Fluid heated from below spontaneously forms convection rolls above a critical Rayleigh number Ra≈1708. This vorticity-streamfunction simulation shows temperature, vorticity, and velocity fields evolving in real time. Adjust Ra and Prandtl number to cross the bifurcation.
Substitution Tilings
Aperiodic tilings generated by replacing each tile with smaller copies: Penrose P3 rhombi (5-fold symmetry), Ammann-Beenker (8-fold), and triangle substitution. No translation period — yet the diffraction pattern has sharp quasicrystalline peaks. Drag to pan, scroll to zoom.
Harmony Lattice — Just Intonation
The Tonnetz (tone network) maps musical intervals as 2D geometry: moving right multiplies by 3/2 (perfect fifth), diagonally by 5/4 (major third). Click nodes to hear pure-ratio tones; select multiple to build just-intonation chords. Triads appear as triangles on the lattice.
Abelian Sandpile Model
Add grains to a grid; any cell with ≥4 grains topples, distributing one grain to each neighbor. Without tuning any parameter, the system self-organizes to criticality — producing power-law avalanches and fractal patterns with exact 4-fold symmetry. Click or drag to add sand.
Stereographic Projection
Project a sphere onto a plane from the north pole — circles on the sphere map to circles on the plane (conformal). Watch latitude/longitude grids, great circles, and loxodromes transform as you drag the sphere. The Riemann sphere identifies ℂ∪{∞} with the sphere.
SIR Epidemics on Networks
Watch a disease spread through Erdős-Rényi, Barabási-Albert scale-free, Watts-Strogatz small-world, and grid networks. R₀=β⟨k⟩/γ governs epidemic threshold. Scale-free networks are far more vulnerable — hubs create superspreader dynamics absent in regular lattices.
Chemical Oscillators
Nonlinear chemistry produces spontaneous oscillations: the Brusselator (Prigogine), Oregonator (Belousov-Zhabotinsky reaction), Van der Pol oscillator, and Repressilator genetic clock. Phase-plane portrait and time series update live. Tune parameters to cross the Hopf bifurcation.
Telegraph Equation
∂²u/∂t² + 2α·∂u/∂t = c²·∂²u/∂x² interpolates between lossless wave propagation and diffusion. Click on the wire to inject Gaussian or square pulses; choose open, matched, or shorted terminations to see reflection, absorption, and inversion. Derived by Lord Kelvin for the transatlantic telegraph.
Solar System — True to Scale
The solar system is almost entirely empty space — this visualization is accurate in both planet sizes and orbital distances simultaneously. Drag to pan from the Sun to Neptune; scroll to zoom. At full zoom-out the Sun is a single pixel. Hover planets for facts.
Semaphore Flags
The visual telegraph — encode any message in flag semaphore, the 19th-century naval signaling alphabet. Click letters in the full alphabet grid, type a message to animate, and watch a figure raise flags through all 26 positions.
Tidal Locking
See how gravity synchronizes a moon's spin with its orbit over time. Toggle between free rotation, tidal braking, and the locked state — watch the "near side" arrow settle toward the planet, just as the Moon always shows Earth the same face.
Devil's Staircase
The Cantor function rises continuously from 0 to 1 yet its derivative is zero almost everywhere — it climbs entirely on a set of measure zero. Adjust the split ratio α to explore fat-Cantor variants, hover for exact values, and watch the construction build level by level.
Chord Progression Explorer
Build and play chord progressions using diatonic harmony. Choose key and scale, click chords to queue them, hear the result via Web Audio, and explore classic progressions like I–V–vi–IV and ii–V–I across major, minor, Dorian, and Mixolydian modes.
Spatial Epidemic Spreading
SEIR disease dynamics on a spatial grid — infection spreads as visible waves rather than uniform mixing. Tune transmission probability, incubation period, mortality, and grid topology; watch the epidemic curve and color-coded grid update together in real time.
Fluid Mixing
Inject colored dye into laminar or turbulent flow and watch how the Reynolds number determines whether streams stay neatly stratified or spiral into chaotic swirls. Click to inject, drag to stir; toggle between flow regimes to feel the difference viscosity makes.
Spiro-Harmonograph
Two damped pendulums — one driving x, one y — trace intricate Lissajous-like curves as their superimposed oscillations decay. Tune frequency ratios, phases, damping, and rotation to reproduce classic harmonograph patterns or discover entirely new ones.
Tensegrity Structures
Buckminster Fuller's floating compression: rigid struts suspended in a web of tension cables, no strut touching another. Perturb the 3D structure and watch it spring back to equilibrium via physics simulation. Adjust stiffness and damping; try the 3-strut simplex, 4-strut, and 6-strut variants.
Musical Temperament
Hear the difference between equal, just, Pythagorean, meantone, and Kirnberger III tunings. Play chords and the circle of fifths; see cents deviations from equal temperament for every note; and listen for the beating that reveals impure intervals.
Whittaker-Shannon Interpolation
The Nyquist-Shannon sampling theorem made visual: discrete samples reconstruct a continuous signal perfectly when the sample rate exceeds twice the signal's highest frequency. Drop below Nyquist and watch aliasing corrupt the reconstruction; draw your own signal to experiment freely.
Abelian Sandpile
Drop grains of sand on a grid — when a cell accumulates 4 grains it topples, cascading to neighbors. The system self-organizes to a critical state producing avalanches with power-law size distributions, a hallmark of self-organized criticality (SOC). Click to drop, or watch the center auto-feed; reveal the identity element of the sandpile group.
Strange Attractors 3D
Six chaotic dynamical systems — Lorenz, Rossler, Thomas, Aizawa, Dadras, Halvorsen — rendered as glowing 3D trails. Trajectories never repeat yet remain confined to a fractal attractor. Drag to rotate the 3D view, tune speed and trail length, and watch multiple initial conditions diverge exponentially while tracing the same strange shape.
Chord Progression Explorer
Build and play chord progressions with music theory guidance. Choose a key and scale (major, minor, Dorian, Mixolydian), click diatonic chords to hear them, double-click to add to your progression, and play it back at any BPM. Common progressions (I-V-vi-IV, ii-V-I, etc.) load instantly with function labels explaining tonic, dominant, and pre-dominant roles.
Diffusion-Limited Aggregation
Particles released from the boundary perform random walks until they stick to a growing cluster. The result is a branching fractal with Hausdorff dimension ~1.71 in 2D. This models snowflake growth, lightning channels, electrodeposition, and mineral dendrites. Adjust stickiness and color modes; watch the aggregate branch and self-avoid.
Reaction-Diffusion Explorer
Gray-Scott model of two reacting chemicals — autocatalysis A+2B→3B and decay B→P — producing Turing patterns: spots, stripes, labyrinths, coral, bubbles. Tune feed and kill rates to navigate the phase diagram; click to seed new reactant B; presets jump to named pattern regimes. The same math underlies leopard spots and zebrafish stripes.
Fourier Drawing Machine
Draw any closed shape — the Discrete Fourier Transform decomposes it into rotating circles (epicycles) that reconstruct your curve. Watch the epicycle arms spin and trace out hearts, stars, trefoils, and Lissajous figures. Slider controls how many frequency components are used; fewer circles produces a smoothed, lossy approximation.
Evolutionary Art
Nine canvases, each encoded as a parametric genome (tree, flow field, orbit, web, bloom, crystal). Click your favorites to select them, then evolve the next generation via crossover and mutation. Over generations the population drifts toward your aesthetic preferences — interactive Darwinian art selection with tunable mutation rate.
Hyperbolic Tiling
Regular {p,q} tilings in the Poincare disk — where (p-2)(q-2)>4 forbids Euclidean realization. Geodesics appear as circular arcs meeting the boundary at right angles. Drag to apply a Mobius transformation and navigate the infinite tiling. Choose hexagons, heptagons, pentagons, or triangles; switch color modes by depth, parity, or angle.
Quantum Tunneling
A Gaussian wavepacket evolves under the 1D Schrodinger equation and hits a rectangular potential barrier. Watch |ψ|² split — part reflects, part tunnels through even when energy is below the barrier height. Live transmission and reflection percentages update as the wave propagates. Tune barrier height, width, and particle energy to explore the tunneling probability.
Cipher Playground
Six classical cryptosystems in one lab: Caesar shift with brute-force cracker, Vigenere polyalphabetic cipher, Atbash reverse-alphabet, randomizable monoalphabetic substitution, frequency analysis tool (compare cipher letter frequencies to English), and a mini Enigma machine with three rotors and reflector. Understand why each cipher was eventually broken.
Orbital Resonance
Two planets in adjustable orbits with tunable periods and eccentricities. Watch conjunction patterns emerge as petals, explore the 1:2 Jupiter resonance, 2:3 Pluto resonance, and Laplace 3:4 configuration. Kepler's equation solved iteratively for realistic elliptical orbits.
Color Wheel Harmony
Interactive HSL color wheel with six harmony schemes: complementary, analogous, triadic, tetradic, split-complementary, and square. Click any hue, adjust saturation and lightness, and see the mathematical relationships behind color theory used in design and fine art.
ASCII Art Generator
Convert text or uploaded images to ASCII art in real time. Choose from detailed, simple, block, binary, or hex character sets. Adjust column width and invert darkness. Renders large text with canvas fonts then rasterizes to characters — export with one click.
Regex Visualizer
Live regular expression tester with pattern breakdown, match highlighting, and group capture display. Includes presets for email, URL, phone, date, IPv4, hex color, word boundaries, and named groups. Tokenizes patterns into color-coded components: quantifiers, anchors, character classes, and groups.
Histogram Equalizer
Image processing playground with eight algorithms: histogram equalization, CLAHE (adaptive), gamma correction, linear contrast, unsharp sharpening, Sobel edge detection, inversion, and grayscale. Upload an image or use built-in samples. Live histograms update in real time.
Bezier Spline Editor
Click to place anchor points and drag control handles to shape smooth cubic Bezier splines. Supports open and closed curves, quadratic and cubic modes, and shows the animated parametric point P(t) traversing the curve. Export the path as an SVG element with one click.
Markov Sentence Forge
Train a Markov chain on any text corpus and generate infinite variations. Choose order 1, 2, or 3 for varying coherence. Word-level or character-level generation. Temperature slider controls creativity. Presets include Alice in Wonderland, Carl Sagan, Shakespeare, haiku, and tech speak. Inspect the top chain transitions live.
Number Base Playground
Live bidirectional converter between binary, octal, decimal, hexadecimal, and any custom base up to 36. Visual bit display with grouping, per-digit positional value expansion, arithmetic in any base with bitwise operations (AND/OR/XOR), and a full ASCII reference table.
Cellular Automata Gallery
Eight automata in one: Wolfram 1D elementary rules (all 256), Conway's Game of Life, Brian's Brain, Seeds, Maze, Anneal, Diamoeba, and HighLife. Each with distinct birth/survival rules producing wildly different emergent behaviors. Adjustable speed and cell size.
Pendulum Wave Sync
N pendulums with carefully tuned frequencies — each making one more oscillation per cycle than the last. All start aligned, then gradually drift into traveling waves, anti-phase patterns, and back to synchrony. Adjustable pendulum count, cycle time, display modes, and motion trails.
L-System Fractal Generator
Lindenmayer systems grow fractal geometry from recursive rewriting rules. Choose from dragon curve, Sierpinski, plant, Hilbert curve, and more — or define your own axiom and production rules. Adjustable iterations, angle, and color depth gradient.
Strange Attractors
Six chaotic dynamical systems — Lorenz, Rössler, Thomas, Dadras, Halvorsen, Aizawa — whose trajectories never repeat yet stay bounded. Choose projection, trail length, color by velocity or time, and tweak parameters live to watch the attractor deform.
Reaction-Diffusion
Gray-Scott model: two chemicals react and diffuse, spontaneously forming Turing patterns — spots, stripes, coral, maze, worms. Click to seed the reaction; presets sweep the parameter space from mitosis to labyrinthine growth.
Abelian Sandpile
Self-organized criticality in action: add grains of sand and watch avalanches cascade when any cell exceeds 3. The system self-tunes to a critical state where individual grains trigger fractal-shaped avalanches of any size.
Penrose Tiling
Aperiodic tiling with perfect 5-fold symmetry using two rhombus tiles. Generated by de Bruijn subdivision of an icosahedral seed — never repeats yet fills the plane. Pan, zoom, and adjust subdivision depth.
Hopf Bifurcation
A fixed point loses stability and births a limit cycle as μ crosses zero. Watch phase portrait, time series, amplitude, and bifurcation diagram update live. Explore Lorenz normal form, Van der Pol, and FitzHugh-Nagumo dynamics.
Continued Fractions
Every real number unfolds into a unique continued fraction [a₀; a₁, a₂, ...] whose convergents are the best rational approximations. Enter any decimal or pick π, φ, √2, e — see the convergent table and log-scale error plot.
Circle Packing
Physics-based relaxation packs circles into circle, square, triangle, or hexagon containers. Watch the density climb toward the Kepler limit of π/√12 ≈ 90.7%. Color by size, index, or contact stress.
Geodesic Dome
Buckminster Fuller's triangulated sphere via icosahedron or octahedron subdivision. Choose frequency 1-6, full sphere or dome cut, color by triangle class or depth. Drag to rotate — watch face counts grow as frequency increases.
Fourier Epicycles
Any closed curve decomposes into a sum of rotating circles via the DFT. Watch epicycles draw hearts, stars, trefoils, and squares — or switch to Draw mode and approximate your own shape. Adjust the number of terms to see convergence.
Scale-Free Network
Grow a Barabási–Albert network via preferential attachment: new nodes connect to hubs with probability proportional to degree. Watch power-law degree distribution emerge, drag nodes, and see how hubs dominate.
Firefly Synchronization
Dozens of fireflies with random natural frequencies couple via a Kuramoto-like rule. Tune the coupling strength and noise to watch spontaneous synchrony emerge — the Mirollo–Strogatz phase transition in real time.
Watts–Strogatz Small World
Rewire a regular ring lattice with probability β. At small β the network gains short paths while keeping high clustering — the small-world regime. Computes average path length and clustering coefficient live.
Conic Sections
Slice a double cone at different angles to produce circles, ellipses, parabolas, and hyperbolas. Explore focus-directrix definitions with eccentricity slider, or switch to 3D cone view to see the cutting plane.
Mechanical Waves & Dispersion
Explore dispersive vs non-dispersive wave propagation: watch a wave packet spread as group and phase velocities diverge, see the dispersion relation ω(k), and feel phonon band structure in a 1D mass-spring chain.
Sandglass Simulation
Cellular automaton granular physics: sand pours through an hourglass neck, forming angle-of-repose piles. Click or button to flip, adjust grain size and friction, watch the arch-clogging instability.
Projective Geometry
Drag points to explore cross-ratio invariance, harmonic conjugates, Desargues' theorem, Pappus' theorem, and Pascal's mystic hexagram — the classical theorems of projective geometry made visual and interactive.
Parabolic Mirror
Trace rays reflecting off a parabolic mirror. Parallel rays converge perfectly at the focus; a point source there reflects into a parallel beam. Compare searchlight, telescope, and off-axis aberration modes.
Power Law Explorer
Generate synthetic power-law data (city populations, word frequencies, earthquakes, wealth) and see how log-log plots reveal the straight-line signature. Toggle linear vs log-log scale to see why heavy-tailed distributions hide in plain sight.
Caustics
Bright curves traced by concentrated reflected light — the shimmering patterns on a coffee cup or pool floor. Reflect rays off circles, ellipses, parabolas, and rippled surfaces; animate the ripple to watch caustics dance.
Mandelbrot & Julia Explorer
Hover over the Mandelbrot set to see its companion Julia set update in real time. Zoom, pin a parameter c, and watch how structure inside vs. outside the Mandelbrot set creates connected vs. Cantor-dust Julia sets.
Graph 4-Coloring
Add nodes, draw edges, and try to color every node so no two neighbors share a color — using at most 4 colors. The Four Color Theorem guarantees it's always possible for planar graphs. Auto-solve or try yourself.
Kepler's Orbital Laws
Animate a planet orbiting the Sun on an adjustable ellipse. Shaded sectors demonstrate equal-area sweeps (Law 2), and live stats verify T²∝a³ (Law 3). Crank eccentricity to near-parabolic orbits.
ODE Solvers: Euler vs RK4
Compare Euler, Midpoint, and Runge-Kutta 4 on classic ODEs — exponential growth, pendulum, logistic, Van der Pol. Increase step size to watch each method diverge from the exact solution at different rates.
Pólya Urn Model
Draw a ball, return it, add more of that color — watch the proportion lock in to a random value. Path dependence in action: early random draws strongly shape long-run outcomes. Each run converges to a different limit.
Zernike Polynomials
Explore the orthonormal wavefront basis of optics: tilt, defocus, astigmatism, coma, trefoil, spherical aberration. Adjust coefficients to build composite wavefronts as used in adaptive optics and eye diagnostics.
Homotopy & Continuous Deformation
Drag the parameter t to watch paths morph continuously into each other. Explore path homotopy, fundamental group π₁, loops that can't be deformed past a hole, and winding numbers in the punctured plane.
Jacobi & Gauss-Seidel Iteration
Solve a 4×4 linear system iteratively. Watch residuals decay on a log plot as Jacobi, Gauss-Seidel, and SOR converge. Edit the matrix entries and compare convergence rates across methods.
Sphere Eversion
Turn a sphere inside-out without tearing — Smale's 1958 impossibility-turned-theorem. Drag through the eversion stages to see self-intersections form and unwind, with inside/outside colors tracking the inversion.
Power Spectrum Analyzer
Build signals from sinusoidal components and compute their FFT in real time. Switch between square waves, chirps, AM modulation, and noise. Compare windowing functions and see leakage reduction in the frequency domain.
Swift-Hohenberg Patterns
Watch stripes, spots, and hexagons spontaneously emerge from noise via the Swift-Hohenberg PDE. Tune the bifurcation parameter ε and nonlinearity to cross the symmetry-breaking threshold and switch between pattern types.
Soliton Collisions (NLS)
Two bright solitons of the focusing nonlinear Schrödinger equation collide and pass through each other unchanged — elastic collision via the split-step Fourier method. Adjust velocities, amplitudes, and nonlinearity g.
Spectral Graph Theory
Explore the Laplacian spectrum of graphs: paths, cycles, random, Petersen, and more. Nodes are colored by the Fiedler eigenvector for spectral bisection. λ₂ measures algebraic connectivity.
Contact Process on Networks
SIS epidemic on random, scale-free, and small-world networks. Watch infections spread and die out around the critical threshold λc = μ/ρ_max. Click nodes to infect or recover them.
Topological Sort Visualizer
Animate Kahn's algorithm step-by-step on DAGs — pipelines, trees, and random dependency graphs. Watch in-degrees count down as nodes enter the queue and get processed in valid topological order.
Galois Field Art
Visualize GF(p) arithmetic as color tables: multiplication, addition, powers, discrete logarithms, and Legendre symbols. Change the prime and color scheme to reveal the symmetry structure of finite fields.
Ramanujan's τ Function
Explore the discriminant modular form Δ(τ) on the upper half-plane alongside Ramanujan's tau function τ(n). See Deligne's bound |τ(p)| ≤ 2p^(11/2) and the action of SL(2,ℤ) on the fundamental domain.
Spherical Harmonics & Sound
Deform a sphere by Y_ℓ^m to visualize quantum angular momentum eigenstates. Drag to rotate; play the resonant tone whose frequency scales as √(ℓ(ℓ+1)). Browse all degrees and orders up to ℓ=5.
Hodge-Helmholtz Decomposition
Decompose any 2D vector field into curl-free (∇φ), divergence-free (∇×A), and harmonic parts via the Hodge theorem. Click to add sources and vortices; watch energy redistribute across components.
Sandpile Music
The abelian sandpile at self-organized criticality, where each toppling cell plays a note. Avalanche sizes follow a power law. Choose a musical scale, drop grains, and hear chaos organize itself into melody.
Airy Functions
Plot Ai(x) and Bi(x), the solutions to y''=xy that arise at quantum turning points. Toggle WKB approximation, mark zeros, and tune the series depth to see convergence vs asymptotic behavior.
Knot Diagrams
Browse animated planar projections of classical knots — trefoil, figure-eight, cinquefoil, torus knots, and the Hopf link. See crossing number, writhe, unknotting number, and toggle rainbow coloring.
Spring Mesh
Click and drag to disturb a 2D spring-mass lattice and watch mechanical waves propagate. Tune stiffness, damping, mass, and impulse strength; toggle fixed boundary conditions to see reflection.
Epidemic Network
Watch an SIR epidemic spread through Erdős–Rényi, Watts–Strogatz, or Barabási–Albert contact networks. Tune β, γ, and R₀; see how network topology determines outbreak size and speed.
Logistic Bifurcation
Explore the full bifurcation diagram of x→rx(1-x), click to set r, and watch the cobweb diagram and time series update. See Feigenbaum's universal constant δ≈4.669 marked on the period-doubling cascade.
Gear Mechanism
Animated involute gear trains — simple pair, compound reduction, planetary gears, and rack-and-pinion. Adjust tooth counts to see speed ratios and torque relationships in real time.
Pendulum Harmonograph
Two decaying pendulums trace intricate Lissajous-family curves. Tune frequency ratios, phase, amplitude, and damping; choose from classic presets or discover your own patterns in static or live drawing mode.
Coupled Oscillators
Two pendulums connected by a spring — explore normal modes, energy beating, and driven resonance. Watch the phase portrait, time series, and energy history as you sweep coupling and driving frequency.
Sierpinski Triangle
Four views of the same fractal: chaos game, recursive subdivision, Pascal's triangle mod 2, and explicit IFS. Hausdorff dimension log₂3 ≈ 1.585 emerges from all four constructions.
Hexagonal Automaton
Cellular automata on a hexagonal grid — hex Life, cyclic, Seeds, snowflake growth, and majority vote rules. Click to paint cells, tune speed, and watch emergent patterns in six-fold symmetry.
Voronoi Diagram
Click to place seed points and watch the plane partition into nearest-neighbor regions. Add random seeds, toggle Delaunay triangulation, or apply Lloyd relaxation to equalize cell areas.
Strange Attractors
Chaotic orbits in 3D that never repeat — choose Lorenz, Rössler, Thomas, or Dadras systems. Drag to rotate the attractor in real time and watch sensitivity to initial conditions in action.
Fourier Series
Rotating epicycles compose square waves, sawtooths, triangles, and pulses. Watch the sum trace the target waveform as you add harmonics — Gibbs phenomenon appears near sharp corners.
Reaction-Diffusion
Gray-Scott model of two chemical species producing spots, stripes, coral, and maze patterns. Click and drag to seed new patterns; switch presets to explore the f-k parameter landscape.
Boids Flocking
Emergent flocking from three local rules — separation, alignment, cohesion. Tune each force independently, add hundreds of agents, and move your cursor as a predator to scatter the flock.
L-System Plants
Lindenmayer systems grow fractal plants, ferns, trees, and dragon curves via recursive string rewriting. Adjust recursion depth and branching angle to morph between organic and geometric forms.
N-Body Gravity
Gravitational choreography between stars and planets — load a solar system, binary star, or galaxy collision scenario. Click to spawn new bodies and watch orbits, captures, and ejections unfold.
Mandelbrot Explorer
Click to zoom deep into the Mandelbrot set — the boundary whose Hausdorff dimension is exactly 2. Scroll-wheel zoom, multiple color palettes, and up to 1000 iterations for fine boundary detail.
Diffusion-Limited Aggregation
Random walkers stick on contact to build a fractal cluster with Hausdorff dimension ≈ 1.71. Tune stickiness, choose color mapping by age or distance, and seed the cluster yourself.
Wave Interference
Place multiple wave sources and watch constructive and destructive interference build standing patterns. Click to add sources, click existing ones to remove them, and sweep wavelength and speed.
Phase Portraits
Click to seed trajectories in the vector fields of pendulums, Van der Pol, Lotka-Volterra, Duffing, and Hopf systems. Fixed points, limit cycles, and separatrices emerge from the flow.
Penrose Tiling
Aperiodic tiling via P3 rhombus deflation — infinitely detailed 5-fold symmetric patterns that never repeat. Increase depth and switch color modes to reveal the quasicrystal structure.
Apollonian Gasket
Circles packed forever by Descartes' theorem — each gap is filled with the unique tangent circle. The residual set has Hausdorff dimension ≈ 1.3057; curvatures are always integers in classic configurations.
Rule 110 Automaton
Elementary cellular automaton proven Turing-complete — flip rule bits to explore all 256 one-dimensional rules, from chaotic Rule 30 to the self-reproducing gliders of Rule 110.
Ulam Spiral
Primes arranged in a square spiral reveal startling diagonal alignments. Animate the sweep and switch color modes to see the mysterious structure hidden in number theory.
Collatz Conjecture
Apply 3n+1 if odd, n/2 if even — does every number reach 1? Visualize individual sequences, overlay hundreds of trajectories, or chart stopping times across a range.
Lissajous Figures
x=sin(at+δ), y=sin(bt) — tune frequency ratios and phase offsets to trace elegant knot-like curves. The 5×5 grid view reveals the full family at a glance.
Spirograph
Hypotrochoids and epitrochoids drawn by a pen on a rolling circle. Tune R, r, and pen distance to produce roses, stars, and looping curves — animated in rainbow gradient.
Chladni Figures
Sand on a vibrating plate migrates to nodal lines where amplitude is zero. Tune the mode numbers m and n to discover the standing wave geometries of cymatics.
Hyperbolic Tiling
The Poincaré disk model of hyperbolic geometry, tiled with {p,q} polygons. Every tile has equal area despite the visual compression near the boundary — infinity packed into a circle.
IFS Chaos Game
Randomly apply affine transforms to converge on fractal attractors: Barnsley fern, Sierpinski triangle, dragon curve, and more. Watch structure emerge from randomness one point at a time.
Elastic Collisions
Perfectly elastic billiard balls conserving energy and momentum. Watch the Maxwell-Boltzmann speed distribution emerge spontaneously — add a heavy ball and observe Newton's cradle dynamics.
Moiré Patterns
Two overlapping periodic grids interfere to create large-scale emergent patterns. Drag to rotate one layer and watch the moiré bands sweep — choose lines, circles, dots, or hexagons.
Sorting Race
Four algorithms compete on the same shuffled array in real time — watch bubble, selection, merge, and quicksort race to the finish and compare operation counts.
Double Pendulum Chaos
Two pendulums started with imperceptibly different angles diverge exponentially — a vivid demonstration of sensitive dependence on initial conditions and deterministic chaos.
Magnetic Pendulum Fractal
A pendulum swings over three magnets and settles near one. Color each launch point by which magnet wins — fractal basins of attraction with infinitely detailed boundaries emerge.
Quantum Wave Packet
A Gaussian wave packet evolves under the Schrödinger equation. Adjust the barrier height and watch tunneling, reflection, and quantum dispersion unfold in real time.
Clifford Attractor
Millions of iterations of a simple trigonometric map trace out a strange attractor. Tune the four parameters to discover entirely different fractal structures hiding in the same equations.
Perlin Noise Terrain
Infinite scrolling landscape generated from fractional Brownian motion — layered Perlin noise octaves create mountains, plains, and oceans with tunable scale and sea level.
Geodesic Dome
An icosahedron subdivided to any frequency projects onto a sphere to form a geodesic structure. Drag to rotate, raise the frequency to approach a perfect sphere — Buckminster Fuller's geometry made interactive.
Laplace Equation Solver
Draw boundary conditions and watch ∇²φ = 0 converge via successive over-relaxation. Field lines trace equipotentials — simulate capacitors, dipoles, or any custom electrostatic boundary value problem.
Lattice Boltzmann Fluid
The D2Q9 Lattice Boltzmann method simulates fluid flow from a mesoscopic collision rule. Draw obstacles and watch vortex shedding emerge — visualized as vorticity, speed, or density.
Rope Physics
Verlet integration with iterative constraint satisfaction simulates a realistic rope. Drag any segment to whip it, pin both ends for a catenary, or let gravity pull it into its natural hanging curve.
Gaussian Primes
Primes in ℤ[i]: complex integers a+bi with a²+b² prime. Color-coded by phase angle or norm, with the unsolved Gaussian Moat Problem — can you walk to infinity on bounded steps?
Fractal Tree
Recursive binary branching with animated wind. Adjust depth, spread angle, and length ratio; switch seasons for changing leaf colors. The same mathematics governs plant morphology and river deltas.
Ackermann Function
A(m,n) grows faster than any primitive recursive function. Trace the deep recursion interactively, inspect the value table, and compare growth curves. A(4,2) has more digits than atoms in the universe.
Modular Arithmetic Clock
Numbers 0…n−1 on a circle, connected to (k×m) mod n. When n is prime and m generates ℤₙ, a single Hamiltonian cycle emerges. Animate through all multipliers and watch the symmetry patterns unfold.
Color Space Explorer
Navigate RGB, HSL, and HSV simultaneously. An interactive hue wheel, 2D cross-sections of each color model, and real-time coordinate translation — see how three spaces map the same perceptual territory.
Trefoil Knot
The simplest non-trivial knot rendered as a 3D tube — drag to rotate. Parametrized by (2+cos 3t)cos 2t, (2+cos 3t)sin 2t, sin 3t. Cannot be untangled without cutting; topologically distinct from a circle.
Neural Activation
Visualize sigmoid, tanh, ReLU, GELU, and Swish activation functions with derivatives. Watch a multi-layer network propagate signals forward — adjust weights, trigger random passes, see activations light up.
Cellular Growth
Extended cellular automata with B/S rule notation — Conway, HighLife, Seeds, Coral, Gnarl, and more. Different birth/survival thresholds produce wildly different morphologies; click the canvas to seed new cells.
L-System 3D
Lindenmayer systems in three dimensions: parallel string rewriting that models plant morphology and fractal geometry. Drag to rotate; adjust iterations to watch branches multiply exponentially.
Fourier Draw
Draw any shape and watch it reconstructed by spinning epicycles. The DFT decomposes your curve into frequency components; each rotating circle represents one harmonic. Adjust how many cycles to use.
Maze Generator
Three classic algorithms — Recursive Backtracker, Prim's, and Kruskal's — generate perfect mazes in real time. Toggle between methods, resize the grid, and solve with BFS to see the shortest path highlighted.
Ising Model
2D Monte Carlo simulation of ferromagnetism on a spin lattice. Near the critical temperature T_c ≈ 2.269 J/k_B, spontaneous magnetization vanishes and critical fluctuations diverge — a continuous phase transition in action.
Hilbert Curve
Space-filling curve with fractal dimension exactly 2. At order n it threads through every cell of a 2ⁿ×2ⁿ grid without crossing itself. Watch it unfold progressively, each level quadrupling the point count.
Riemann Zeta Zeros
ζ(½+it) traced in the complex plane. The Riemann Hypothesis conjectures all non-trivial zeros lie on Re(s)=½. Watch the curve wind through the origin at each known zero — 14.135, 21.022, 25.011…
Pascal's Triangle mod p
By Lucas' theorem, C(m,n) mod p produces self-similar Sierpiński patterns for prime p. Color each residue class independently and explore the fractal structure that emerges from simple arithmetic.
4D Tesseract
A hypercube projected from 4D through 3D to your screen. Sixteen vertices, 32 edges, 8 cubic cells. Drag to rotate in 3D space; sliders control independent XW, YW, ZW rotations in the fourth dimension.
Kepler Orbit Simulator
Newtonian gravity produces elliptical orbits with the Sun at one focus. Watch Kepler's second law — equal areas swept in equal times — illustrated in real time. Add multiple bodies with different eccentricities.
Stereographic Projection
Maps the Riemann sphere onto the plane from the north pole. Circles on the sphere project to circles on the plane — a conformal map that underlies Möbius transformations and complex analysis.
Erdős–Rényi Random Graph
In G(n,p) each edge appears with probability p. A giant connected component emerges abruptly near p_c = 1/n — a percolation threshold. Sweep p and watch the discontinuous transition unfold.
A* Pathfinder
Draw walls on a grid, then watch A* or Dijkstra navigate. A* uses a heuristic to prioritize nodes near the goal, exploring far fewer cells than Dijkstra while still guaranteeing the shortest path.
Fibonacci Phyllotaxis
Golden angle seed packing — nature's most efficient arrangement.
Levy Flight
Heavy-tailed random walks with infinite variance — how albatrosses forage.
Polar Roses
Rhodonea curves r = cos(k theta) — rational k unlocks infinite petal families.
Sieve of Eratosthenes
Watch the ancient prime-finding algorithm mark composites in real time.
Torus Knot
A (p,q)-knot winding around a torus — trefoil, cinquefoil, and beyond.
Heat Diffusion
Paint heat and watch it spread via the 2D heat equation. Interactive.
Cloth Simulation
Verlet integration + constraint relaxation. Click to grab and tear cloth.
Halvorsen Attractor
Cyclic 3D strange attractor with three-fold symmetry and chaotic orbits.
Ant Colony Optimization
Stigmergic pathfinding: ants discover shortest routes via pheromone trails.
Snowflake Growth
Reiter's dendritic solidification model on a hexagonal lattice.
Pinball Physics
Elastic ball collisions, bumpers, flippers, and physics-based pinball simulation.
Langton's Ant
A Turing-complete cellular automaton: three simple rules produce emergent highways and chaos.
Monte Carlo π
Random points reveal π — watch the estimate converge as the circle fills with dots.
Graph Coloring
4-Color Theorem: any planar graph can be colored with just 4 colors. DSatur algorithm live.
Seashell Spiral Generator
Raup's 4-parameter model morphs between nautilus, cone snail, ammonite, and helix.
Conway's Game of Life
Draw cells, customize B/S rules, and watch emergent gliders, oscillators, and chaos.
Traveling Salesman
Nearest-neighbor heuristic plus 2-opt local search — watch the tour shrink in real time.
Binary Search Tree
Insert, search, delete — animated traversals and AVL self-balancing with live rotation.
Riemann Surfaces
Domain coloring of √z, log z, sin z, Newton fractals — hue = argument, brightness = magnitude.
Waveform & Fourier Transform
Draw a waveform and see its frequency spectrum. Select harmonics to reconstruct the signal.
Fluid Pressure & Pascal's Principle
U-tube manometer, hydraulic press, and depth explorer — drag to feel Pascal's law in action.
Grand Explorer ★ Lab 1000
Milestone lab: deep fractal zoom, 7 strange attractors, bifurcation diagrams, automata, and emergent swarms — all in one.
Thomas' Cyclical Attractor
A cyclically symmetric strange attractor with adjustable dissipation — watch chaos bloom in 3D.
Lloyd's Relaxation
Voronoi cells converge to a centroidal tessellation through iterated Lloyd steps — step by step or watch it flow.
Drum Membrane
2D wave equation on a circular drum. Bessel function vibration modes, click to excite.
Voronoi Music
Each Voronoi cell plays a musical note. Click to add seed points, drag to position them — the tessellation updates live and every cell rings its pitch. Five scales, four wave types, reverb.
Gray-Scott Reaction Diffusion
Two chemical species with one feedback loop — ∂u/∂t = Dᵤ∇²u − uv² + f(1−u). Tune feed and kill rates to grow coral, spots, stripes, worms, or mazes. Click to inject reactant and paint new patterns.
Gravity Simulator
N-body Newtonian gravity with softened Plummer potential. Drag to launch bodies with velocity; Shift-drag for heavy stars. Preloaded with figure-8 three-body orbit, binary star, and solar system scenarios.
Quantum Circuit Simulator
Drag-and-drop quantum gates (H, X, Y, Z, CNOT, Measure) onto 3-qubit wires. Real-time statevector updates show amplitude and phase for all 8 basis states — build Bell states, teleportation circuits, and Grover steps.
Hidden Markov Model
Visualize a 3-state HMM with editable transition and emission matrices. Generate observation sequences and watch the Viterbi algorithm decode the most likely hidden state path — color-coded accuracy at each step.
Belief Propagation
Factor graph with a loop — loopy sum-product belief propagation runs step by step. Click variable nodes to clamp observations and watch marginal beliefs converge (or not) as messages flow around the cycle.
VAE Latent Space Explorer
Navigate a 2D latent space learned by a variational autoencoder. Move the cursor or drag sliders to interpolate between digit-like shapes — the decoder continuously renders what lives at each (z₁, z₂) coordinate.
Self-Attention Visualizer
Type tokens, adjust Q/K/V weight matrices by clicking cells, and watch the attention heatmap update live. Temperature controls sharpness; two heads let you compare what each one attends to.
Normalizing Flow
Chain invertible transforms — affine coupling, radial flow, rotation, shear — to morph a Gaussian into multimodal shapes. Watch the log-Jacobian track volume changes and animate the flow step by step.
HP Lattice Protein Folding
Monte Carlo energy minimization on a 2D HP lattice model. H (hydrophobic) residues attract; watch the chain collapse and fold as temperature anneals. Edit the sequence and track the best conformation found.
Auction Mechanism Design
First-price sealed-bid vs. Vickrey (second-price) auction with up to 8 bidders. Compare truthful bidding, bid-shading, and overbidding strategies. Simulate 1000 trials to verify the Revenue Equivalence theorem.
Z₂ Lattice Gauge Theory
Flip Z₂ links on a 2D lattice and watch plaquette Wilson loops respond. Metropolis MC sweeps thermalize the system; scan β to see the confinement phase transition at β_c ≈ 0.44 — the self-dual point.
Topological Defects
XY model spin field with Monte Carlo dynamics. Vortices and anti-vortices appear, annihilate, and bind as temperature crosses the Kosterlitz-Thouless transition at T_KT ≈ 0.89 — watch the defect density diverge.
Abelian Sandpile
Chip-firing on a grid: cells topple when they hold 4+ grains, cascading into avalanches. Drop grains at the center and watch power-law avalanche sizes emerge — self-organized criticality without any tuning.
Gray-Scott Reaction-Diffusion
Two reacting chemicals produce spots, stripes, labyrinthine mazes, and mitosis-like division. Tune feed rate F and kill rate k with sliders to explore the full pattern-formation phase diagram.
Watts-Strogatz Small World
Rewire a ring lattice with probability p and watch average path length L(p) collapse while clustering C(p) stays high — the small-world sweet spot underpinning social and biological networks.
Logistic Map Cobweb
Cobweb diagram for x_{n+1} = r·x_n·(1−x_n). Drag r from 2.5 to 4 to see fixed point → period-2 → period-4 cascade → chaos. Bifurcation diagram and Lyapunov exponent update live.
2D Ising Model at Criticality
Metropolis algorithm on a 64×64 spin lattice. Temperature slider crosses T_c = 2/ln(1+√2) ≈ 2.269 where susceptibility and specific heat diverge — Onsager's exact phase transition in action.
Voter Model on Networks
Each node copies a random neighbor's opinion at each step. Compare consensus time scaling on complete graphs (~N) vs rings (~N²) vs random graphs. Add zealots who never change to prevent full consensus.
Lorenz Attractor (3D)
RK4 integration of the Lorenz equations with σ=10, ρ=28, β=8/3. Drag to rotate the 3D butterfly attractor; launch a second trajectory to see exponential divergence — Lyapunov exponent λ₁ ≈ 0.906.
Contact Process / SIS Epidemic
Infected cells spread at rate λ and recover at rate 1. Cross the phase transition at λ_c ≈ 1.648 (directed percolation universality class) to switch from extinction to endemic survival.
Kuramoto Synchronization
N=50 oscillators with Lorentzian natural frequencies couple via sin(θ_j − θ_i). Raise coupling K past K_c ≈ 1.6 and watch the order parameter r leap from zero to a synchronized state.
Diffusion Limited Aggregation
Particles random-walk until they stick to the growing cluster, producing a fractal with dimension D_f ≈ 1.71. Box-counting measures the dimension live as the DLA cluster branches and grows.
Quantum Harmonic Oscillator
Visualize wavefunctions ψₙ(x) for energy levels n=0..10 using Hermite polynomials. Show probability densities |ψ|², energy level spacing Eₙ=(n+½)ℏω, and animate quantum beating in superpositions.
2D Random Walk
Multiple walkers diffuse simultaneously; plot ⟨r²⟩ ∝ t live. Switch to Lévy flight (power-law step sizes, α=1.5) and watch superdiffusion: ⟨r²⟩ ∝ t^{2/α} grows dramatically faster.
Partition Function Explorer
Interactive thermodynamics: Z=Σe^{−βEᵢ} for discrete energy levels. Watch free energy F=−kT ln Z, average energy ⟨E⟩, entropy S, and heat capacity Cv evolve with temperature.
1D Cellular Automaton (Wolfram)
All 256 elementary CA rules. Rule 30 generates chaos used in Mathematica's RNG; Rule 90 grows a Sierpiński triangle; Rule 110 is Turing-complete; Rule 184 models traffic flow.
Fourier Epicycles
Draw any closed path with your mouse; watch it decomposed into rotating circles (epicycles) via DFT. Adjust the number of Fourier terms to control reconstruction fidelity. Presets: heart, star, trefoil.
KdV Soliton Collision
Numerical integration of the KdV equation u_t+6uu_x+u_xxx=0. Two solitons of different speeds collide and pass through each other completely unchanged — a signature of exact integrability.
Mandelbrot Deep Zoom
Click to zoom into the Mandelbrot set with smooth escape-time coloring. Hover to see the Julia set for parameter c. The boundary has Hausdorff dimension exactly 2 (Shishikura 1998).
Entropy & Maxwell's Demon
Gas particles in two chambers; Maxwell's Demon opens a gate to sort fast from slow particles. Track entropy decrease vs bits of memory erased — Landauer's principle: each erasure costs kT ln2.
Double Pendulum Chaos
RK4 integration of the double pendulum equations. Launch multiple trajectories with tiny initial differences and watch them diverge exponentially. Lyapunov exponent λ estimated live from trajectory separation.
Mean Field Phase Transition
Landau theory F(m)=a(T−Tc)m²+bm⁴−Hm. Watch the free energy double-well develop below Tc, the order parameter m*(T) curve, and susceptibility χ diverge at the critical point.
Shannon Channel Capacity
Interactive C = B·log₂(1+S/N). Sliders for bandwidth and SNR show the fundamental limit on reliable communication. Color-coded capacity surface reveals why doubling bandwidth beats doubling power.
Huffman Coding
Enter any text and watch the optimal prefix-free tree build live. Greedy merging of lowest-frequency nodes, codeword assignment, compression ratio vs ASCII, and Shannon entropy comparison.
Genetic Algorithm — TSP
Evolve a population of tours on 20 cities via tournament selection, order-1 crossover, and swap mutation. Watch fitness improve over generations and compare against the nearest-neighbor heuristic.
Neural Network Learns XOR
A 2-2-1 MLP learns XOR via backpropagation. Watch all weights, gradients, and the decision boundary evolve in real time. Loss curve and per-example predictions update each step.
Conway's Life Pattern Library
Game of Life with a library of famous patterns: Gosper glider gun, LWSS, pulsar, R-pentomino, acorn. Insert patterns, draw cells, step or run. Watch infinite population emerge from the gun.
Kolmogorov Complexity
Approximate K-complexity via LZ78 compression ratio. Compare random, repetitive, structured, and π-digit strings. Most strings are incompressible — the Berry paradox shows K is uncomputable.
Hamming Code Error Correction
Encode 4 data bits into Hamming(7,4). Flip bits to inject errors, then watch syndrome calculation pinpoint the error position and automatically correct it. Compare efficiency with repetition codes.
Turing Machine Simulator
Visual Turing machine with scrolling tape and highlighted transitions. Preloaded programs: increment binary, 3-state busy beaver (writes 6 ones), palindrome checker, unary addition. Step or run.
Information Bottleneck
The IB tradeoff: compress X into T while preserving information about Y. Vary β to trace the IB curve in the information plane I(X;T) vs I(T;Y). Blahut-Arimoto iterations, cluster assignment matrix.
Ant Colony Optimization
ACO Ant System solving TSP on 30 cities. Pheromone trails strengthen good paths, evaporation forgets bad ones. Watch edge intensities glow as ants converge. Compare against nearest-neighbor baseline.
Poincaré Disk Model
Hyperbolic geometry in the Poincaré disk. Draw geodesics (circles orthogonal to boundary), place triangles where angle sum is less than π, and measure hyperbolic distances. Toggle a {7,3} heptagonal tessellation.
Prime Number Spiral (Ulam)
Ulam spiral: integers arranged in a square spiral with primes highlighted, revealing striking diagonal alignments. Switch to the Sacks spiral — primes cluster on arcs of prime-rich polynomials.
Continued Fractions & Golden Ratio
Enter any number to see its continued fraction expansion [a₀; a₁, a₂, ...]. Watch convergents home in on the target. The golden ratio φ = [1;1,1,1,...] is the most irrational number. Explore the Stern-Brocot tree.
Knot Invariants Explorer
Browse knot diagrams — unknot, trefoil, figure-eight, cinquefoil, and the granny knot. See crossing number, unknotting number, Seifert genus, and Jones polynomial V(t). Reidemeister moves illustrated.
Projective Geometry & Pascal's Theorem
Pascal's mystic hexagram: drag 6 points on an ellipse and watch the three pairs of opposite sides always meet on a line. Also: conic sections as cone slices and vanishing points in perspective.
Kepler Conjecture & Circle Packing
Compare square lattice packing (π/4 ≈ 78.5%) versus hexagonal packing (π/√12 ≈ 90.7%) — the densest possible, proved by Hales in 1998 after 387 years. Animate random packing and measure density live.
Geodesics on Curved Surfaces
Trace geodesics on a sphere (great circles), torus ((p,q)-curves that are closed or space-filling), and hyperboloid (ruling lines that are simultaneously straight and geodesic). Drag to rotate.
Penrose Aperiodic Tiling
Generate Penrose P3 rhombus tilings via Robinson triangle deflation at up to 7 levels. Never periodic, yet 5-fold symmetric — the mathematical basis for quasicrystals (Nobel Prize 2011).
Four Color Theorem
Color planar maps with at most 4 colors so no adjacent regions share a color. Click regions to paint, run greedy auto-coloring, and see conflict detection live. Choose from region, grid, or Voronoi maps.
Riemann Surfaces & Branch Cuts
Domain coloring of √z, log z, z^(1/3), z², and sin z. Hue encodes argument, brightness encodes magnitude. Branch cuts appear as discontinuities. Animate a loop around the branch point to see sheet-switching.
Evolutionary Fitness Landscape
Wright's NK model: evolve a population on a rugged genotype fitness landscape. Tune N (genome length) and K (epistasis) to go from smooth single-peak to rugged multi-peak landscapes. Compare natural selection vs random drift.
Boids Flocking Simulation
Reynolds' 1987 emergent flocking: separation, alignment, and cohesion rules create realistic flock behavior from purely local interactions. Add predators that flocks evade, toggle trails, and tune each rule weight live.
Hodgkin-Huxley Neuron
Numerically solve the 1952 Nobel Prize equations for action potential firing: C_m·dV/dt = I − g_Na·m³h(V−E_Na) − g_K·n⁴(V−E_K) − g_L(V−E_L). Watch threshold behavior, refractory period, and repetitive firing.
Hardy-Weinberg & Genetic Drift
Wright-Fisher model of allele frequency evolution. Infinite N gives Hardy-Weinberg equilibrium; finite N shows random fixation and loss. Compare N=10, 100, 1000 simultaneously. Add selection and mutation.
Lotka-Volterra Predator-Prey
Phase portrait of dx/dt = αx − βxy, dy/dt = δxy − γy. Tune α, β, γ, δ and watch closed orbits in phase space. Add stochastic noise, logistic prey growth, and multiple initial conditions to explore the basin structure.
Membrane Diffusion & Osmosis
Particle simulation of Fick's Law diffusion across a semipermeable membrane with adjustable channel proteins. Watch osmotic equilibration, tune diffusion coefficient and temperature, and see the Goldman equation membrane potential.
Red Queen & Coevolution
Host-parasite coevolution under matching-alleles, gene-for-gene, and quantitative arms-race models. See Red Queen oscillations — allele frequencies cycle endlessly as each species adapts to the other's current most common genotype.
Neural Spike Train Statistics
Model spike trains as Poisson processes or leaky integrate-and-fire neurons. Visualize raster plots for 20 neurons, compute inter-spike interval histograms, Fano factor, and CV. Compare regular, irregular, and burst firing patterns.
HP Protein Folding Model
Lau-Dill hydrophobic-polar model on a 2D lattice. Edit an H/P sequence and watch Monte Carlo search for the minimum-energy conformation. Energy = non-adjacent H-H contacts. Observe the folding funnel and energy landscape.
Trophic Cascade
Yellowstone-inspired food web: remove wolves → coyote explosion → elk boom → vegetation collapse. Restore the apex predator and watch the system recover. Four-trophic-level ODE model with biomass time series.
Vocal Formants
F1–F2 vowel space explorer with real-time synthesis. Click the formant plot to morph between vowels, hear the resonances, and see how the vocal tract shapes speech sounds.
Cochlear Traveling Wave
The basilar membrane inside the cochlea acts as a biological Fourier analyzer. Watch the traveling wave peak at the characteristic frequency place — a tonotopic map from 20 Hz to 20 kHz.
Acoustic Impedance
When sound crosses a boundary between media, reflection and transmission depend on the impedance mismatch. Explore why 99.9% of sound bounces off water, and how ultrasound gel solves this.
Binaural Beats
Feed a slightly different frequency to each ear and the brainstem perceives a phantom beat at the difference frequency. Explore delta, theta, alpha, beta, and gamma brainwave bands — headphones required.
Pipe Organ Physics
Open and closed pipes support different harmonic series. Visualize pressure standing waves inside the pipe, play individual modes, and use the mini-keyboard to hear how pipe length determines pitch.
FM Synthesis
Frequency modulation synthesis: a carrier modulated by a modulator creates rich sideband spectra via Bessel functions. Explore the DX7 sounds that defined 1980s music — bells, brass, pianos, and metallic clangs.
Sonic Crystal
A periodic array of cylinders creates a phononic crystal with acoustic bandgaps — frequencies where sound cannot propagate by Bragg scattering. Watch waves pass or decay depending on whether the frequency falls in the gap.
Comb Filter
Adding a delayed copy of a signal to itself creates periodic peaks and notches in the frequency response — like teeth on a comb. Explore feedforward and feedback variants, flanging, and the Karplus-Strong connection.
Acoustic Radiation Pressure
Intense sound waves exert a steady force on particles — the basis of acoustic levitation and tweezers. Gor'kov potential landscapes show trap locations; drag particles and watch them snap back to pressure nodes.
ADSR Envelope
Attack, Decay, Sustain, Release — the four parameters that sculpt how synthesizer notes evolve. Drag the envelope handles, play a piano keyboard, and hear how ADSR transforms any waveform's character.
PageRank Random Walk
A random surfer walks the web. PageRank is the stationary distribution — watch empirical visit frequencies converge to the true values as you control the damping factor and teleportation probability.
Minimum Spanning Tree
Kruskal's and Prim's algorithms animated step by step — each decision explained. Watch how sorting edges by weight and greedily avoiding cycles builds the minimum-cost connected subgraph.
Network Centrality
Degree, betweenness, closeness, and eigenvector centrality computed and compared on the same graph. Node size encodes importance — drag nodes and see how position affects every measure.
Community Detection
Louvain modularity optimization finds communities by maximizing intra-group edges minus the random expectation. Adjust resolution γ to find coarse or fine-grained structure.
Cascade Failure
Power-grid style cascade: overload one node, watch failure propagate as load redistributes to survivors. Threshold controls resilience — see how a single failure can collapse the grid.
Network Motifs
Triangles, feed-forward loops, and mutual dyads — the recurring subgraph patterns that appear far more often than chance in real networks. Compare counts to random graph baselines.
Link Prediction
Which missing edges are most likely to form? Common Neighbors, Jaccard, Adamic-Adar, and Preferential Attachment scores rank candidate links — reveal the top prediction and watch the graph evolve.
Graph Laplacian & Spectral Partitioning
L = D − A encodes graph structure. The Fiedler vector (second-smallest eigenvector) gives the optimal graph bisection — spectral clustering visualized with eigenvalue spectrum display.
Max Flow / Min Cut
Edmonds-Karp algorithm finds augmenting paths via BFS. Watch flow fill each edge and the min-cut emerge — the max-flow min-cut theorem made visible, with bottleneck annotations at each step.
Hypergraph Synchronization
Kuramoto oscillators coupled through hyperedges — groups of 3+ nodes interacting simultaneously. Higher-order coupling enables abrupt synchronization transitions impossible with pairwise interactions alone.
Network Robustness
Scale-free networks survive random failures but collapse under targeted hub attacks. Remove nodes and watch the giant component shrink — the robustness curve plots the stark difference.
Brewster's Angle
Fresnel equations for s- and p-polarized light at a dielectric interface. At Brewster's angle, reflected light is perfectly s-polarized — Rₚ vanishes exactly. Interactive charts show reflectance vs angle for any n₁/n₂.
Chromatic Aberration
Wavelength-dependent refraction causes color fringing in lenses — quantified by the Abbe number V = (nd−1)/(nF−nC). Explore axial and lateral CA, spot diagrams for each wavelength, and how finesse V determines severity.
Fabry-Pérot Etalon
Resonant optical cavity with multiple-beam interference — sharp Airy transmission peaks separated by the free spectral range. Tune reflectivity, spacing, and tilt angle; watch Haidinger fringes form in real time.
Optical Tweezers
A focused laser beam traps dielectric particles via gradient force — the same technique that won the 2018 Nobel Prize. Move the beam to manipulate particles; Brownian fluctuations scale with temperature via ⟨Δx²⟩ = kT/κ.
Rainbow Formation
Descartes' 1637 geometric optics of raindrops: the minimum deviation angle D_min gives the rainbow at ~42°. Trace rays through a single drop, fan multiple wavelengths, and see the dispersion curve D(b) with its stationary minimum.
Eikonal & Wavefronts
The eikonal equation |∇S|² = n² governs ray optics in inhomogeneous media. Huygens construction propagates wavefronts from point, plane, and slit sources through gradient-index media and lens-shaped regions.
Birefringence
Crystal optics: ordinary and extraordinary rays travel at different speeds, accumulating phase retardation Γ = 2π·Δn·d/λ. Jones matrix formalism tracks polarization through wave plates on the Poincaré sphere.
Aperture Diffraction
Fraunhofer diffraction patterns are the 2D Fourier transform of the aperture function. Compare circles, rectangles, slits, hexagons, annular rings, and stars — each shape stamps a unique signature in the far field.
Dispersion & Prism
White light decomposed by wavelength-dependent index via the Sellmeier equation. Trace colored rays through BK7, SF11, and FK51 glass prisms; plot n(λ) curves and watch angular spread scale with Abbe number.
Aberration of Starlight
Bradley's 1728 discovery: Earth's 30 km/s orbital speed shifts apparent star positions by up to 20.5 arcseconds. Relativistic vector aberration tilts the telescope axis — stars trace annual ellipses across the sky.
Wavefront Aberrations
Zernike polynomial decomposition of optical aberrations — defocus, astigmatism, coma, spherical, trefoil. Tune each coefficient and watch the pupil function phase map and PSF update live; Strehl ratio tracks wavefront quality.
Von Kármán Vortex Street
Lattice-Boltzmann D2Q9 simulation of alternating vortex shedding behind a cylinder. Tune Reynolds number and watch the wake transition from steady to oscillating — Strouhal number St ≈ 0.2 throughout.
Hele-Shaw Viscous Fingering
Saffman-Taylor instability: air injected into viscous oil forms fractal fingers. Darcy's law drives the interface; surface tension B = γb²/12μU controls fingering — lower surface tension yields more branches.
Rayleigh-Bénard Convection
Fluid heated from below, cooled from top forms convection rolls at Ra > Ra_c = 1708. Boussinesq approximation simulation with live Nusselt number tracking convective heat enhancement.
Kelvin-Helmholtz Instability
Shear between two fluid layers rolls up into billowing vortices — seen in Jupiter's clouds and ocean thermoclines. Growth rate σ = k·ΔU/2; unstable when Richardson number Ri < 1/4.
Stokes Creeping Flow
Exact analytic solution for Re → 0 flow past a sphere: ψ = Ur²sin²θ(1 − 3R/2r + R³/2r³). Stokes drag F = 6πμRU. Fully reversible, time-symmetric — streamlines envelope the sphere with no wake.
Bernoulli & Venturi Effect
P + ½ρv² = const along a streamline. Continuity A₁v₁ = A₂v₂: flow through a constriction accelerates and pressure drops. Visualize pressure, velocity, and dynamic pressure fields in a Venturi tube.
Marangoni Flow
Surface tension varies with temperature (dσ/dT < 0), driving flow from hot to cold regions — the "tears of wine" effect. Draw heat sources; watch Marangoni stress drive surface currents away from hot spots.
Turbulent Energy Cascade
Kolmogorov 1941: energy injected at large scales cascades to small scales with E(k) ∝ k^(−5/3) in the inertial subrange. Live spectrum plot shows the −5/3 slope; vorticity visualization reveals the multi-scale structure.
Taylor-Couette Flow
Viscous flow between rotating cylinders. Exact Couette solution v_θ = Ar + B/r becomes unstable at Taylor number Ta_c ≈ 1712, forming toroidal Taylor vortices, then wavy vortices, then turbulence.
Soap Film Optics
Thin-film interference: path difference Δ = 2nh·cosθ creates iridescent colors as the film drains. Gravity, Marangoni flow, and evaporation thin the film from top — color sequence traces Newton's ring order.
Möbius Strip
One-sided, one-boundary surface: x=(1+v·cos(u/2))cos(u). Non-orientable — an ant traverses both "sides" without crossing an edge. Cut along the centerline; twist controls; ant path animation.
Hyperbolic Plane
Poincaré disk model of H²: geodesics are circular arcs orthogonal to the boundary. Explore {p,q} tilings — {7,3} heptagonal, {5,4} pentagonal — where infinitely many congruent polygons tile the disk.
Gaussian Curvature
K = κ₁·κ₂, the product of principal curvatures. Live heatmap on torus, monkey saddle, and bumpy sphere. Theorema Egregium: K is intrinsic — unchanged by bending without stretching.
Parallel Transport
A vector parallel-transported around a latitude circle on S² acquires holonomy angle α = 2π·sin(θ) = solid angle enclosed. Geometric (Berry) phase: intrinsic to the curved geometry, not the path speed.
Hopf Fibration
π: S³ → S² — each point on S² lifts to a circle (fiber) in S³. Fibers over a latitude circle form a torus. Stereographic projection to ℝ³ reveals Villarceau circles. Linking number between any two fibers = 1.
Torus Topology
(p,q)-torus curves wind p times toroidally and q times poloidally. When gcd(p,q)=1: a knot. When gcd>1: that many linked components. Fundamental group π₁(T²) = ℤ×ℤ generated by these two loops.
Conformal Maps
Analytic maps f: ℂ→ℂ preserve angles where f′(z)≠0. Side-by-side domain/image grid shows z², eᶻ, Joukowski z+1/z, log z, and Möbius transforms deforming grids conformally.
Geodesics on S²
Great circles are the geodesics of S²: shortest paths, intersections with planes through the origin. Fan from a point or random great circles. Distance d(p,q) = arccos(p·q) — the central angle.
Linking Number
Gauss linking integral lk(α,β) = (1/4π)∮∮(α−β)·(α′×β′)/|α−β|³ is a topological integer. Hopf link (lk=1), unlinked (lk=0), and higher linking configurations with animated 3D curves.
Klein Bottle
Non-orientable closed surface with χ=0, no boundary, and no inside/outside. Cannot embed in ℝ³ without self-intersection. Figure-8 immersion rendered in 3D with transparent faces.
Ising Model
2D Ising model with Metropolis algorithm. Watch ferromagnetic domains form below T_c ≈ 2.269 J/k. Live magnetization history, temperature sweep through the continuous phase transition.
2D Percolation
Site and bond percolation on a square lattice. Cluster labeling via union-find. Giant spanning cluster appears at p_c ≈ 0.5927 — a geometric second-order phase transition with fractal clusters.
2D Random Walk
Multiple walkers, glowing trails, live MSD ⟨r²⟩ = 4Dt plot versus theory. Einstein's 1905 insight: diffusion arises from random molecular kicks. Adjust walkers, step size, and speed.
Entropy & Gas Mixing
Two ideal gases (blue & red) mix after barrier removal. Watch entropy S = k ln Ω rise toward maximum. Boltzmann H-theorem and the arrow of time from microscopic reversibility.
Van der Waals Phase Diagram
P-V isotherms of the Van der Waals equation. Maxwell equal-area construction marks the liquid-gas coexistence line. Spinodal region highlighted. Sweep through the critical temperature.
Brownian Motion
Langevin equation simulation: m·ẍ = -γẋ + noise. MSD grows as 4Dt with D = kT/γ (Einstein relation). Fluctuation-dissipation theorem connecting thermal noise to viscous drag.
q-State Potts Model
Generalization of Ising to q spin states. For q ≤ 4: continuous transition. For q > 4: first-order (discontinuous) with latent heat. Colorful domain coarsening with Metropolis dynamics.
Voter Model
Agents copy a random neighbor's opinion each step. Coarsening to consensus on 2D grids (time ∝ N² ln N). Add zealots for a non-trivial stationary state. Switchable topology.
Schelling Segregation
Agents move if too few neighbors share their type. Even mild preference (33%) produces strong macro-segregation — a tipping-point phase transition from individual tolerance to collective separation.
Block-Spin Renormalization
Kadanoff block-spin RG on the 2D Ising model. Coarse-grain by majority rule at multiple scales. At T_c, the system is scale-invariant — a visual fixed point of the RG flow.
FPUT Recurrence
Fermi-Pasta-Ulam-Tsingou: a chain of nonlinear oscillators that should thermalize but instead almost-periodically returns to its initial state — the mystery that launched nonlinear dynamics. Watch energy mode amplitudes over time.
Calkin-Wilf Tree
An infinite binary tree that enumerates every positive rational exactly once. Each node p/q has children p/(p+q) and (p+q)/q. Click nodes to trace paths and find Stern-Brocot equivalents — a bijection ℕ → ℚ⁺.
Diamond-Square Terrain
Fractal heightmap generation via iterative midpoint displacement. Roughness exponent H controls how fast noise decreases — H=1 is smooth, H=0 is rough. Fractal dimension D = 3−H. Watch the terrain emerge step by step.
Hydrogen Wavefunctions
Visualize |ψ_{n,l,m}|² as a 2D probability density cross-section using real radial and spherical harmonic formulas. Controls for quantum numbers n (1–4), l, m. Count nodes, see orbital shapes, track energy levels.
Phonon Dispersion
1D diatomic crystal: two alternating masses M and m on springs. Shows acoustic and optical branches of ω(k). Click any k-point to see actual atomic motions animated. Controls: mass ratio, spring constant.
Tight-Binding Band Structure
Electrons hopping between lattice sites: E(k) = ε₀ − 2t·cos(ka) in 1D, with Fermi surface and filling. Switch to 2D square lattice for the full band colormap — half-filling gives a nested square Fermi surface.
Grover's Quantum Search
Amplitude amplification for N=8–64 items: oracle inverts target amplitude, diffusion operator boosts it. Watch probability concentrate on the target in ⌊π√N/4⌋ iterations — a provable quadratic speedup over classical O(N).
XY Model — BKT Transition
2D spins rotating freely, colored by angle. Below T_BKT≈0.89 vortex pairs are bound (quasi-long-range order); above they unbind and proliferate. A topological phase transition with no local order parameter — Nobel Prize 2016.
Active Matter / Vicsek Model
Self-propelled particles align with neighbors within radius r plus noise η. At low noise, collective flocking emerges — a non-equilibrium order-disorder transition. Track the order parameter as noise increases past the critical point.
Electron Drift & Hall Effect
Electrons in a 2D conductor under E and B fields: Lorentz force curves trajectories, building a transverse Hall voltage. Displays drift velocity, Hall angle, cyclotron frequency, and R_H = 1/(ne) as you vary field strengths.
Rankine Vortex
Fluid vortex with solid-body rotation inside radius R and irrotational flow outside. Watch streamlines, velocity profile v(r), and vorticity distribution. Adjust core size and strength to see pressure field and free-surface dip.
Tent Map & Chaos
Piecewise linear chaos: x→μ·min(x,1-x). At μ=2 the map is topologically conjugate to the shift on binary sequences — maximally chaotic. Watch orbit diagrams, sensitive dependence, and the Lyapunov exponent log(μ).
Hamiltonian Cycles
Find Hamiltonian cycles on graphs using backtracking and random restarts. Visualize the search process, path extension, and backtracking live. Vary graph density to see how quickly solutions appear near the phase transition.
Circle Inversion Fractals
Möbius transformations via circle inversion: z→r²/z̄. Repeatedly invert a set of circles to generate Apollonian gaskets and limit sets of Schottky groups. Adjust circle radii and positions to sculpt intricate fractal foams.
Cahn-Hilliard Phase Separation
Spinodal decomposition: ∂φ/∂t = ∇²(φ³−φ−κ∇²φ). A conserved order parameter separates into two phases via coarsening — domains grow as t^(1/3). Watch the characteristic bicontinuous morphology emerge and coarsen.
Chirikov Standard Map
Area-preserving twist map: p→p+K·sin(θ), θ→θ+p. At K=0 invariant tori fill phase space; as K increases KAM tori break, islands form, and chaos floods the phase portrait. The canonical model of Hamiltonian chaos.
Vector Field Explorer
Explore 2D autonomous ODE systems with nullclines, fixed points, and integrated trajectories (RK4). Choose from saddle nodes, spirals, Van der Pol, Lotka-Volterra, Duffing, FitzHugh-Nagumo, and homoclinic orbits.
Langevin Dynamics
Brownian particles in energy landscapes driven by mẍ = −∇V − γẋ + ξ(t). Watch thermal fluctuations kick particles over barriers, building up the Boltzmann distribution P∝e^{−V/kT}. Explore double wells, washboards, and rugged landscapes.
Anosov Cat Map (3D)
Arnold's cat map on the 2-torus: (x,y)→(2x+y, x+y) mod 1 — hyperbolic automorphism with eigenvalues (3±√5)/2. Watch images shred exponentially, explore periodic orbits, and see the mixing in 3D phase-space representation.
Fokker-Planck Equation
Evolution of probability distributions under drift and diffusion: ∂P/∂t = −∂(AP)/∂x + D∂²P/∂x². Watch distributions spread, drift, and relax to Boltzmann equilibrium in a potential well. Visualize the probability current.
Optical Caustics
Light rays refracted by a wavy surface concentrate into bright caustic curves — singularities of the ray map. The fold (A₂) and cusp (A₃) catastrophes appear as stable caustic shapes. Adjust wave amplitude and lens shape.
Bell Inequality Explorer
Test quantum entanglement vs. classical hidden variables — run CHSH experiments and watch the S parameter exceed the classical bound of 2, reaching the Tsirelson limit 2√2 ≈ 2.828.
Catalan Ballot Paths
Lattice paths from (0,0) to (n,n) that never cross above the diagonal — the Catalan numbers in action. Animate random valid paths, display all C_n Dyck paths, and explore the ballot problem combinatorially.
Floquet Parametric Resonance
A pendulum with a vibrating pivot obeys the Mathieu equation. Explore the Ince-Strutt stability diagram and watch how periodic driving near 2ω₀/n causes exponential amplitude growth — parametric resonance.
Bessel Drumhead Modes
Animated normal modes of a circular membrane: u_{mn}(r,θ,t) = J_m(j_{mn}r)cos(mθ)cos(ωt). Select any (m,n) mode, watch Bessel functions create nodal circles and diameters, visualized as a 3D surface.
Contact Angle & Wetting
Young's equation cos θ = (γ_SG − γ_SL)/γ_LG determines whether a liquid spreads or beads up. Tune surface energies, add Wenzel roughness, and watch the droplet cross-section morph from superhydrophilic to lotus-like.
Monodromy & Analytic Continuation
Place branch points in the complex plane for √z, ∛z, or log z, then animate a loop around them to watch analytic continuation permute Riemann sheets — the monodromy group visualized geometrically.
Stochastic Differential Equations
Simulate geometric Brownian motion, Ornstein-Uhlenbeck, CIR, and custom SDEs with Euler-Maruyama. Watch many sample paths diverge or mean-revert, and see the evolving probability distribution in real time.
Saffman-Taylor Viscous Fingering
A less viscous fluid displacing a viscous one in a Hele-Shaw cell grows unstable fingers via a Laplacian instability. Linearized Floquet-mode analysis with tunable viscosity ratio and surface tension.
Sphere Tessellation
Explore geodesic icosphere subdivisions, UV grids, octahedral meshes, and cube spheres — drag to rotate in 3D. Track Euler characteristic V−E+F=2 and compare face-area uniformity across tessellation types.
Ricci Flow on Surfaces
Hamilton's Ricci flow ∂g/∂t = −2Ric smooths Riemannian metrics toward constant curvature. Watch curvature distributions evolve and homogenize numerically, verifying the uniformization theorem in 2D.
Recurrence Plot
Visualize when a dynamical system returns near its past states — revealing hidden temporal structure, determinism, and chaos through the geometry of recurrence matrices.
Delay Embedding (Takens)
Reconstruct a strange attractor from a single scalar time series using Takens' 1981 theorem — one observable contains the full attractor topology.
Baker's Map
The prototypical chaotic map: stretch, compress, and fold the unit square. Watch smooth initial distributions develop fractal microstructure under iteration.
Smale Horseshoe
Smale's 1960 geometric construction of chaos — step through the stretch-and-fold mechanism that creates a Cantor-set invariant set with full symbolic dynamics on {0,1}^ℤ.
Poincaré Return Map
Reduce a continuous flow to a discrete map by recording where trajectories cross a hyperplane — revealing chaos, periodicity, and attractor structure.
Multifractal Spectrum
Explore the singularity spectrum f(α) — how a fractal measure's local Hölder exponent varies across space, described by the Legendre transform of the mass exponent τ(q).
Bernoulli Shift
The simplest chaotic system: shift a binary sequence left by one. Visualize how the doubling map reads binary digits, and why sensitive dependence on initial conditions is unavoidable.
Natural Invariant Measure (SRB)
Compute the SRB measure empirically — the fraction of time a generic orbit spends in each region. Compare with exact formulas like the arcsine law for the logistic map at r=4.
Correlation Dimension (GP)
Estimate the fractal dimension of a strange attractor from a time series using the Grassberger-Procaccia algorithm — the slope of the log-log correlation integral C(r) ~ r^D₂.
Symbolic Dynamics
Encode orbits as sequences of symbols (L/R) determined by which interval is visited at each step. Explore the kneading invariant, topological entropy, and grammar complexity.
Prime Rain
Watch numbers fall as colored columns — primes glow as a single hue, composites reveal their factor structure as stacked color bands. Each number's arithmetic fingerprint made visible.
Newton Basin of Attraction
Run Newton's method for complex polynomials — color each starting point by which root it converges to. The fractal boundary between convergence basins is infinitely intricate.
DNA Random Walk
Map DNA sequences to 2D walks — each nucleotide steers a direction. Compare genomes visually by their geometric fingerprints; hidden periodicities emerge as spirals and loops.
Newtonian Gravity Field
Place massive bodies and watch test particles trace orbits — slingshots, captures, figure-8 orbits, and gravitational chaos. Click to add bodies, drag to set velocity.
Tensor Visualizer
Watch how 2×2 matrices deform space — stretch, shear, rotate, or reflect a vector field. Eigenvectors glow in gold: the special directions that only scale, never rotate.
Autocorrelation Explorer
Draw any signal and watch its autocorrelation and power spectrum update live. The Wiener-Khinchin theorem says they are Fourier transform pairs — discover hidden periodicities.
Platonic Solid Nets
Drag to rotate the five Platonic solids in 3D and see their unfolded flat nets. Euler's formula V−E+F=2 holds for all of them — a topological miracle.
Schwarzschild Orbits
Trace light and matter geodesics around a black hole. See orbital precession, the photon sphere at r=3M, and light capture — all from the exact Schwarzschild metric.
Cellular Noise Explorer
Worley noise: color each pixel by its distance to the nearest random seed point. F1, F2, and F2−F1 modes create cells, foam, cracked mud, and stone textures.
Wireworld Computer
Wireworld simulates electron flow with 4 simple cell-state rules — capable of universal computation. Build AND gates, OR gates, clocks, and flip-flops from first principles.
Sylvester–Gallai Theorem
Click to place points and watch ordinary lines (through exactly two points) appear in gold. Any non-collinear point set must have at least one — Kelly's elegant 1948 proof in action.
Weyl Equidistribution
The sequence {nα} mod 1 fills [0,1) uniformly when α is irrational. Compare golden ratio vs rational fractions — see period-2 orbits vs dense Weyl sequences in real time.
Hypocycloid Roller
Watch a small circle roll inside a larger one, tracing hypocycloids: deltoid (k=3), astroid (k=4), and exotic quasi-periodic curves for irrational radius ratios.
Prime Zeta Function
P(s) = Σ p⁻ˢ over all primes — visualised on the real axis and as a complex heatmap. P(1)=∞ (diverges!), P(2)≈0.4522. Watch partial sums converge, prime by prime.
Apollonius Sphere Packing
Four mutually tangent spheres seed a recursive 3D packing via Descartes' Circle Theorem. Drag to rotate. The fractal dimension is ≈2.47 — every interstice filled forever.
Schrödinger's Cat State
Quantum superposition of two coherent states |α⟩+|−α⟩ shown via the Wigner function — negative values (blue) prove quantum coherence. Dial up decoherence and watch the fringes vanish.
Ford Circles & Spheres
For each fraction p/q in lowest terms, a circle of radius 1/(2q²) sits tangent to the x-axis. Two circles touch iff they're Farey neighbours. A number-theoretic fractal landscape.
Origami Waterbomb Tessellation
Rigid origami with one degree of freedom: drag the fold angle and watch the flat sheet corrugate into 3D. Used in deployable solar panels and medical stents.
Catmull–Clark Subdivision
The algorithm behind Pixar: each subdivision step inserts face, edge, and vertex points to smooth a coarse mesh toward a C² bicubic B-spline surface. Drag to rotate.
Ballot Problem & Catalan Paths
Lattice paths from (0,0) to (n,n) that never cross the diagonal are counted by Catalan numbers C(n). The reflection principle bijects bad paths to those from (−1,1) — a stunning geometric proof.
Cardiac Action Potential
Voltage-gated ion channel dynamics in a cardiac muscle cell. Tune Na⁺, K⁺, and Ca²⁺ conductances to explore the 5 phases of the cardiac AP — upstroke, plateau, repolarization. The long plateau (~200 ms) prevents tetanic contraction.
Double-Scroll Attractor
Chua's circuit — the simplest autonomous electronic circuit that produces chaos. A piecewise-linear nonlinear element drives trajectories to wind around two unstable equilibria, never repeating. Switch between x–z, x–y, y–z, and rotating 3D projections.
Kauffman NK Boolean Network
N binary genes each regulated by K random Boolean functions, updated synchronously. At K=1 networks freeze; at K=2 (the "edge of chaos") attractors cycle with length ≈√N; at K≥3 chaos reigns. Watch attractor detection and Hamming distance divergence live.
3D Ising Model
Ferromagnetic spins on a cubic lattice via Metropolis Monte Carlo. Red=up, blue=down. Near T_c≈4.51 J/k_B the correlation length diverges and domains of all scales appear — the fingerprint of a second-order phase transition. Track magnetization and energy live.
Jarzynski Equality Demo
Drag a particle through a potential and watch the work distribution converge to e^{−βΔF}. The Jarzynski equality recovers free energy differences from non-equilibrium paths — fast or slow — via exponential averaging.
Frenzy Automaton
Multi-state totalistic cellular automaton: a cell advances when its neighbor count exceeds a threshold. Adjust states, threshold, and neighborhood to create spiraling waves, turbulence, and excitable medium dynamics.
Lagrangian Mechanics
Double pendulum solved via Euler-Lagrange equations with RK4 integration. Watch chaotic trajectories trace out sensitive-dependence-on-initial-conditions while the generalized coordinates T, V, and Lagrangian L=T−V update live.
Diffusion Equation Explorer
Interactive 1D and 2D heat equation solver. Paint hot and cold regions, choose Neumann, Dirichlet, or periodic boundary conditions, and watch thermal energy spread according to ∂u/∂t = D∇²u.
Random Walks in 1D, 2D, 3D
Visualize random walks across dimensions. All show ⟨r²⟩ ∝ t scaling, yet 1D and 2D walkers return to origin almost surely while 3D walkers escape with ~66% probability — Pólya's 1921 recurrence theorem in action.
Navier-Stokes Fluid
Real-time fluid simulation via Jos Stam's stable fluids method. Mouse over to inject velocity and dye; toggle vorticity or velocity vector views. Adjust viscosity and diffusion to explore laminar vs turbulent flow regimes.
Schrödinger Potential Well
Quantum eigenstates solved by shooting method for square, harmonic, double, and Morse wells. Visualize wavefunctions, energy levels, and time evolution. Watch superposition states slosh as e^{−iEt/ℏ} phases evolve.
Berry Phase
Geometric phase on the Bloch sphere: as a spin-½ state traces a loop in parameter space, it acquires a phase γ = −Ω/2 equal to half the subtended solid angle — independent of traversal speed, purely geometric.
Knot Theory Explorer
Rotate and explore 3D knot and link embeddings: unknot, trefoil, figure-8, cinquefoil, Hopf link, and Borromean rings. Each shows crossing number, Jones polynomial, Alexander polynomial, and genus. Drag to rotate.
2D Fourier Transform
Generate patterns, view their 2D DFT in frequency space, then paint a filter mask to block or pass frequencies. See how low-pass blurs, high-pass sharpens, and notch filters remove periodic stripes — all in real time.
Spring-Mass Chain
N coupled masses on springs with adjustable k, m, and damping. Excite individual normal modes or random superpositions, drive at resonance frequencies, and watch the power spectrum reveal the complete mode structure.
Phyllotaxis — Golden Angle
Sunflower seed packing via the golden angle θ ≈ 137.508°. Drag the angle away from golden to see radial arms emerge — rational angles create visible spokes, while 2π/φ² is the "most irrational" and maximally uniform.
Rucklidge Attractor
A 3D strange attractor discovered by A.M. Rucklidge (1992) while studying magnetoconvection. Two coiling scroll wings with chaotic transitions, rendered as a glowing trail in rotating 3D with adjustable κ and λ parameters.
Rabinovich–Fabrikant Attractor
Proposed by Rabinovich & Fabrikant (1979) to model stochasticity in modulated plasma waves. Features intricate looping orbits with multiple basins of chaos — adjust α and γ to find the chaotic regime.
Arneodo Attractor
A 3D chaotic system related to Shilnikov homoclinic bifurcations — chaos emerges from a spiraling orbit that returns to a saddle-focus equilibrium. Four parameter presets reveal different attractor topologies.
Wigner Quasi-Probability
The Wigner function W(q,p) is the quantum phase-space analog of classical probability — but it can go negative, a direct signature of non-classical states. Visualize coherent, Fock, cat, squeezed, and thermal states in real time.
Lindblad Decoherence
The Lindblad master equation governs quantum decoherence: watch a qubit's Bloch vector spiral inward as dephasing, decay, and excitation channels destroy quantum superpositions. Track purity and von Neumann entropy live.
Bond Percolation Phase Transition
Open each edge of a square lattice with probability p. At the critical threshold p_c = 0.5, a spanning cluster first appears — a second-order phase transition. Watch the giant cluster form at criticality with real-time cluster coloring.
Complex Ginzburg–Landau
The CGLE governs spatiotemporal dynamics near Hopf bifurcations: oscillating reactions, lasers, convection. Tune b and c to traverse the Benjamin–Feir instability boundary and watch ordered spirals dissolve into spatiotemporal turbulence.
Hindmarsh–Rose Bursting Neuron
The HR model (1984) captures the full range of neuronal dynamics: quiescence, tonic spiking, and chaotic bursting. Increase applied current I to drive transitions between regimes. View phase planes, time traces, 3D attractor, and ISI histograms.
Bouali Attractor
A 3D autonomous system by Safieddine Bouali (1999) with butterfly-like wings and torus-knot structure near the transition to chaos. Adjust four parameters to explore distinct chaotic topologies, rendered as a rotating glowing trail.
Dadras–Momeni Attractor
A 5-parameter chaotic system introduced by Dadras & Momeni (2009) with topologically distinct attractors. Can exhibit butterfly-type and scroll-type chaos within the same parameter family. Drag to rotate in 3D.
Site Percolation at Criticality
Occupy vertices of a square lattice with probability p. The critical threshold p_c ≈ 0.5927 (no known closed form — open problem) marks where an infinite cluster first appears. Explore cluster statistics, spanning probability, and power-law cluster size distributions.
Lü–Chen Unified Attractor
Lü & Chen (2002) discovered a bridge parameter α that continuously interpolates between the Lorenz (α=0), Lü (α=0.8), and Chen (α=1) attractors. Sweep α to watch the attractor topology morph between three distinct chaotic families.
Stommel Thermohaline
Stommel's classic 2-box ocean model showing how temperature and salinity compete to drive overturning circulation. Adjust forcing to trigger bistability and flip between TH and SA circulation modes. Phase portrait shows multiple equilibria.
Quantum Kicked Rotor
Quantum chaos meets Anderson localization: a quantum rotor kicked periodically diffuses classically then freezes due to quantum interference — dynamical localization. Compare ⟨p²⟩ growth between quantum (saturating) and classical (linear diffusion) realms.
Oregonator BZ Oscillator
The Field-Körös-Noyes reduced model of the Belousov-Zhabotinsky reaction: three species (HBrO₂, Br⁻, Ce⁴⁺) in a stiff nonlinear ODE that generates sustained chemical oscillations. Explore the 3D phase-space limit cycle and tune stoichiometry.
Mathieu Stability Chart
The Ince-Strutt diagram: stability tongues of the Mathieu equation d²x/dt² + [a−2q cos(2t)]x = 0. Used in Paul ion traps, parametric amplifiers, and Faraday waves. Click anywhere to simulate trajectories at that (a,q) pair.
Allen-Cahn Phase Field
Gradient flow of the Ginzburg-Landau energy: ∂φ/∂t = ε²Δφ − W'(φ). Watch diffuse interfaces sharpen, small bubbles shrink by curvature, and Ostwald ripening coarsen the domain pattern. Draw custom initial conditions.
Dendritic Solidification
Kobayashi's phase-field model of crystal growth with anisotropic surface tension. Tune undercooling and symmetry order to grow 4-fold, 6-fold, or exotic dendritic arms — the same physics that forms snowflakes and metallic microstructure.
Bogdanov-Takens Bifurcation
A codimension-2 organizing center where saddle-node, Hopf, and homoclinic bifurcations all collide. Click the 2D parameter space (μ₁,μ₂) to explore all four qualitatively distinct phase portraits in this single normal form.
Husimi Q-Function
Quantum phase space portraits: Q(α) = |⟨α|ψ⟩|²/π, the positive-definite cousin of the Wigner function. Compare Fock states (rings), coherent states (Gaussians), Schrödinger cat states (two blobs), and squeezed states (ellipses).
Kaplan-Yorke Dimension
Live computation of the Kaplan-Yorke (Lyapunov) dimension D_KY = j + Σλᵢ/|λⱼ₊₁| for chaotic attractors. Select Lorenz, Rössler, Chen, or Sprott attractors and watch the fractal dimension emerge from the Lyapunov spectrum in real time.
Supercritical vs. Subcritical Hopf
Compare the two generic routes to oscillation: supercritical (smooth continuous birth of limit cycle, amplitude ~ √μ) vs. subcritical (unstable cycle, bistability, hysteresis). Phase portraits, radial flow diagrams, and bifurcation curves update live.
Hard-Disk Boltzmann Gas
N elastic hard disks in a box: watch the Maxwell-Boltzmann velocity distribution emerge spontaneously from any initial condition via ergodic exploration. Color shows speed, histogram tracks how quickly the system forgets its initial state.
K-d Tree Visualization
Explore the k-dimensional tree: a spatial data structure that recursively partitions space by alternating axis splits. Move your mouse to query nearest neighbors and watch how the search prunes entire subtrees, visiting only O(log n) nodes instead of all n points.
Diffraction Grating
Multi-slit interference: N slits create sharp principal maxima at d·sin(θ)=mλ, with a single-slit sinc² envelope. Tune the number of slits, slit spacing, wavelength (380–700nm), and slit width to see how the intensity pattern I ∝ sinc²(β)·[sin(Nδ/2)/sin(δ/2)]² responds.
Bishop Random Walk
A chess bishop moves only diagonally — it is forever trapped on squares of one color, a hard parity invariant. Watch the visit heatmap reveal which half of the board is accessible, and why the bishop's walk is equivalent to two independent 1D random walks on ±1 coordinates.
Strange Attractor Gallery
Six classic chaotic attractors: Lorenz, Rössler, Thomas, Halvorsen, Aizawa, and Chen-Lee. Thousands of particles trace the fractal geometry of each system's invariant set — the strange attractor where trajectories never repeat but remain forever bounded in phase space.
Rayleigh-Bénard Convection
A fluid layer heated from below spontaneously forms convection rolls above the critical Rayleigh number Ra_c ≈ 1708. Tune Ra = gαΔTd³/νκ and watch the Boussinesq equations drive temperature plumes and velocity cells emerge from a purely conductive state.
Iterated Prisoner's Dilemma
Eight strategies — Tit-for-Tat, Always Defect, Grim Trigger, Pavlov, Forgiving TfT, and others — compete in a round-robin tournament. Adjust the payoff matrix and noise level, then run the tournament to see which strategy achieves the highest total score across all pairings.
Kaleidoscope Symmetry
Draw in one sector and watch N-fold dihedral symmetry D_N replicate your strokes across all sectors simultaneously. Auto-mode drives a Spirograph-like Lissajous path through the symmetry group; adjust N from 2 to 16 to explore wallpaper-group-like reflective tilings.
Lorenz First-Return Map
Plot successive local maxima of z(t) against each other: z_{n+1} vs z_n. The resulting tent-like map is the geometric proof that the Lorenz system is chaotic — it mimics a 1D expanding map where arbitrarily close initial conditions diverge exponentially. Tune σ, ρ, β to distort the tent.
Quantum Tunneling
Numerically solve the 1D time-dependent Schrödinger equation iℏ∂ψ/∂t = Hψ for a Gaussian wave packet approaching a rectangular potential barrier. Watch Re(ψ), Im(ψ), and |ψ|² evolve; the live transmission coefficient T ≈ exp(−2κL) updates as you change E/V₀ and barrier width.
Chip-Firing / Abelian Sandpile
Self-organized criticality in action: drop chips onto a grid; any cell with ≥4 chips fires, distributing one to each neighbor. The system self-tunes to a critical state where avalanche sizes follow a power law — no external tuning required. Click to trigger local avalanches and watch them cascade.
Moran Process
Birth-death evolutionary dynamics on a finite population: one individual is chosen proportional to fitness to reproduce, one chosen uniformly to die. Watch mutants (fitness r) compete against residents. The exact fixation probability ρ = (1−1/r)/(1−1/rᴺ) is derived analytically — confirm it empirically by running hundreds of replicates.
Wright-Fisher Drift
Each generation the entire population is resampled by binomial draws from the current allele frequency — pure genetic drift. Watch multiple trajectories wander to fixation or loss, time-scale set by N (fix ~ 2N·ln(2N) generations). Add selection s and mutation μ; compare empirical fixation rate to the neutral prediction p₀.
Spectral Graph Laplacian
Generate a stochastic block model graph and compute its Laplacian L = D − A. The Fiedler vector (λ₂ eigenvector) encodes the best graph bisection — Cheeger inequality λ₂/2 ≤ h(G) ≤ √(2λ₂). Run spectral clustering: k-means on the first k eigenvectors recovers community structure invisible to adjacency alone.
Born Approximation
First-order quantum scattering: the differential cross section dσ/dΩ ∝ |Ṽ(q)|² where Ṽ is the Fourier transform of the potential and q = k₀ − k is the momentum transfer. Choose Yukawa, Gaussian, square-well, or screened Coulomb potentials; watch both the linear cross section and 3D polar scattering diagram update as you tune energy and range.
Bravais Lattices
All 14 three-dimensional Bravais lattices — the complete classification of periodic crystal structures. Drag to rotate each lattice (triclinic, monoclinic, orthorhombic, tetragonal, cubic, hexagonal, rhombohedral). Corner atoms in blue, centering atoms (body-center, face-center, base-center) in gold. Lattice parameters a, b, c, α, β, γ shown.
Ptolemaic Deferent & Epicycles
The geocentric solar system as Ptolemy described it (~150 AD): each planet rides an epicycle (small circle) whose center rides a deferent (large circle) around Earth. Toggle planets, add up to 5 epicycles per planet, and watch retrograde loops emerge from pure circular motion — the Almagest system that predicted planetary positions for 1400 years.
Ornstein–Uhlenbeck Process
The mean-reverting stochastic process dX = κ(θ−X)dt + σ dW — the continuous-time limit of AR(1) and the velocity process of Brownian motion with friction. Watch multiple paths relax toward the long-run mean θ; the stationary distribution converges to N(θ, σ²/2κ), shown as a live histogram against the theoretical Gaussian.
Mutation–Selection Balance
Allele frequency dynamics under the interplay of natural selection (coefficient s), dominance h, forward mutation rate μ, and back-mutation rate ν. The deterministic recursion Δp tracks convergence to equilibrium p̂ ≈ μ/hs. Launch multiple trajectories from different starting frequencies; equilibrium line shown in yellow.
Entropy Coding
Build a Huffman code from scratch: adjust symbol probabilities (random, geometric, or uniform), watch the optimal prefix tree assemble, and compare code lengths against the Shannon bound −log₂p. The fundamental theorem: H ≤ L̄ < H + 1 bits per symbol. See efficiency approach 100% and redundancy shrink toward zero as distributions sharpen.
Bragg Diffraction
X-ray diffraction from crystal planes: constructive interference when 2d·sinθ = nλ (Bragg's Law, 1913). Tune wavelength, lattice spacing, and glancing angle; the reflected beam glows when the Bragg condition is exactly satisfied. A second beam off the plane below shows the path-difference geometry that Von Laue and Bragg used to unlock crystal structure determination.
Radiolaria Growth
Watch siliceous marine microorganisms grow their intricate skeletal lattices in real time. Tunable spine count, branching depth, and rotational symmetry produce the dazzling geometric forms that fascinated Ernst Haeckel. Click anywhere to seed a new organism; each one grows its own unique radially symmetric architecture.
Marbling Fluid
Ebru — the ancient Turkish art of painting on water. Drag colored ink across a viscous surface and watch it marble into flowing, swirling patterns. The simulation models incompressible fluid deformation: each brush stroke perturbs the velocity field and advects color through it.
Lagrange Points
In the rotating two-body problem five special equilibrium points emerge where gravity and centrifugal force exactly balance. Drop test particles and watch them orbit or escape; L4 and L5 (the Trojan points) trap objects in stable horseshoe orbits while L1, L2, L3 are unstable saddle points.
Faraday Instability
Vertically vibrate a fluid layer and standing wave patterns emerge spontaneously — below threshold the flat surface is stable; above it, beautiful lattice patterns (squares, hexagons, stripes) lock to half the driving frequency via parametric resonance. Tune amplitude and frequency to explore the pattern zoo.
Gierer-Meinhardt Patterns
The Gierer-Meinhardt activator-inhibitor system (1972) explains how biological morphogenesis produces spots, stripes, and labyrinths from uniform initial conditions. A fast-diffusing inhibitor stabilizes a slow-diffusing activator into Turing patterns — the same mechanism behind zebrafish stripes and seashell patterns.
Coral Reef Growth
A diffusion-limited aggregation model of coral colony expansion. Nutrient particles drift in from the ocean, sticking where they touch the growing structure and building fractal branching architectures. Tune growth rate, nutrient flux, and branching probability to grow plate, brain, or staghorn coral formations.
Jerk System Chaos
The simplest possible continuous chaotic system: a third-order ODE with a single nonlinearity that produces a strange attractor. Tune the parameter a to sweep through period-doubling bifurcations to full chaos. The phase portrait traces the Rössler-like attractor that Sprott found hiding in this minimal jerk equation.
Richtmyer-Meshkov Instability
When a shockwave crosses a density interface, mushroom-shaped spikes of heavy fluid penetrate the light fluid and vice versa. This instability drives mixing in supernova explosions, inertial confinement fusion, and supersonic combustion. Watch the interface corrugate, roll up, and cascade into turbulence.
Sand Castle Physics
Wet sand holds together through capillary bridges — thin water menisci between grains that generate attractive forces. Too dry and grains flow apart; too wet and cohesion disappears. Explore the optimal water content that gives wet sand its remarkable strength and lets you build towering structures that dry sand can never achieve.
Figure-Eight Choreography
Chenciner and Montgomery proved that three equal masses can chase each other in a stable figure-eight orbit — a choreographic solution of the three-body problem. Watch the ballet: each body passes through the same points in sequence, separated by one-third of the period. Perturb the initial conditions to see how fragile this beautiful symmetry is.
Heisenberg Spin Chain
Classical XXZ Heisenberg spin chain simulated with Metropolis Monte Carlo. Watch spin configurations evolve in real-time on a space-time diagram — tune the exchange coupling J, anisotropy Δ, magnetic field, and temperature to explore ferromagnetic ordering, antiferromagnetic checkerboard patterns, and spin-wave excitations.
Topological Edge States
The SSH (Su-Schrieffer-Heeger) model is the simplest topological insulator. When inter-cell hopping w exceeds intra-cell hopping v, the winding number jumps to 1 and protected zero-energy edge states appear at the chain boundaries. Watch the bulk-boundary correspondence in action — the winding path in k-space either encircles the origin or it doesn't.
Quantum Walk on a Lattice
A quantum particle walks on a 2D lattice using a discrete-time coin operator — Hadamard, Grover, or DFT. Unlike a classical random walk where spread grows as √t, quantum interference causes ballistic spreading proportional to t, producing a striking butterfly-shaped probability distribution. Watch coherent quantum transport unfold in real time.
NMR Bloch Sphere Precession
The Bloch equations describe how a nuclear spin magnetization vector precesses in magnetic fields — the physics behind MRI. Apply a static B₀ field for Larmor precession, a resonant RF pulse B₁ to tip the magnetization, and watch T₁ longitudinal and T₂ transverse relaxation bring it back to equilibrium. The FID signal traces out as the magnetization spirals home.
Lid-Driven Cavity Flow
The classic CFD benchmark: a fluid-filled square box with a moving top lid. Simulated with the Lattice Boltzmann method (D2Q9), the flow develops a primary recirculation vortex plus corner vortices whose size and strength depend on Reynolds number. Switch between speed, vorticity, and streamline views to see different aspects of the flow structure.
Turbulent Jet Mixing
A high-velocity jet injected into still fluid spreads by entraining ambient fluid — a canonical problem in turbulence. The particle simulation captures self-similar spreading (half-width ∝ x) and centerline velocity decay (u_c ∝ 1/x), with tunable turbulence intensity, entrainment coefficient, and passive scalar mixing visualized through tracer colors.
Onsager Exact Ising 2D
Lars Onsager's 1944 exact solution of the 2D Ising model was a landmark in statistical mechanics — the first rigorous proof of a phase transition. Compare the Wolff cluster Monte Carlo simulation against Onsager's exact magnetization m(T) = (1 − sinh⁻⁴(2J/kT))^(1/8) and the logarithmically divergent specific heat at T_c = 2J/ln(1+√2) ≈ 2.269.
DNA Base-Pair Melting
The Peyrard-Bishop-Dauxois model treats DNA as a chain of nonlinear oscillators coupled by Morse potentials — each base pair has a displacement from equilibrium that fluctuates thermally. As temperature rises through the melting point, base pairs progressively open. GC pairs (stronger) melt at higher temperatures than AT pairs, producing a sequence-dependent sigmoidal melting curve.
Diffusion on Fractal
Random walks on the Sierpinski gasket diffuse anomalously: mean-square displacement grows as t^(2/d_w) where the walk dimension d_w = ln5/ln2 ≈ 2.32 exceeds the Euclidean value of 2. Watch hundreds of walkers spread across the fractal lattice and measure the subdiffusive exponent — a direct signature of the fractal geometry slowing down transport.
Islamic Geometric Art
Islamic geometric art achieves infinite complexity from finite symmetry rules — star polygons, rosettes, and girih tiles tile the plane by the same mathematics as crystallographic groups. Generate 6-, 8-, 10-, and 12-fold star patterns with adjustable interlacing angles and color palettes, from Alhambra gold to lapis lazuli blue. The patterns that covered medieval mosques encoded abstract algebra centuries before it was formalized.
Treap
A randomized BST that satisfies both the binary search tree property (by key) and the heap property (by random priority). Watch nodes insert and rotate into place — expected O(log n) height, no rotations needed after deletion, elegant probabilistic balance.
Hungarian Algorithm
Optimal assignment of workers to jobs at minimum total cost. Step through row/column reduction and augmenting paths to see how the algorithm finds a perfect matching in O(n³) — visualized as a bipartite graph and cost matrix side by side.
Chandler Wobble
Earth's rotation axis wobbles relative to its crust with a ~433-day period (predicted 305 days for a rigid Earth — real Earth is non-rigid) combined with an annual term. Watch the polar motion trace spiraling Lissajous-like paths in the polar coordinate plane.
Mach Bands
Where a luminance ramp meets a flat region, the visual system creates illusory bright and dark stripes — caused by lateral inhibition in the retina. Toggle the luminance profile to see the second-derivative amplification that creates bands not present in the image.
Wythoff's Game
Two piles of stones — remove any amount from one pile, or equal amounts from both. The P-positions (losing for the mover) follow the golden ratio: they are exactly the pairs (⌊nφ⌋, ⌊nφ²⌋). Play against the optimal AI and explore the golden-ratio structure of the game board.
Cayley Graph
A Cayley graph encodes a group's algebraic structure geometrically: vertices are group elements, colored arrows are left-multiplication by generators. Explore cyclic groups, dihedral groups, quaternions, and symmetric groups — every vertex looks identical, a visual proof of vertex-transitivity.
Spin Echo (NMR)
Nuclear spins precess at slightly different frequencies — inhomogeneous broadening fans them out and kills the signal. A 180° refocusing pulse reverses the dephasing, regenerating an echo. The Hahn echo principle underlies all MRI and NMR spectroscopy.
Milankovitch Cycles
Three orbital parameters modulate Earth's insolation over tens of thousands of years: eccentricity (~100 kyr), axial obliquity (~41 kyr), and precession (~23 kyr). Watch their superposition drive the combined summer-insolation forcing that paces the ice ages.
Myelin & Saltatory Conduction
Myelin sheaths insulate axons, leaving bare Nodes of Ranvier where action potentials regenerate. The AP jumps node to node (saltatory conduction) rather than spreading continuously — achieving 70–120 m/s vs 0.5–2 m/s and using 100× less energy.
Contrast Sensitivity Function
The human visual system is bandpass for spatial frequency — most sensitive near 3–5 cycles/degree, falling off at both extremes. Explore Gabor patches, the Campbell-Robson chart, and the CSF curve that defines what you can and cannot see.
Stern-Brocot Tree
Every positive rational number appears exactly once in this infinite binary tree — each node is the mediant of its two ancestors. Adjust depth to explore how all fractions emerge from 0/1 and 1/0 through successive mediants.
Apollonian Gasket
Infinitely nested tangent circles where every gap is filled by the largest possible inscribed circle. Descartes' Circle Theorem (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²) governs each new curvature — generates a fractal with Hausdorff dimension ≈1.305.
Calcium Wave
Intracellular calcium waves propagate via CICR (calcium-induced calcium release): cells fire when Ca²⁺ exceeds threshold, amplify the signal, then enter a refractory period. Click to trigger release events and watch waves collide and annihilate.
Ice-Albedo Feedback
A positive feedback loop: more ice raises albedo, absorbs less sunlight, creates more ice. The energy-balance model shows two stable states — ice-free and snowball Earth — separated by an unstable equilibrium. Adjust solar forcing to find the tipping points.
Wilson-Cowan Oscillator
Excitatory and inhibitory neural populations coupled in the classic Wilson-Cowan model. Tune coupling weights to produce fixed points, limit-cycle oscillations, or bistability. Nullclines in the phase portrait reveal the geometry of each regime.
Thermohaline Conveyor
The Stommel 2-box model of ocean overturning: temperature and salinity differences drive circulation between tropical and polar boxes. Increase freshwater forcing to collapse the AMOC — a classic example of a tipping-point bifurcation in climate dynamics.
Waveguide Modes
Each TE or TM mode in a rectangular waveguide has a cutoff frequency below which only evanescent (non-propagating) fields exist. Visualize the transverse field vector patterns, explore cutoff conditions, and see how phase velocity diverges near cutoff.
Radix Sort
LSD radix sort processes one digit place at a time with a stable counting sort — distributing numbers into ten buckets then collecting them in order. Runs in O(nk) with no comparisons at all. Step through each pass to see exactly why the order is preserved.
Pascal Triangle mod n
Color each entry of Pascal's triangle by its remainder mod n. When n is prime, Kummer's theorem produces Sierpinski-like self-similar patterns; composite n yields qualitatively different structures. Toggle the modulus to watch symmetry appear and shatter.
Totient Function Graph
Euler's φ(n) counts integers up to n coprime to n. Primes cluster on the top envelope φ(p)=p−1; highly composite numbers plunge to low ratios. Color by primality, smallest prime factor, or φ(n)/n ratio — and see the density converge to 6/π² on average.
Hydrogen Orbital
Visualize the electron probability density |ψ|² for hydrogen's quantum states. Adjust principal quantum number n and angular momentum l to explore s, p, d orbitals and the nodal surfaces that emerge from the Schrödinger equation.
Zeeman Splitting
Watch spectral lines split when a magnetic field is applied. The interaction of orbital and spin magnetic moments with the external field lifts energy degeneracy, revealing the quantum structure hidden in atomic spectra.
Rabi Oscillation
A two-level quantum system driven by resonant electromagnetic radiation. Population oscillates between ground and excited states at the Rabi frequency — the heartbeat of quantum optics, NMR, and qubit control.
FFT Butterfly
The Cooley-Tukey fast Fourier transform decomposes a DFT into a recursive butterfly network. Trace how N log N complex multiplications replace N² — the algorithm that enabled digital signal processing.
Z-Plane Poles & Zeros
Design digital filters by placing poles and zeros on the complex z-plane. See the frequency response update in real time as you drag poles toward the unit circle — stability and resonance made geometric.
Barabási-Albert Network
Preferential attachment: new nodes connect to existing nodes proportional to their degree, generating scale-free networks with power-law degree distributions. The rich get richer — and hubs emerge inevitably.
SIR Epidemic on Network
Watch an infectious disease spread through a contact network. Susceptible-Infected-Recovered dynamics on graph structure shows how topology shapes outbreak size — and why herd immunity thresholds depend on network heterogeneity.
Voronoi Polycrystal
Random seed points generate a Voronoi tessellation mimicking the grain structure of metals. Each cell is a crystal grain; boundaries form where grains meet. Adjust grain count and watch how microstructure affects material properties.
Spinodal Decomposition
A Cahn-Hilliard phase-field simulation of a binary mixture quenched into instability. Small fluctuations amplify spontaneously at all scales, forming interconnected domains that coarsen over time — no nucleation required.
Mackey-Glass Equation
A delay differential equation that transitions from stable equilibrium to periodic oscillations to chaos as the time delay τ increases. Originally a model of blood cell regulation, now a benchmark for chaotic time series.
Rayleigh–Plateau Instability
Surface tension breaks a falling liquid jet into discrete droplets. Watch perturbations with wavelength λ > 2πr grow exponentially while short-wave ripples decay — Lord Rayleigh's 1878 prediction that explains every dripping faucet.
KdV Solitons
Korteweg–de Vries solitons are nonlinear water waves that maintain shape indefinitely and pass through each other with only a phase shift. Adjust speeds and watch collisions — the particle-like behavior that launched soliton physics in 1895.
Coulomb Crystal
Charged ions in a harmonic trap crystallize into geometric shells via Coulomb repulsion — exact solutions to Thomson's problem. Tune trap strength and temperature to watch the phase transition from plasma to crystal, as seen in real ion traps.
Optical Lattice
Counterpropagating laser beams trap ultracold atoms in a periodic standing-wave potential. Watch atoms tunnel between sites, crystallize at low temperature, and exhibit Mott insulator physics — the backbone of quantum simulation experiments.
Morse Potential
The Morse potential V(r) = Dₑ(1−e^{−a(r−rₑ)})² models real diatomic bond stretching with a finite number of quantized vibrational levels and anharmonic spacing — exactly what infrared spectroscopy measures in molecules.
Elastic Membrane
Simulate the 2D wave equation on a stretched membrane — click to pluck and watch circular wavefronts, interference, and boundary reflections. The math underlying Chladni patterns, drum physics, and vocal cord vibration.
Fisher–KPP Wave
The Fisher–KPP traveling wave models gene spread, invasive species, and neural activity. Any initial disturbance asymptotes to the minimum speed c* = 2√(rD). Toggle the Allee effect to see populations collapse below a critical density threshold.
Fractal Basins of Attraction
Newton's method applied to zⁿ = 1 in the complex plane reveals fractal basin boundaries with Hausdorff dimension 2. Every neighborhood of a boundary point contains initial conditions converging to all n roots — deterministic chaos at the boundary.
Mode Coupling & Energy Transfer
Two coupled oscillators exchange energy at a Rabi frequency set by coupling κ and detuning Δω. At resonance, energy transfers completely — Rabi oscillation. Add nonlinearity to trap energy in one mode and watch the beating collapse.
Pendulum Lattice (Sine-Gordon)
A chain of torsion-coupled pendulums simulates the sine-Gordon equation, supporting topological kink solitons, anti-kinks, and breathers. Click to launch kinks and watch them pass through each other — the winding number (topological charge) is conserved.
Faraday / Parametric Pendulum
A pendulum with oscillating pivot obeys the Mathieu equation and shows parametric instability inside Arnold tongues. The stability map reveals resonance zones — and the inverted pendulum can be stabilized above a critical drive frequency.
Stochastic Ising (Metropolis)
Metropolis–Hastings Monte Carlo sampling of the 2D Ising model near the exact critical point Tc = 2/ln(1+√2) ≈ 2.269. Watch the order–disorder phase transition, diverging correlation length, and critical exponents β=1/8, γ=7/4 in real time.
4D Hypercube Projection
A tesseract rotating through six independent planes of 4D space, projected to 2D via perspective. Drag to rotate in 3D; control XW, YW, ZW, XY rotation speeds. Color by depth, axis, or rainbow mode.
Clifford Torus
The flat torus in S³ ⊂ ℝ⁴ — zero intrinsic curvature but living in 4D. Stereographic projection from S³ to ℝ³ turns it into the familiar donut, yet in 4D it is perfectly symmetric. Fundamental to the Hopf fibration.
Apollonian Circle Packing
Recursive fractal built via Descartes' Circle Theorem: given three mutually tangent circles, two more fit the gap with integer curvatures. Hausdorff dimension ≈ 1.306. Color by curvature, generation, or spectrum.
Circle Limit — Hyperbolic Tiling
Poincaré disk model: the entire infinite hyperbolic plane inside a circle. Choose any valid {p,q} Schläfli symbol where (p-2)(q-2)>4 — infinitely many hyperbolic tilings exist, unlike Euclidean geometry's three.
Involute Gear Geometry
The involute of a circle — string unwinding from a cylinder — creates gear tooth profiles with constant velocity ratio at all contact points. Euler's elegant solution drives almost all modern mechanical transmissions.
Spectral Painting
Superimposed eigenvectors of the grid Laplacian create standing wave interference patterns — the normal modes of a rectangular drumhead. Mix modes, animate phases, and watch Chladni-like nodal lines emerge.
Riemann Drum — Isospectrality
Two differently-shaped polygonal drums with identical vibration spectra — Gordon, Webb & Wolpert's 1992 answer to Kac's question "Can you hear the shape of a drum?" (Answer: No.) Visualize matching eigenmodes side by side.
Calabi-Yau Slice
2D projection of the complex algebraic surface z₁ⁿ + z₂ⁿ = 1 in ℂ². Calabi-Yau manifolds are Ricci-flat Kähler manifolds (Yau 1978) used to compactify the 6 extra dimensions in string theory.
Lorenz Ribbon
Multiple nearby trajectories in the Lorenz system trace a luminous ribbon through the butterfly strange attractor. Watch sensitive dependence on initial conditions stretch and fold the ribbon while the attractor (dim ≈ 2.06) remains.
Julia Set Morphing
A continuous journey through the parameter space of Julia sets J_c = {z : f_c^n(z) bounded}. The set transitions between connected, dendrite, and Cantor-dust phases as c crosses the Mandelbrot boundary. Click to pin c.
Symplectic Phase Flow
Liouville's theorem made visible: Hamiltonian flow preserves phase-space volume exactly. Watch clouds of particles evolve under harmonic, pendulum, Duffing, Morse, and double-well potentials — stretching but never compressing.
Cardiac Fibrillation
Spiral wave reentry in a 2D excitable cardiac medium. Watch normal planar waves degenerate into chaotic fibrillation and see how defibrillation shocks can restore synchrony.
XOR Perceptron
Why a single-layer perceptron cannot solve XOR — and how adding one hidden layer solves it. Interactive visualization of the decision boundary learning process.
Gyroscope Precession
Euler equations for a spinning top: precession, nutation, and the counterintuitive behavior of angular momentum under gravity and torque.
Renormalization Group Flow
Fixed points and RG flow in coupling-constant space. Watch how irrelevant operators flow to zero and relevant operators drive phase transitions at the Wilson-Fisher fixed point.
Lattice Vibrations
Phonons in a 1D diatomic chain: acoustic and optical branches, zone boundary modes, and the formation of band gaps from alternating mass or spring constants.
Spiral Waves
Self-organizing spiral waves in a 2D excitable medium (FitzHugh-Nagumo model). Rotate, collide, and annihilate spiral pairs — the same dynamics underlying cardiac arrhythmias.
EEG Brain Rhythms
Simulated EEG oscillations from alpha, beta, theta, and delta bands. Interactive Fourier decomposition shows how brain states emerge from superimposed neural rhythms.
Neural Network Loss Landscape
3D visualization of a neural network loss landscape: saddle points, flat regions, and sharp vs. flat minima. Explore how optimizer trajectories navigate high-dimensional parameter spaces.
Soddy Circles
Descartes' Circle Theorem and Soddy's construction: given three mutually tangent circles, exactly two circles are tangent to all three. The curvatures satisfy (k₁+k₂+k₃+k₄)²=2(k₁²+k₂²+k₃²+k₄²).
Neural Field Bump Attractor
Continuous attractor dynamics in a 1D neural field (Wilson-Cowan with Mexican-hat connectivity). Working memory as a persistent bump of activity that can be moved by transient inputs.
Turing Reaction-Diffusion
Gray-Scott model of two reacting chemicals: spots, stripes, coral, maze, and worm patterns. Tune feed and kill rates to explore the parameter space of biological morphogenesis.
Fourier Epicycles
Draw any closed curve and watch it reconstructed with rotating circles (Fourier series). Each circle adds a harmonic frequency — increase harmonics to recover finer details.
Boolean Network Dynamics
Kauffman random Boolean networks model gene regulatory dynamics. K=1 → ordered (frozen), K=2 → critical (complex attractors), K≥3 → chaotic. Observe attractor cycles in the state-space timeline.
Phase Portrait Explorer
Launch trajectories through pendulum, Van der Pol, Duffing, Lotka-Volterra, Hopf, and saddle systems. RK4 integration reveals fixed points, limit cycles, and separatrices.
Magnetic Pendulum
Three magnets create fractal basins of attraction. Tiny changes in starting position lead to wildly different final magnets — a vivid demonstration of sensitive dependence and chaos.
Ulam Prime Spiral
Stanislaw Ulam's 1963 doodle: integers arranged in a spiral reveal that primes cluster on diagonal lines corresponding to prime-generating quadratic polynomials.
Fluid Pressure & Pascal's Law
Drag the hydraulic piston to see Pascal's Law in action. A small force on the narrow piston produces a large force on the wide piston — the principle behind hydraulic jacks and brakes.
Penrose Quasicrystal
Penrose tiling via Robinson triangle substitution: five-fold symmetry, no periodicity, ratio of tile types equals φ (golden ratio). The mathematical structure behind quasicrystals (Nobel 2011).
Boids Swarm Simulation
Craig Reynolds' 1987 flocking algorithm: separation, alignment, and cohesion produce emergent murmuration. Click to place a predator and watch the flock evade.
Resistor Network Solver
Build grid, ladder, Wheatstone bridge, or cube resistor networks. Click two nodes, hit Solve, and watch Kirchhoff's Current Law compute voltages and currents via nodal analysis.
Hyperbolic Crochet
Daina Taimina's mathematical crochet: each row adds extra stitches at ratio r, creating negative Gaussian curvature that ruffles and folds — physically modeling the hyperbolic plane.
2D Ripple Tank
Click to place wave sources and watch interference patterns emerge in real time. Adjust wavelength, damping, and color mode to explore constructive and destructive interference.
Portfolio Efficient Frontier
Markowitz mean-variance optimization: Monte Carlo sampling reveals the efficient frontier, Capital Market Line, max-Sharpe portfolio, and minimum-variance point across random asset universes.
LIGO Gravitational Wave Detector
Simulate a Michelson interferometer detecting spacetime strain. Trigger binary black hole and neutron star merger chirps, tune arm length and noise floor, watch the signal unfold.
Scale Tuning Lab
Compare Equal Temperament, Just Intonation, Pythagorean, and Meantone tuning systems. Visualize cents deviations, the Pythagorean comma, and play intervals to hear the difference.
Hidden Attractors
Leonov & Kuznetsov's 2011 discovery: chaotic attractors not connected to any unstable equilibrium. Explore Sprott A (no equilibria), Thomas, and Chua hidden systems in 3D.
Epidemic Network Spreading
SIR epidemic dynamics on Erdős-Rényi, Barabási-Albert, Watts-Strogatz, and lattice networks. Tune β, γ, vaccination coverage, and watch R₀ determine outbreak fate.
Lévy Flights
Random walks with heavy-tailed step lengths. Switch between Brownian motion and Lévy flights (α=0.5–2) to see how superdiffusion creates clustering with rare long jumps — the foraging strategy of albatrosses and sharks.
Sinai Billiards
A circle obstacle in a square table creates provably chaotic dynamics. Trace multiple trajectories from nearly-identical starting points and watch exponential divergence — Sinai's 1970 proof that billiards can be ergodic.
Kármán Vortex Street
Flow past a cylinder generates alternating vortex shedding — the phenomenon behind bridge oscillations, singing wires, and the Tacoma Narrows collapse. Control Reynolds number to see laminar→turbulent transition.
Central Limit Theorem
Sample from uniform, exponential, bimodal, or custom distributions. Watch the sampling distribution of the mean converge to Gaussian as n grows — the most important theorem in statistics, visualized live.
Quantum Tunneling
A wavepacket hits a potential barrier and partially transmits. Adjust barrier height and width to see tunneling probability change exponentially — the mechanism behind alpha decay, tunnel diodes, and STM.
Voronoi Drum Machine
Click to place points that become Voronoi cells — each cell is a drum pad. Cells trigger sounds based on area (pitch) and neighbor count (timbre). Rearrange points to compose spatial rhythms.
Origami Fold Simulator
Step-by-step crane fold with interactive crease pattern. Click each fold line to apply mountain or valley folds. The flat-foldability conditions (Kawasaki's theorem, Maekawa's theorem) are verified in real time.
Diffusion on Graphs
Heat equation on a network: inject concentration at any node and watch it spread according to the graph Laplacian. Compare star, ring, complete, and random graphs — eigenvectors of L determine the diffusion modes.
Collatz Conjecture 3D
Visualize the Collatz tree in 3D: n→n/2 (even) or 3n+1 (odd). Branches colored by stopping time, rotatable. All known integers eventually reach 1 — but nobody can prove it for all n.
Crystal Symmetry Groups
Interactive explorer of the 17 wallpaper groups. Drag a motif tile and see it tiled with full symmetry: reflections, rotations, glides. Switch between p1, p2, pm, pg, cm, p4m and other groups.
Soap Film Minimal Surfaces
Interactive Plateau problem: drag control points to define a wire frame boundary and watch the soap film relax to a minimal surface via mean-curvature flow. Catenoid, Enneper, and helicoid presets.
Phase Diagram Explorer
Interactive P-T phase diagrams for water, CO₂, and a Lennard-Jones fluid. Click to place the system state and watch which phase (solid/liquid/gas/supercritical) it occupies. Triple point and critical point annotated.
Spectral Clustering
Generate 2D point clusters (blobs, rings, moons). Compute the graph Laplacian, visualize its eigenvectors, then cluster via k-means in eigenspace. See why spectral methods find non-convex clusters that k-means misses.
Solar Wind & Magnetosphere
Earth's magnetic dipole deflects the solar wind into a bow shock. Parker spiral for interplanetary field lines. Visualize magnetopause standoff distance as a function of solar wind pressure.
Ising Renormalization Group
Block-spin renormalization of the 2D Ising model. Watch the coupling constant flow toward fixed points under repeated coarse-graining. See the critical point as an unstable fixed point of the RG map.
Airy Diffraction Pattern
Fraunhofer diffraction through a circular aperture produces Airy rings. Adjust wavelength and aperture size. Rayleigh criterion for angular resolution shown. Two-source version demonstrates telescope resolving power.
Buddhabrot Nebula
The Buddhabrot renders escape trajectories of c values near the Mandelbrot boundary, creating a Buddha-shaped nebula. Three color channels for different escape time ranges. Progressive refinement rendering.
ODE Solver Comparison
Solve the same stiff ODE with Euler, RK2, RK4, and adaptive RK45. Visualize step-size adaptation, local truncation error, and global error accumulation. See why high-order methods win for smooth problems.
Pendubot Control
An under-actuated double pendulum (motor only at base). Demonstrate swing-up via energy-based control, then balance at the unstable upright equilibrium with LQR. Toggle controller on/off to watch it fall.
Graph Eigenvalue Zoo
Explore how graph structure determines spectrum: path, cycle, complete, bipartite, Petersen, hypercube, random. Visualize adjacency eigenvalues as an energy level diagram. Cheeger inequality bounds shown.
Elastic Instability & Buckling
Compress a slender column and watch it snap through the Euler buckling instability. Adjust slenderness ratio, boundary conditions, and load eccentricity. Post-buckling equilibrium path traced in a load-deflection diagram.
Topology Surface Zoo
Interactive gallery of surfaces classified by genus and orientability: sphere (g=0), torus (g=1), Klein bottle, projective plane, double torus (g=2). Euler characteristic χ = 2−2g displayed. Cut-open views reveal handle structure.
Lempel-Ziv Compression
Type or paste text and watch LZ77/LZ78 parse it into literal/reference tokens in real time. Compression ratio and dictionary visualized. See why repetitive text compresses well but random text doesn't — Kolmogorov complexity in action.
Firefly Entrainment
Pulse-coupled oscillators synchronize like firefly swarms. Each firefly fires when its phase reaches 1, then all neighbors get a kick ε. Watch incoherent blinking lock into synchrony via Mirollo-Strogatz theory.
Thomas' Cyclically Symmetric Attractor
Thomas' system: ẋ=sin(y)−bx, ẏ=sin(z)−by, ż=sin(x)−bz. At b≈0.208 the attractor is chaotic but retains cyclic symmetry. Rotate in 3D, vary b from order to chaos.
Feynman Diagram Builder
Drag and connect propagator lines and vertices to build QED Feynman diagrams. The tool computes the Feynman rules contribution (symbolic) and checks loop order. Educational introduction to perturbation theory.
Ice Nucleation & Crystallization
Supercooled water droplet with fluctuating clusters. Classical nucleation theory: free energy barrier ΔG = 4πr²γ − (4/3)πr³Δμ. Watch subcritical embryos dissolve and supercritical nuclei grow into an ice crystal.
Topological Insulator Edge States
1D SSH model (Su-Schrieffer-Heeger chain): alternate hopping t₁ and t₂. Trivial phase (t₁>t₂): gap, no edge states. Topological phase (t₁
NMR Spin Echo
Bloch equations for a spin-1/2 in a magnetic field. Apply π/2 pulse to tip M into the xy-plane, watch free precession, then π pulse refocuses dephased spins into a spin echo. T1/T2 relaxation adjustable.
Wetting & Contact Angle
Young's equation γ_SV = γ_SL + γ_LV·cos(θ). Drag the surface energy sliders to see droplets go from superhydrophilic (θ→0) to superhydrophobic (θ→180°). Cassie-Baxter and Wenzel states on rough surfaces.
2D Sand Automaton
Falling sand simulation with multiple materials: sand, water, fire, stone, plant, acid. Left-click to draw, right-click to erase. Each material follows simple local rules that produce emergent complex behavior.
Brownian Ratchet & Feynman's Paradox
Feynman's pawl-and-ratchet thought experiment. Thermal fluctuations drive a ratchet wheel — but only when there's a temperature gradient. Visualize why a ratchet in thermal equilibrium can't do work (second law).
Kelvin-Helmholtz Instability
Two fluid layers shear past each other. Small perturbations grow into rolling vortex billows — seen in cloud bands, ocean thermoclines, and Jupiter's atmosphere. Pseudo-spectral simulation with configurable density ratio and shear.
SIS Epidemic on Networks
Susceptible-Infected-Susceptible dynamics on configurable networks (ER, BA scale-free, regular lattice, small-world). Epidemic threshold λ_c = 1/λ_max (spectral radius). Watch herd immunity and endemic equilibrium.
Best Rational Approximations
The continued fraction algorithm produces the best rational approximations to any real number. Visualize convergents on the number line, Stern-Brocot tree path, and Farey sequence. See why π ≈ 355/113 is so accurate.
Zero-Knowledge Proof: Graph Coloring
Interactive demonstration of the ZK proof for 3-colorability. Peggy commits to a random permutation of her 3-coloring. Victor picks an edge; Peggy reveals the two endpoint colors. Repeat 100x: soundness error → (2/3)^100.
Acoustic Metamaterial Band Gap
1D spring-mass chain with periodic heavy/light mass alternation. Band structure computed via Floquet-Bloch theory: pass bands and stop bands. Visualize wave propagation and evanescent decay in the gap frequency range.
Moiré Bilayer & Magic Angle
Two graphene-like hexagonal lattices rotated by angle θ. At the magic angle (~1.1°) the moiré superlattice has flat bands — the mechanism behind unconventional superconductivity in twisted bilayer graphene (Cao et al. 2018).
SAT Solver Visualization
DPLL algorithm solving 3-SAT instances. Watch unit propagation, pure literal elimination, and backtracking on the decision tree. Generate random 3-SAT near the phase transition (α≈4.27) where instances are hardest.
Maxwell's Demon Game
Play as Maxwell's Demon: sort fast (red) and slow (blue) gas molecules between two chambers by operating a trapdoor. Track temperature difference vs. your information cost. Landauer's principle: each bit erased dissipates kT ln 2.
Brownian Ratchet & Feynman's Paradox
Feynman's pawl-and-ratchet: thermal fluctuations can't drive directed motion at equilibrium. Asymmetric potential with thermal noise. Add a temperature gradient to extract work. Second law visualized.
Kelvin-Helmholtz Instability
Two fluid layers shearing past each other. Perturbations grow into rolling vortex billows — seen in cloud bands, the ocean thermocline, and Jupiter's atmosphere. Adjust shear velocity and density contrast.
Vortex Ring Dynamics
Thin-core vortex rings propagate via self-induction. Two coaxial rings: leapfrogging when same-sign, mutual annihilation when opposite. Biot-Savart law for induced velocity. Smoke ring physics.
Tidal Resonance & Bay of Fundy
Shallow water wave resonance in a closed basin. Natural period T = 4L/√(gH). Bay of Fundy has T≈12h matching the M2 tidal forcing → 16m tides. Tune length and depth to find resonance.
3D Population Dynamics
Three-species food web: Lotka-Volterra with a top predator. Phase space trajectory in 3D (prey, predator, super-predator). Coexistence, limit cycles, and chaotic dynamics depending on coupling strengths.
Fisher Information Geometry
Visualize the statistical manifold of Gaussian distributions. The Fisher metric makes it a hyperbolic space (Poincaré half-plane). Geodesics = paths of minimum KL divergence. Natural gradient descent shown.
Platonic Solids & Duality
Interactive 3D viewer of all 5 Platonic solids plus their duals (tetrahedron↔tetrahedron, cube↔octahedron, dodecahedron↔icosahedron). Morphing slider between solid and dual. Euler characteristic V−E+F=2 displayed.
Erdős-Rényi Phase Transition
Add edges to a random graph one by one. At p = 1/(n−1) a giant connected component suddenly emerges — the percolation phase transition. Component size distribution plotted live.
Doppler Effect Simulator
Moving source emitting sound waves. Drag the source to change velocity and direction. Wavefront compression (blueshift) and expansion (redshift) visualized. Mach cone forms when v > c_sound.
3D Lissajous Curves
x=A sin(at+δ), y=B sin(bt), z=C sin(ct+φ). Vary frequency ratios and phases to create closed orbits, torus knots, and Lissajous figures in 3D. Drag to rotate. When ratios are rational, curves close; irrational → dense on a torus.
Quantum Chaos & Level Statistics
Energy level spacing statistics of a classically chaotic billiard (stadium) vs integrable system (circle). Wigner-Dyson distribution (GUE/GOE) for chaos vs Poisson for integrable — the quantum signature of classical chaos.
Protein Secondary Structure
Ramachandran plot: allowed φ-ψ backbone dihedral angle regions for alpha-helix and beta-sheet. Drag a residue and watch it snap into helical or sheet geometry in the 3D backbone viewer.
Neuron Morphology & Cable Theory
Hodgkin-Huxley action potential propagating along a dendritic cable. Rall's cable theory: space constant λ = √(r_m/r_a) determines signal decay. Adjust membrane resistance, axial resistance, and stimulation site.
3D Attractor Basin Explorer
Newton's method basins of attraction for z³−1=0 in the complex plane — three roots create a fractal boundary. Extend to 4 and 5 roots. Zoom and explore the fractal basin structure.
Holographic Diffraction Grating
Simulate a diffraction grating: constructive interference at d·sin(θ)=mλ. Add slits one by one and watch the diffraction pattern sharpen. Compare single-slit, double-slit, and multi-slit patterns.
Plasma Waves & Dispersion
Dispersion relations for plasma waves: Langmuir, ion-acoustic, whistler, and Alfvén modes. Plot ω-k diagram, group and phase velocities. Electromagnetic cutoffs and resonances in a magnetized plasma.
Evolutionary Game Theory
Replicator dynamics for 2×2 games: Prisoner's Dilemma, Hawk-Dove, Stag Hunt, Coordination. Phase portrait on the simplex with fixed points and basins. Watch strategy frequencies evolve.
2D Wave Equation Explorer
Wave equation on a 2D rectangular membrane: u_tt = c²∇²u. Click to excite modes. Visualize normal modes (Chladni patterns), superposition, and standing wave nodal lines. Circular and square domain options.
Ergodic Billiard Trajectories
Compare billiards in integrable (circle, ellipse) vs ergodic (stadium, Sinai) domains. Plot Poincaré section (x, v_x at y=0) to see KAM tori vs chaotic sea. Berry-Tabor vs Bohigas-Giannoni-Schmit conjecture.
Chemical Potential & Phase Coexistence
Grand canonical ensemble: chemical potential μ controls particle number. Maxwell construction for liquid-gas coexistence from van der Waals EOS. Spinodal decomposition and the miscibility gap.
Hyperbolic Geodesics
Geodesics in the Poincaré disk model of hyperbolic geometry: circular arcs perpendicular to the boundary. Click two points to draw the unique geodesic. Parallel postulate fails — infinitely many 'parallels' through a point. Triangle angle sum < 180°.
Schelling Segregation Model
Thomas Schelling's 1971 model: agents move if fewer than t% of neighbors match their type. Even mild preferences (t=33%) produce sharp segregation. A micro-to-macro cautionary tale about emergent discrimination.
Active Nematics
Rod-shaped active particles align with neighbors but have no head/tail distinction. Topological defects (+½ and −½ disclinations) spontaneously create and annihilate. The physics of bacterial films, cytoskeletal flows, and bird flocks.
Riemann Zeta Function
ζ(s) in the complex plane via analytic continuation. Domain coloring reveals zeros on the critical line Re(s)=½. Zoom into the first non-trivial zeros (≈14.13i, 21.02i, 25.01i). Euler product ↔ Dirichlet series comparison.
Surface Tension & Bubble Dynamics
Plateau's laws: soap films meet in threes at 120°. Young-Laplace equation ΔP = 4γ/r for a bubble. Rayleigh-Plesset collapse of a cavitation bubble: pressure spike as radius → 0. Drag bubble endpoints to see film relax.
Cosmic Inflation & CMB
de Sitter expansion: comoving horizon shrinks, then expands. Quantum fluctuations freeze out and become the CMB power spectrum. Inflation solves flatness, horizon, and monopole problems — visualized in one animation.
Diffusion Maps & Manifold Learning
Data points on a Swiss roll, torus, or sphere. Diffusion kernel K_ij = exp(-d²/ε). Eigenvectors of the normalized diffusion operator reveal intrinsic geometry. Compare to PCA which fails on nonlinear manifolds.
Topological Sort & DAG
Build a dependency graph (DAG) by clicking to add nodes and edges. Kahn's algorithm and DFS-based topological sort run step by step. Cycle detection highlights the invalid edge. Applications: build systems, course prerequisites.
Organ Pipe & Standing Waves
Open and closed organ pipe resonances: f_n = n·v/2L (open), f_n = (2n-1)·v/4L (closed). Click to excite a harmonic; hear it via Web Audio API. Chladni-like amplitude profile along the pipe.
Turing Completeness: Rule 110
Stephen Wolfram's Rule 110 cellular automaton is Turing complete (Cook 2004). Build gliders, background patterns, and collisions. See how simple local rules produce universal computation — the threshold of complexity.
Spin Ice & Magnetic Monopoles
Pyrochlore spin ice: Ising spins on corner-sharing tetrahedra obey ice rules (2-in, 2-out). Defects are emergent magnetic monopoles connected by Dirac strings. Flip spins to create/annihilate monopole pairs.
Cellular Potts Model
Glazier-Graner model of biological cell sorting. Cells minimize surface energy; differential adhesion drives Steinberg's sorting hierarchy. Simulate cancer invasion, tissue organization, and cell migration with configurable adhesion matrix.
Density Functional Theory
1D Kohn-Sham DFT for a diatomic molecule. Self-consistent field iteration: solve Schrödinger, compute density, update exchange-correlation potential, repeat. Watch electron density converge. HOMO/LUMO gap displayed.
Simplicial Homology
Build a simplicial complex by clicking to add vertices, edges, and triangles. Compute H₀ (connected components), H₁ (loops), H₂ (voids) in real time. Euler characteristic χ = β₀ − β₁ + β₂ updates live.
Genetic Drift & Fixation
Wright-Fisher model with selection, mutation, and drift. Multiple allele frequency trajectories; fixation probability P = 2s for new mutation (Kimura). Compare neutral drift, selection, and bottleneck effects.
Polarization & Jones Calculus
Jones vectors for linear, circular, elliptical polarization. Multiply by Jones matrices for wave plates (λ/2, λ/4), polarizers, rotators. Poincaré sphere visualization of polarization state. Malus's law interactively.
Quaternion Julia Sets
4D quaternion Julia sets sliced through 3D hyperplanes. Rotate the 3D cross-section in real time. Explore the w=0, w=0.5, w=1 slices of quaternion iteration q → q² + c for different c values.
Chemical Reaction Network
Build a CRN by adding species and reactions. Compute the stoichiometry matrix, deficiency δ = n−l−s. Deficiency Zero Theorem: if δ=0 and weakly reversible, unique stable equilibrium guaranteed. Simulate via mass-action ODEs.
Spherical Voronoi Diagram
Voronoi tessellation on the surface of a sphere (Fortune's algorithm extended). Click to add generators. Color cells by area. Applications: airport catchment areas, virus capsid geometry, plant cell patterns.
Cusp Catastrophe Surface
The cusp catastrophe: V(x) = x⁴/4 − ax²/2 − bx. Control parameters (a,b) sweep through the bifurcation set. Watch the equilibrium surface fold and the discontinuous jump at the cusp. Zeeman's catastrophe machine.
Turbulent Boundary Layer
Velocity profile evolution from laminar to turbulent: viscous sublayer (u⁺=y⁺), log layer (u⁺=2.5 ln y⁺ + 5.5), outer layer. Rouse suspended sediment profile. Drag-adjustable Reynolds number; streak visualization of near-wall turbulence.
Entropy Production & Irreversibility
Comparison of reversible vs irreversible processes: isothermal expansion, heat conduction, mixing. Clausius inequality dS ≥ δQ/T animated. Entropy production rate σ = J·∇(1/T) for heat flow visualized.
Chaos Synchronization
Two identical Lorenz systems with unidirectional coupling x'=x+ε(x-x'). Below coupling threshold: desynchronized chaos. Above threshold: complete synchronization — identical trajectories despite chaos. Lyapunov exponent conditioned on coupling.
Optical Fiber Modes
Step-index fiber: solve characteristic equation for LP modes. Cut-off conditions V_c = 2.405, 3.832, ... Number of modes N ≈ V²/2. Mode field profiles and dispersion curves. Single-mode vs multimode regime with configurable V-number.
Persistent Homology
Point cloud → Vietoris-Rips filtration → persistence diagram. Watch β₀ and β₁ birth/death as ε grows. Bars in persistence barcode encode topological lifetime. See how noise creates short bars, signal creates long ones.
Neural Oscillations & Gamma Rhythm
Wilson-Cowan E-I network: excitatory and inhibitory populations coupled via mean field. Bifurcation from fixed point to limit cycle (gamma, 40Hz). Phase plane with nullclines; power spectrum of E activity.
Electromagnetic Induction
Faraday's law: ε = −dΦ/dt. Move a bar magnet through a coil; induced EMF and current displayed. Lenz's law: opposing force slows the magnet. Eddy current braking in a conducting plate. Adjustable coil turns and magnet strength.
Stochastic Resonance
A subthreshold signal plus noise drives a bistable system. SNR peaks at optimal noise level D* — counterintuitive: adding noise improves signal detection. Models sensory neurons, stock prices, climate bistability.
Prime Number Theorem
π(x) ∼ x/ln(x) and Li(x) = ∫dx/ln(x). Live comparison of actual prime count vs PNT approximation vs Li(x). Error term oscillates around zeros of ζ(s). Chebyshev's θ(x) = Σ_{p≤x} ln p ∼ x.
Floquet Theory & Parametric Resonance
Mathieu equation: ẍ + (a − 2q cos 2t)x = 0. Stability diagram (Ince-Strutt chart) in (a,q) space: tongues of instability emanating from a=n². Click to place parameters; watch amplitude grow or stay bounded. Kapitza pendulum as application.
Morse Theory & Topology
Height function on a surface: track critical points (minima, saddles, maxima) as the level set sweeps up. Morse inequalities: #minima − #saddles + #maxima = χ. Handle attachment at each critical point changes topology.
IFS Fractals & Attractor
Build your own iterated function system: add affine contractions with configurable rotation, scaling, translation. Watch the attractor converge via the chaos game. Measure Hausdorff dimension via box counting.
Acoustic Holography
Phased array of acoustic emitters: adjust phases to steer a focused pressure node to any 3D position. Acoustic levitation principle — trapped particles hover at pressure minima. Bessel beam and vortex beam presets.
Viral Quasispecies Evolution
RNA virus with mutation rate μ near the error threshold μ_c = 1/L. Below: master sequence dominates. Above: quasispecies cloud with error catastrophe. Eigen's master equation on a fitness landscape.
Heat Pump & Refrigeration Cycle
Reverse Carnot cycle on a P-V and T-S diagram. Coefficient of performance COP = Q_c/W. Animate the compressor, condenser, expansion valve, and evaporator. Compare ideal vs realistic Rankine cycle.
Nonlinear Optics: SHG
Second harmonic generation: two photons ω combine to make one at 2ω. Phase matching condition Δk = k(2ω) − 2k(ω) = 0 controls efficiency. Coupled wave equations for E(ω) and E(2ω); quasi-phase matching via periodic poling.
Immune Network & Clonal Selection
Jerne's immune network theory: antibody concentrations evolve via stimulation/suppression. Clonal selection amplifies matching clones. Memory cells persist; cross-reactivity creates attractor landscape. Toggle antigen presence.
Vibrating String Modes
Plucked string via d'Alembert solution: initial condition decomposed into traveling waves. Adjust pluck position and watch harmonics. Compare fixed-fixed (guitar) vs fixed-free (clarinet) boundary conditions. Timbre from harmonic envelope.
Order Book & Market Microstructure
Limit order book with bid/ask queues. Submit market orders, limit orders, cancellations. Watch price impact, spread, and depth. Mid-price as Brownian motion; adverse selection cost for market makers.
Symplectic Integrators
Leapfrog/Störmer-Verlet vs Euler integration of a Hamiltonian system (pendulum, Kepler orbit). Symplectic methods conserve energy drift-free for exponentially long times. Phase portrait shows Euler spiral vs leapfrog closed orbit.
Fluid Sloshing Dynamics
Shallow water equations in a rectangular tank. Tilt or shake and watch sloshing resonance build. Natural frequencies ω_n = √(gkn tanh(kn·h)). Tuned liquid damper application.
Quantum Eraser Experiment
Which-path information destroys interference. Mark one slit with a polarizer; pattern vanishes. Erase the which-path info; pattern returns. Delayed-choice variant. Wave-particle duality visualized.
Domino Wave Dynamics
Chain reaction of falling dominoes. Wave velocity v ∝ (spacing/height)^α. Optimal spacing maximizes wave speed. A small domino can topple one 1.5× taller — energy amplification demonstrated.
Network Motifs & Feed-Forward Loops
Milo et al. 2002: count 3-node subgraphs in biological networks vs random. Feed-forward loop (FFL) is over-represented in transcription networks — implements sign-sensitive delay and pulse generation.
Aizawa Attractor
A toroidal strange attractor with beautiful 3D structure. ẋ=(z-b)x-dy, ẏ=dx+(z-b)y, ż=c+az-z³/3-(x²+y²)(1+ez)+fz·x³. Rotate in 3D, vary parameters, compare Lyapunov exponents.
Polymer Physics: Chain Models
Freely-jointed chain, worm-like chain (WLC), and Gaussian chain models. End-to-end distance R = N^ν·l (Flory exponent ν≈0.588 in 3D). Force-extension curve via Langevin function. DNA stretching.
Lattice QCD Toy Model
1D lattice gauge theory: Wilson action S = -β Σ Re(U_p). Monte Carlo updates via Metropolis algorithm. Plaquette expectation value vs β: confinement and deconfinement phases. String tension from Polyakov loops.
Rayleigh-Taylor Instability
Heavy fluid on top of light fluid: interface is unstable. Fingers of heavy fluid sink; bubbles of light fluid rise. Growth rate σ = √(gk(ρ₁-ρ₂)/(ρ₁+ρ₂)). Most unstable wavelength with surface tension.
Color Theory & Perception
CIE 1931 chromaticity diagram. Mix RGB primaries and see where they land. Metamers: different spectra, same color. Opponent channels (L-M, S-(L+M), L+M). Color blindness simulation.
Takens Embedding Theorem
Recover strange attractor topology from a 1D time series via delay embedding: x(t), x(t+τ), x(t+2τ). Optimal delay τ from mutual information minimum. False nearest neighbors. Lorenz, Rössler, ECG examples.
Voronoi Pulse
Dynamic Voronoi diagrams with glowing color gradients — click to reseed, cells pulse continuously
Double Pendulum
Chaotic double pendulum with colorized trail, adjustable mass and length sliders, reset button
Henon Attractor
Henon map strange attractor with draggable a and b parameters, colorized by iteration count
Lorenz Butterfly
Lorenz attractor 3D projection with auto-rotation, multiple trajectories, and sigma/rho/beta sliders
Abelian Sandpile
Click to drop sand grains and watch self-organized criticality — avalanches cascade through the grid
Forest Fire
Forest fire cellular automaton — trees grow, lightning strikes, and fires spread in tunable waves
Ising Model 2D
2D Ising spin model with temperature slider — watch ferromagnetic phase transition at Tc near 2.269
Boids Ocean
Underwater boids flocking simulation with fish shapes — click to scatter, adjust cohesion and alignment
Neural Oscillators
Kuramoto model of coupled neural oscillators — raise coupling K and watch synchronization emerge
Rumor Spreading
SIR-style rumor propagation on a small-world network — click nodes to seed, tune spread and forgetting rates
2D Turbulence
Simplified Navier-Stokes fluid turbulence with vortex injection and vorticity coloring
Hex Maze
Hexagonal grid maze via recursive backtracker, solved live with animated A* pathfinding
Diffusion-Limited Aggregation
Random walkers stick to a growing crystal — fractal branching emerges from chaos
Eden Growth Model
Random cluster growth from center with fractal boundary, colored by time of addition
Ballistic Deposition
Particles fall and stick to a rough surface — log-log roughness growth plot included
Random Graph Evolution
Erdős–Rényi graph gains edges in real time — giant component emerges at p = 1/n
Scale-Free Network
Barabási–Albert preferential attachment — hubs emerge as the network grows live
Opinion Dynamics
Deffuant bounded-confidence model — agents cluster into opinion groups, histogram live
Sand Dunes
Barchan dune migration via wind transport and avalanche — dunes form and migrate
Plasma Instability
Two-stream plasma instability — phase space portrait shows vortex formation as instability grows
Convection Rolls
Rayleigh-Bénard convection rolls in a heated fluid layer
Turing Patterns II
Reaction-diffusion with adjustable activator/inhibitor — spots, stripes, labyrinths
Gray-Scott II
Gray-Scott reaction-diffusion with spot, stripe, mitosis, and coral preset modes
Lattice Gas Automaton
HPP lattice gas — particles collide on a grid, fluid behavior emerges from simple rules
Q-State Potts Model
Generalized Ising model with q spin states — domain coarsening at variable temperature
Directed Percolation
Bonds flow downward only — watch active vs absorbing phase transition near the critical point
Branching-Annihilating Walks
Particles branch and annihilate in pairs — parity-conserving critical dynamics
Threshold Automata
Bootstrap percolation — cells activate when enough neighbors are active, cascades unfold
Stochastic Sandpile
Self-organized criticality with noise — randomized toppling builds power-law avalanches
Elastic Billiards
Newtonian elastic collisions on a frictionless table — click to throw balls
Stadium Billiard
Bunimovich stadium billiard showing chaotic vs regular trajectories — click to launch balls
Lorentz Gas
Point particle bouncing off circular scatterers — ergodic diffusion with trail visualization
N-Body Galaxy
Gravitational N-body simulation — spiral arms emerge, click to add stars, pan and zoom
Epicycles
Ptolemaic epicycles reconstruct any shape via Fourier decomposition — draw your own curve
3D Strange Attractor Explorer
Six strange attractors in 3D — Lorenz, Chen, Halvorsen, and more, colorized by velocity
Dadras Attractor
5-parameter chaotic Dadras system in auto-rotating 3D — tune parameters to reshape the attractor
Sprott Attractors
All 19 Sprott attractors (A through S) in 3D — select any, colored by z-axis height
Rössler Bifurcation
Live 3D Rössler attractor alongside a real-time bifurcation diagram as parameter c sweeps
Hyperchaos
Hyperchaotic Rössler 4D system with two positive Lyapunov exponents — toggle projection planes
Colpitts Oscillator
Chaotic Colpitts LC oscillator — 3D phase portrait with Q and γ parameter sliders
Glycolytic Oscillator
Biochemical oscillations in glycolysis — phase portrait with nullclines and enzyme kinetics sliders
Selkov Model
Two-variable autocatalytic oscillator — animated limit cycle with multiple trajectories
Spatial Predator-Prey
Lotka-Volterra on a 2D grid — population waves and spiral patterns with diffusion
Spatial SIR Epidemic
Infection spreading on a 2D grid — click to seed, watch wavefronts, adjust β/γ
SEIR Epidemic Model
ODE phase portrait and time series — R₀ slider shows endemic equilibrium vs extinction
SIR on a Network
Force-directed network with S/I/R colored nodes — click to infect, Erdős-Rényi, scale-free, small-world
Game of Life+
Conway's Game of Life with draw mode, speed control, and pattern library: glider, pulsar, Gosper gun
Brian's Brain
Three-state cellular automaton — firing, refractory, ready — chaotic traveling waves
Langton's Ant Extended
Multiple ants with different rule strings — emergent highways and spirals in multiple colors
Turmite
2D Turing machine with configurable transition tables — spiral, snowflake, highway presets
Cyclic Automata
Spiral waves emerge as cells cycle through N states eating the next
Majority Vote
Each cell adopts neighbors' majority opinion — watch consensus form or fragment
Axelrod Tournament
Classic strategies compete in iterated Prisoner's Dilemma — TFT dominates
Rock Paper Scissors CA
Spatial RPS dynamics with spiraling territorial fronts and noise
Spatial Prisoner's Dilemma
Cooperators vs defectors on a grid — cluster dynamics determine survival
Evolutionary Game Theory
Hawk, Dove, and Bourgeois compete until an evolutionarily stable strategy emerges
Wealth Distribution
Fair random exchanges spontaneously produce Pareto inequality — Gini live
Pedestrian Dynamics
Social force model — crowds navigate bottlenecks with density heatmap
3D Flocking Boids
Boids flock in 3D space with separation, alignment, and cohesion forces
Termite Mound
Stigmergic agents cluster wood chips via pheromone — complexity from simplicity
★ Lab #1500 — Grand Symphony
A milestone celebration: Lorenz attractor, reaction-diffusion, and boids in one grand canvas
Bee Waggle Dance
Honey bee figure-eight dance encoding direction and distance to food source
Cardiac Spiral Wave
Spiral wave of electrical excitation in heart tissue via FitzHugh-Nagumo model
Excitable Media
BZ reaction-inspired CA with target waves and spiral waves — click to seed
Actin Polymerization Waves
Cytoskeletal wave dynamics in cells — activator-inhibitor traveling waves on membrane
Min Protein Oscillations
MinD/MinE pole-to-pole oscillations select cell division site in E. coli
Golden Ratio Spiral
Logarithmic spiral φ = 1.618... with Fibonacci rectangles and nautilus overlay
Golden Angle Phyllotaxis
Seeds placed at 137.5° golden angle reveal emergent Fibonacci spirals
Penrose P3 Tiling
Aperiodic rhombus tiling with 5-fold symmetry — click to subdivide fractally
Quasicrystal Diffraction
5/7/8/10/12-fold quasicrystal — real-space tiling alongside simulated electron diffraction
Aperiodic Tiling
Penrose rhombus substitution tiling — no repeating period, infinite non-periodic detail
Escher Symmetry Groups
All 17 wallpaper groups visualized with animated pattern examples — cycle through the complete classification
Wave Function Collapse
Constraint-propagation procedural generation — watch tiles collapse from superposition one by one
Nonogram Solver
Animated Picross puzzle solver — constraint propagation fills in rows and columns step by step
Maze Algorithms
Five maze generation algorithms side-by-side — DFS, Prim's, Kruskal's, Wilson's, Aldous-Broder
Pathfinding Compare
Draw walls and compare A*, Dijkstra, BFS, DFS, and Greedy on an interactive grid simultaneously
Binary Tree Visualizer
Animated BST and AVL tree insert, delete, search with balance factors and rotation labels
Graph Traversal
Build your own graph and watch BFS and DFS animate node discovery order in real time
Hash Table Visualizer
Animated insert, lookup, and delete showing separate chaining vs open addressing and load factor
Red-Black Tree
Animated red-black tree insertion with color-flips, rotations, and black-height tracking
3D Fractal Tree
Recursive 3D branching tree with draggable rotation, depth, angle, and taper sliders — each generation a different color
3D L-System
3D turtle-graphics L-system renderer with plant and tree presets, axiom/rules display, and drag-to-rotate
Space Colonization
Organic tree growth via space colonization — branches grow toward random attractor points and consume them
Marching Squares
Animated metaball isosurfaces extracted by the marching squares algorithm with adjustable threshold
Signed Distance Field
Interactive SDF builder — drag shapes, apply union/intersection/difference with smooth blending visualization
Sphere Tracing
Real-time ray marching renderer of 3D SDF scenes with soft lighting and ambient occlusion approximation
Tectonic Plates
Voronoi-based plate simulation — drifting plates collide to build mountains and diverge to form rift zones
River Networks
Flow accumulation on a procedural heightmap reveals drainage basins and river networks emerging from terrain
Wind Erosion
Aeolian sand transport simulation — saltation and avalanching form migrating dunes in real time
1D Schrödinger Equation
Animated quantum wavepacket in a potential well via split-operator method — real, imaginary, and probability density
2D Schrödinger
Quantum wavepacket in a 2D box — click to place potential barriers and watch diffraction and interference
Hydrogen Orbitals
Probability density cross-sections for hydrogen atom orbitals 1s through 3d, colored by quantum phase
Feynman Path Integral
All paths from A to B weighted by e^{iS/ℏ} — see constructive interference pick out the classical trajectory
Relativistic Particle
Particles at fractions of c with live length contraction, time dilation, and Lorentz factor γ
Special Relativity
Twin paradox, length contraction, and time dilation shown simultaneously with adjustable velocity slider
Lorentz Transform
Two reference frames with events in both — watch simultaneity break down as you animate the boost
Minkowski Diagram
Draw worldlines on a spacetime diagram — light cones, simultaneity lines, adjustable boost velocity
Black Hole Orbits
Test particle orbits in Schwarzschild spacetime — ISCO, photon sphere, and inspiral with adjustable angular momentum
Fourier Transform
Draw a time-domain signal and watch the live FFT spectrum update — reconstruct with adjustable components
Discrete Fourier
DFT as rotating phasors summing to reconstruct a signal — N components, adjustable frequencies, magnitude spectrum
Hilbert Curve II
Animated space-filling Hilbert curve up to order 8, colored by position
Worley Noise
Cellular noise with animated feature points — F1, F2, and F2-F1 modes
Simplex Noise
2D simplex noise with octaves, persistence, lacunarity and animated time flow
Curl Noise
Thousands of particles swirling through divergence-free curl noise flow fields
Turbulent Noise
Kolmogorov turbulence spectrum — particles advected through field with energy plot
Fractal Noise (fBm)
Fractional Brownian motion explorer — Hurst exponent controls terrain roughness
Value vs Perlin Noise
Side-by-side comparison of value noise and Perlin gradient noise artifacts
Domain Warping
Iterated domain warping f(p+f(p+f(p))) reveals organic flowing structures
fBm Landscape
Procedural terrain from fractional Brownian motion with animated cloud layer
Ridged Multifractal
Sharp mountain ridges via Musgrave's ridged multifractal noise — animated flythrough
Topological Phases
Phase transitions on a lattice with winding numbers and edge states
Sierpinski Variations
Morphing fractals: Sierpinski, Apollonian gasket, Koch snowflake
Cyclic Cellular Automaton
Rock-paper-scissors automaton producing mesmerizing spiral waves
Gray-Scott Reaction
Reaction-diffusion chemistry: spots, labyrinths, coral, and waves
Voronoi Noise
Animated Voronoi cells with multiple distance metrics
Stippling Art
Weighted Voronoi stippling with Lloyd relaxation
Maze Generation
Watch four classic maze algorithms build labyrinths in real time
Flow Field
Particle systems following curl noise through color-coded fields
Epicycles & Fourier
Draw any shape; watch Fourier epicycles reconstruct it
Clifford Attractor
Strange attractor fractal clouds with interactive parameter control
Quantum Harmonic Oscillator
Wave functions and probability densities for quantum energy levels
Quantum Zeno Effect
How frequent measurement freezes quantum decay
Double Pendulum Chaos
Butterfly effect: near-identical pendulums diverge into chaos
Driven Oscillator
Resonance curves and amplitude response of a driven spring
Resonance Tower
Stack of springs driven to resonance — energy propagation
Fractal Antenna
Koch snowflake antenna and multi-band resonance
Wave Interference
Drag point sources and watch 2D interference patterns form
Thin Film Optics
Soap bubble iridescence from thin-film interference
Light Polarization
E-field oscillation, Malus's law, and polarizer filters
Laser Cavity Modes
Standing waves, longitudinal modes, and gain curves in a laser cavity
Neural Network
Animated forward pass through configurable deep network layers
Gradient Descent
SGD, momentum, and Adam racing down a 3D loss landscape
Backpropagation
Step-by-step forward and backward pass with gradient pulses
Attention Heatmap
Transformer self-attention weights visualized as animated heatmap
Decision Boundary
Live logistic regression boundaries as you draw training data
SVM Playground
Max-margin hyperplane, support vectors, and kernel tricks
K-Nearest Neighbors
Interactive KNN with live decision regions and neighbor visualization
K-Means Clustering
Watch Lloyd's algorithm converge on animated point clouds
PCA Explorer
Principal components, covariance ellipses, and explained variance
Activation Functions
Gallery of neural activation functions with animated derivatives
Markov Chain
Random walker on states with animated transition probabilities
Coupon Collector
How many tries to collect all N coupons? Race to find out
Gambler's Ruin
Random walkers racing between ruin and riches
Branching Process
Galton-Watson trees growing, exploding, or going extinct
Power Law Explorer
Heavy tails on linear vs log-log scale — Zipf rank-frequency
Sorting Algorithms
Race bubble, merge, quick, and heap sort side by side
Pathfinding
Draw mazes; watch A*, Dijkstra, BFS, DFS find the exit
Binary Heap
Tree and array views with animated insert and extract-min
Dynamic Programming
Watch DP tables fill cell by cell for classic problems
Trie Explorer
Type words to grow a prefix tree; search highlights subtrees
Music Synthesizer
Play a glowing keyboard with ADSR envelope and live waveform
Chord Progressions
Hear and see I-IV-V and jazz progressions on a piano keyboard
Waveform Shaper
Draw a transfer curve and hear your custom distortion effect
Spectral Analyzer
Real-time FFT spectrum with glowing bars and harmonic control
Rhythm Sequencer
16-step drum machine with kick, snare, hi-hat in a neon grid
Fourier Synthesis
Build any waveform by mixing harmonics — hear it change live
Standing Waves
Harmonics on a string — animated nodes and audible tones
Instrument Harmonics
Harmonic spectra of violin, piano, flute, guitar — hear the difference
Celestial Mechanics
Kepler orbits with equal-area sweeps and eccentricity control
Solar System
All 8 planets with accurate relative periods and sizes
Navier Lattice
Lattice Boltzmann fluid simulation with vortices, jets, and turbulence emerging from discrete velocity collision rules.
Euler Vortex
Point vortex dynamics under the 2D Euler equations — click to add vortices and watch them braid around each other.
Spinor Field
A field of quaternion spinors visualized as arrows: rotate 720° to return home, unlike any ordinary vector.
Berry Phase
Geometric phase on the Bloch sphere: a quantum state traces a closed loop and acquires a phase equal to half the enclosed solid angle.
Adiabatic Loop
Quantum adiabatic evolution of a two-level system, tracking dynamic and geometric Berry phase accumulation in real time.
Quantum Holonomy
Parallel transport of a tangent vector on a curved surface — the holonomy angle reveals the enclosed Gaussian curvature.
Symplectic Map
Phase space portraits of area-preserving maps: click to seed orbits and watch KAM tori, resonance islands, and chaotic seas emerge.
Lyapunov Field
Lyapunov exponent landscape where each pixel encodes the chaos measure of a dynamical system at that parameter point.
Julia Morph
Julia sets morphing continuously as the complex parameter c traces paths through the Mandelbrot set boundary.
Möbius Flow
Continuous Möbius transformations flowing over the complex plane — circles and lines map to circles, conformal symmetry made visible.
Toda Lattice
Nonlinear spring-mass chain with exact soliton solutions — localized waves that pass through each other unchanged, revealing integrability.
Fermi-Pasta-Ulam
Energy pumped into one mode of a nonlinear chain refuses to thermalize — it returns. The FPU recurrence that mystified physicists for decades.
Lorentz Map
1D iterated maps with cobweb diagrams and bifurcation portraits — watch period-doubling cascades lead to chaos as a parameter varies.
Homoclinic Orbit
A trajectory that escapes and returns to the same saddle point — the intersection of stable and unstable manifolds hides a horseshoe of chaos.
Separatrix Web
Energy contours of 2D Hamiltonians form a web of separatrices dividing phase space — click to launch trajectories through the portrait.
Floquet Spectrum
Quasi-energy bands of a periodically driven quantum system — scan driving amplitude to watch Arnold tongues and bandwidth collapse emerge.
Mode Locking
The circle map shows Arnold tongues where a forced oscillator locks to rational winding numbers — the devil's staircase in parameter space.
Devil's Staircase
The Cantor function rises from 0 to 1 while remaining constant almost everywhere — a continuous function that increases only on a measure-zero set.
Continued Fractions
Every real number unfolds into a tower of integers. Convergents give the best rational approximations, tracing a path through the Stern-Brocot tree.
Calkin-Wilf Tree
A binary tree that enumerates every positive rational exactly once — explore the structure, highlight paths, and compare with the Stern-Brocot tree.
Wigner Crystal
Electrons crystallize into a triangular lattice when Coulomb energy dominates kinetic energy — watch the phase transition as density lowers and r_s increases.
Quantum Corral
Electron standing waves trapped inside a ring of Fe atoms on Cu(111), reproducing the iconic IBM STM image. Bessel function modes confined by the corral.
Scanning Tunneling Microscopy
STM surface topology scan rendered in 3D perspective. Tunneling current I ∝ exp(-2κd) maps surface height with atomic resolution.
Atomic Orbital Viewer
3D probability density clouds for s, p, d, and f orbitals. Drag to rotate and explore phase lobes of hydrogen wavefunctions ψ_nlm.
Hydrogen Wavefunction
Radial probability density r²|R_nl|² for all (n,l) states up to n=6. Visualize how electron shells emerge from quantum numbers.
Spherical Harmonics
|Y_lm|² plotted as radial displacement on a sphere — the angular eigenfunctions of L² and L_z. Sliders for l and m, drag to rotate.
Angular Momentum Addition
Clebsch-Gordan decomposition of j₁ ⊗ j₂ into irreducible representations, visualized as a color-coded coefficient matrix with block structure.
Larmor Precession
A magnetic dipole precesses around an external B field at ω_L = γB. Animate the precession cone for proton, electron, or neutron — the basis of NMR and MRI.
Cyclotron Motion
Charged particles spiral in a uniform magnetic field at ω_c = qB/m. Watch circular orbits and see Landau level energies quantize on the side panel.
Landau Level Spectrum
The Landau fan diagram E_n = ℏω_c(n+½) vs magnetic field, alongside the density of states showing broadened peaks — the physics behind quantum Hall effect.
DNA Double Helix
Animated 3D double helix with A-T and G-C base pairs rotating in perspective. Drag to spin, adjust speed and pitch — the molecule of life made visceral.
Neural Spike Raster
Integrate-and-fire neuron network with live raster plot. Each row is a neuron; dots are spikes. Adjust drive, coupling, and noise to see synchrony emerge.
Reaction-Diffusion 3D
Gray-Scott reaction-diffusion rendered as an isometric height map — concentration becomes terrain. Switch presets to see coral, spots, stripes, and worms.
Turing Pattern Formation
Spots, stripes, labyrinths, and coral grow from a flat sheet via Turing instability. Alan Turing's 1952 prediction rendered live with tunable parameters.
Morphogen Gradient
The French flag model: a morphogen diffuses from a source and degrades, creating a concentration gradient that assigns cell fates by threshold crossing.
Replicator Dynamics
Evolutionary game theory on the 3-strategy simplex. Edit the payoff matrix, choose rock-paper-scissors or hawk-dove, and watch orbits trace the simplex.
Genetic Drift
Wright-Fisher model with multiple populations. Allele frequencies wander and fix by chance alone — watch population size control the pace of stochastic loss.
Neutral Evolution
Kimura neutral theory in action: synonymous vs nonsynonymous substitutions, the molecular clock, and the site frequency spectrum across simulated populations.
Phylogenetic Tree
Interactive radial and cladogram trees for vertebrates, primates, plants, and eukaryotes. Click leaves for taxon info; toggle rotation and layout modes.
Codon Wheel
All 64 codons arranged in the classic genetic code wheel. Click any codon to see its amino acid, one-letter code, biochemical properties, and synonymous codons.
N-Body Gravity
Click to add masses and watch gravitational orbits form in real time. Collisions merge bodies; sliders control G, mass, and trail length.
Roche Limit
Tidal forces shred a satellite into a ring when its orbit dips inside the Roche limit. Adjust mass ratio and density to see the disruption threshold.
Accretion Disk
Spiraling gas falls onto a compact object. Inner edge glows white-hot, Doppler shift colors approaching and receding material blue and red.
Black Hole Shadow
Simplified photon orbit visualization around a Kerr black hole. Adjust spin to see the shadow deform; the photon sphere glows at r = 3M.
Stellar Evolution
A star lives and dies before your eyes — molecular cloud to main sequence to white dwarf, neutron star, or black hole depending on its mass.
Hertzsprung-Russell Diagram
Interactive temperature-luminosity diagram with real named stars. Hover for data, click to see stellar type, animate an evolutionary track.
Stellar Nucleosynthesis
Periodic table colored by cosmic origin — Big Bang, cosmic rays, AGB stars, massive stars, supernovae, and neutron star mergers. Click for fusion chains.
Supernova Remnant
Sedov-Taylor blast wave expanding into the ISM. Filaments, knots, a pulsar wind nebula, and a light echo evolve as the remnant sweeps up the interstellar medium.
Pulsar Timing Array
Five millisecond pulsars act as galactic clocks. Watch the rotating lighthouse beams and correlated timing residuals reveal a gravitational wave background.
Gravitational Waves
Binary black hole inspiral, merger, and ringdown. Spacetime grid stretches and squeezes in plus polarization while the chirp signal builds to coalescence.
Goldbach Comet
Visualize Goldbach's conjecture — count the ways each even number splits into two primes and watch the comet shape emerge.
Mersenne Primes
Explore Mersenne primes (2^p−1), animate the Lucas-Lehmer primality test, and see all 51 known giants on a log scale.
Number Partition
Partition integers into Young tableaux, explore the explosive growth of p(n), and see Ramanujan's asymptotic formula in action.
Integer Sequences
Tour famous OEIS sequences — Fibonacci, Catalan, Bell, primes — with ratio convergence and log-scale growth plots.
Fourier Transform
Add frequency components and watch the DFT spectrum update live. Build square waves, sawtooths, and custom waveforms.
Discrete Cosine Transform
See how the DCT decomposes signals into cosine bases. Interactive JPEG compression demo with quality slider on 8×8 blocks.
STFT Spectrogram
Synthesize chirps, AM/FM signals, and beats — watch the rolling time-frequency spectrogram update in real time.
Convolution Demo
Animate the sliding kernel interpretation of convolution. Mix box, Gaussian, triangle, and decay shapes and see the output live.
Z-Transform Poles & Zeros
Place poles and zeros on the complex unit disk and instantly see the resulting digital filter frequency response and phase.
Laplace Transform
Explore s-domain pole-zero maps for RC, RLC, PID, and custom systems. Compare step response, impulse response, and Bode plots.
Shannon Entropy Visualizer
Drag probability bars to shape a discrete distribution and watch H(X) update live — see how far you are from maximum entropy.
Binary Symmetric Channel
Tune the crossover probability and see Shannon capacity C = 1 − H(p), with a live animation of bits flipping through the noisy channel.
Shannon Capacity Limit
C = B log₂(1 + S/N): explore the capacity surface over bandwidth vs SNR, and see where BPSK, QAM, and other modulations operate.
Error Correction Demo
Send text through a noisy channel, compare raw vs repetition-coded output, and build BER vs noise curves in real time.
Hamming (7,4) Code
Encode 4 data bits into a 7-bit Hamming codeword, inject single-bit errors, and watch syndrome detection correct them automatically.
Turbo Decoder
Visualize iterative belief propagation between two SISO decoders exchanging extrinsic information — the secret behind near-Shannon-limit codes.
LDPC Tanner Graph
Explore LDPC bipartite factor graphs with variable and check nodes. Click nodes to trace connections, then run belief propagation.
Mutual Information I(X;Y)
Adjust a 2D joint distribution and watch I(X;Y) = H(X) + H(Y) − H(X,Y) computed live alongside joint heatmap and marginals.
KL Divergence Explorer
Shape two distributions with sliders and see D_KL(P||Q) and D_KL(Q||P) diverge — a visceral demonstration of information-theoretic asymmetry.
Rate-Distortion Theory
Trace the R(D) = ½ log₂(σ²/D) curve for a Gaussian source, see where JPEG, MP3, and PCM operate, and explore the compression tradeoff.
PageRank Visualizer
Watch iterative power method converge on a directed graph — nodes sized by PageRank score, add/remove edges live.
Small-World Network
Watts-Strogatz model with rewiring slider — see clustering coefficient and average path length shift in real time.
Minimum Spanning Tree
Kruskal's and Prim's algorithms animated step-by-step on a random weighted graph — toggle and compare.
Network Centrality Map
Same graph rendered by degree, betweenness, closeness, or eigenvector centrality — nodes sized and colored by score.
Knapsack DP Visualizer
Dynamic programming table built cell by cell — watch optimal values propagate and see which items are chosen.
Sorting Algorithm Visualizer
Bubble, insertion, merge, quick, and heap sort animated with live comparison/swap counters and speed control.
Pathfinding Algorithms
Draw walls on a grid and animate A*, Dijkstra, BFS, and DFS — explored cells and final path shown in distinct colors.
B-Tree Visualizer
Insert keys and watch the B-tree split and rebalance — configurable minimum degree t from 2 to 5.
Hash Collision Demo
Insert strings and visualize chaining vs linear probing — see load factor and collision count update live.
Suffix Array Explorer
Enter a string to see all sorted suffixes, the SA index, LCP array, and pattern matching highlighted interactively.
Crystal Structure Factor
Interactive Fourier synthesis of crystal diffraction patterns — tune basis atoms and see reciprocal-space spots and real-space electron density.
Fluid Vortex Dynamics
Point vortex system with Hamiltonian mechanics — click to add vortices and watch them orbit, pair, and translate under mutual induction.
Molecular Orbital Viewer
LCAO visualization of H₂, N₂, O₂ bonding and antibonding orbitals — adjust bond length and watch wavefunctions reshape.
Auction Theory
First-price vs second-price (Vickrey) auction bidding dynamics with revenue equivalence theorem simulation.
Attention Field
Transformer self-attention visualization — explore query/key/value geometry and how tokens attend to each other.
Type Theory Explorer
Propositions-as-types, dependent types, and proof terms — navigate the Curry-Howard correspondence interactively.
Maxwell's Equations
Animated curl and divergence of E and B fields across five scenarios — point charges, dipoles, EM waves, and capacitors.
Turing Patterns
Gray-Scott reaction-diffusion system generating biological spots, stripes, and labyrinths — tune feed and kill rates live.
Crystal Phonons
Lattice dispersion relations for monatomic and diatomic crystals — visualize acoustic and optical phonon branches with animated atom motion.
Wave Packet Revival
Quantum wave packet spreading, collapse, and fractional revival in an infinite square well — watch the autocorrelation revive.
Totalistic Cellular Automata
Outer totalistic rules beyond elementary CA, including Life-like and continuous rules with interactive rule editor.
Eigenvalue Methods
Power iteration, inverse iteration, and QR algorithm visualized step-by-step on 2×2 and 3×3 matrices.
Polyhedra Explorer
Rotating 3D Platonic and Archimedean solids with face, edge, and vertex inspection.
Neural ODE
Continuous-depth network as a learned vector field, showing trajectory integration and phase portrait.
SPH Fluid
Smoothed particle hydrodynamics simulation of 2D fluid with pressure, viscosity, and interactive forces.
Arithmetic Dynamics
Iteration of rational maps z→z²+c over the complex plane and p-adic Mandelbrot-like parameter spaces.
Boltzmann Gas
Maxwell-Boltzmann speed distribution emerging live from elastic particle collisions in 2D.
Outer Totalistic CA
2D outer totalistic rules (Game of Life family) with interactive rule editor and classic pattern library.
Nuclear Decay Chains
Interactive decay chain explorer for U-238, Th-232, and U-235 with half-lives and Bateman activity curves.
Tensor Networks
MPS and MERA diagram builder showing tensor contraction, bond dimension, and entanglement entropy.
Möbius Strip Flow
Vector field flow on a Möbius band illustrating non-orientability and parallel transport on a non-orientable surface.
Hypercube Projection
Rotating 4D tesseract projected to 2D with perspective, orthographic, and 3D/4D/5D dimension modes.
Lévy Flights
Compare Gaussian random walks vs Lévy stable distributions with tunable α, revealing superdiffusion and heavy tails.
SIR Epidemic Model
Interactive SIR/SEIR/SIRS epidemic dynamics with R₀ slider, vaccination threshold, and phase portrait.
Gradient Descent Landscape
2D loss landscape explorer with animated SGD, Momentum, Adam, and RMSProp trajectories on classic test functions.
Galois Field Arithmetic
GF(2ⁿ) addition and multiplication tables, primitive polynomials, field calculator, and extension tower structure.
Catalan Paths
Five manifestations of Catalan numbers: Dyck paths, parenthesizations, polygon triangulations, ballot problem, and binary trees.
Riemann Surface Branch Cuts
Complex functions √z, log(z), and z^(1/3) visualized as colored sheets with interactive branch cut explorer.
Random Matrix Eigenvalues
GUE, GOE, and Poisson eigenvalue spacing distributions with Wigner surmise and level repulsion visualization.
Dirichlet Series
Partial sums of ζ(s), L-functions, and Dirichlet characters visualized in the complex plane with zero-hunting.
Conformal Map Explorer
Joukowski, Möbius, z², and exp(z) maps with live grid distortion showing angle-preserving conformal structure.
Genetic Algorithm TSP
Traveling salesman problem solved via crossover and mutation, with live tour animation and fitness evolution plot.
Heisenberg Spin Chain
1D quantum spin chain with magnon dispersion, spin-spin correlations, and entanglement entropy scaling.
Lorenz Manifold
Stable and unstable manifolds of the Lorenz attractor, Poincaré sections, and sensitive dependence on initial conditions.
Tropical Curves
Tropical polynomial amoebas, max-plus arithmetic, and piecewise-linear geometry over the tropical semiring.
Persistent Homology Live
Draw point clouds and watch Vietoris-Rips filtration build topology — persistence diagrams and barcodes update live.
Market Clearing
Supply and demand curve intersection with price discovery dynamics, surplus visualization, and demand/supply shocks.
Continuous CA
Smooth cellular automata with continuous states — Smooth Life, reaction-diffusion, Lenia, and wave propagation rules.
Navier-Stokes Vorticity
2D vorticity-streamfunction fluid solver with lid-driven cavity, adjustable Reynolds number, and interactive vortex injection.
Knot Invariants
Trefoil, figure-eight, and torus knots with Gauss codes, Reidemeister moves, and Alexander and Jones polynomials.
Feigenbaum Tree
Period-doubling bifurcation diagram with Feigenbaum constants δ≈4.669 and α≈2.502, cobweb, and orbit explorer.
Betti Numbers
Interactive simplicial complexes with live β₀/β₁/β₂ computation, Euler characteristic, and boundary matrix.
Contact Geometry
Contact structures on ℝ³, Legendrian knots, Reeb vector field, and overtwisted vs tight visualization.
Mean Field Ising
Self-consistency m=tanh(βJzm), phase transition at T_c=Jz/k, hysteresis, and free energy landscape.
Schrödinger Cat State
Quantum superposition Wigner function, decoherence dynamics, and Q-function phase space visualization.
Derangement Explorer
Hat-check problem: D_n via inclusion-exclusion, animated permutations, and convergence to 1/e.
Spectral Clustering
Graph Laplacian eigenvectors, Fiedler vector bisection, and k-way clustering on generated point clouds.
Chromatic Polynomial
Deletion-contraction algorithm on interactive graphs, chromatic polynomial evaluation, and graph coloring.
Birth-Death Process
M/M/1 queue simulation, stationary distribution π_n=ρⁿ(1−ρ), Gillespie algorithm sample paths.
Lattice Gauge Theory
U(1) gauge field on 2D lattice, Wilson loop, plaquette action, and Monte Carlo updates with confinement.
Kolmogorov Cascade
Energy cascade from large to small eddies with k^(-5/3) spectrum
Turbulence Spectrum
Live power spectrum of turbulent flow — the Kolmogorov -5/3 law
Vortex Stretching
Angular momentum amplifies vorticity — the key mechanism of 3D turbulence
Reynolds Decomposition
Mean flow plus fluctuations — Reynolds stress and turbulence modeling
Intermittency & Multifractal
Anomalous scaling exponents and the She-Leveque multifractal spectrum
Energy Cascade 3D
Richardson's cascade — big whirls feeding little whirls in 3D
SLE & Percolation
Schramm-Loewner evolution and conformal invariance at criticality
Lyapunov Spectrum
Divergence of nearby trajectories and the geometry of chaos
Kolmogorov Microscale
Length, time, and velocity scales where viscosity dissipates turbulent energy
Structure Functions
S_p(ℓ) scaling exponents — K41 vs anomalous turbulence statistics
Spin Foam
Loop quantum gravity networks and spacetime discretization
Bose-Einstein Condensate
Phase transition where bosons collapse into a macroscopic quantum ground state
Quantum Zeno Effect
Frequent measurement freezes quantum decay — a watched pot never decays
Decoherence
Quantum superposition dissolving into classical mixture via environment coupling
Entanglement Entropy
S = −Tr(ρ_A log ρ_A) — bipartite entanglement and Schmidt decomposition
Path Integral
Feynman's sum over all paths — each weighted by e^(iS/ℏ)
Feynman Diagram Builder
Interactive QED diagram construction with symbolic amplitude expressions
Renormalization Group Flow
Wilsonian RG flow lines, fixed points, and running coupling constants
Higgs Field
Mexican hat potential and spontaneous electroweak symmetry breaking
Dark Energy Expansion
Friedmann equation — scale factor a(t) for matter, radiation, and Λ-dominated universes
Wave Diffraction
Single and double slit Fraunhofer diffraction — adjust wavelength, slit width, and separation to see interference fringes in real color.
Soap Film Geometry
Minimal surfaces spanning wire frames — rotating polyhedra with faces colored by mean curvature, iridescent like real soap films.
Gravitational Lens
Einstein ring lensing — click to move the lens mass, tune Einstein radius and source offset to create arcs and full rings.
Fluid Vortex
Click to plant point vortices in 2D flow — watch particles spiral and vortices mutually advect via Kirchhoff's equations.
Quantum Tunneling
Gaussian wavepacket hitting a potential barrier — Schrödinger equation via finite differences, live transmission and reflection fractions.
Double Pendulum
Chaotic double pendulum with glowing trail history — add shadow copies to watch Lyapunov divergence unfold in real time.
Sandpile Automaton
Abelian sandpile model — drop grains, watch avalanches cascade at all scales. Self-organized criticality (Bak-Tang-Wiesenfeld 1987).
Spiral Galaxy
Lin-Shu density wave model — stars orbit at differential speeds while spiral arms co-rotate as persistent density patterns.
Sunflower Phyllotaxis
Golden angle seed packing — nudge the divergence angle away from 137.508° to watch Fibonacci spirals give way to spokes and gaps.
Hexagonal Automaton
Cellular automaton on a hex grid with four rule sets — click to paint cells, watch crystals grow, gliders drift, and life-like patterns emerge.
Elastic Collisions
Hard-sphere elastic collisions with exact impulse resolution — conservation of kinetic energy and momentum, color-coded by speed.
Elastic Wave Propagation
2D FDTD grid simulation of longitudinal and transverse elastic waves. Click to emit pulses.
Standing Wave Modes
String and drum membrane standing waves with adjustable modes. Nodes, antinodes, and WKB.
Complex Domain Coloring
Domain coloring of z², sin(z), Γ(z), ζ(z) and more. Hue encodes argument, brightness encodes magnitude.
Airy Functions
Ai(z) and Bi(z) solutions to y''=zy with WKB approximation overlay and classical turning point.
Tschirnhaus Curves
Lissajous, Tschirnhaus cubic, rose curves, folium of Descartes, and strophoid with live parameter sliders.
Catenary Chains
Hanging chain under gravity traces y = a·cosh(x/a). Compare with parabola, visualize tension forces.
Roulette Curves
Epicycloids, hypocycloids, and trochoids from rolling circles. Adjustable R, r, d ratios.
Winding Number
Draw a closed curve and see each region colored by its topological winding number.
Maxwell's Demon
Two-chamber gas simulation where a demon sorts hot and cold particles, illustrating the entropy paradox.
Limaçon Curves
Polar curve r = b + a·cos(θ): cardioid, trisectrix, inner loop, and convex forms with animated tracing.
Rose Curves — Polar
Animate r = cos(n/d·θ) through rational ratios — watch petal counts shift as n and d sweep the family.
Spirograph Generator
Full hypotrochoid spirograph with R, r, d sliders, multi-color traces, and live parametric equations.
Superellipse Explorer
Lamé curve |x/a|ⁿ+|y/b|ⁿ=1 animates from star to diamond to circle to squircle to square as n sweeps.
Epitrochoid Art
Layered epitrochoids build Guilloché-style patterns; randomize for endlessly varied ornamental curves.
Bernoulli Lemniscate
(x²+y²)²=2a²(x²−y²): the ∞-shaped curve where the product of distances from two foci stays constant.
Brachistochrone Race
Race a ball on cycloid vs straight line vs parabola — the cycloid always wins by Bernoulli's 1696 challenge.
Tautochrone Pendulum
Huygens 1659: release balls at any height on a cycloid and all arrive at the bottom at the same time.
Evolute Curves
Animated normal lines sweep out evolutes of the ellipse, parabola, cycloid, and astroid in real time.
Strophoid Curves
Right strophoid and oblique variants with angle-trisection construction and animated point tracing.
Cardioid Caustic
Light rays reflecting inside a circle converge to a cardioid caustic — the heart shape in your coffee cup.
Deltoid Rolling
A circle of radius R/3 rolling inside a larger circle traces a 3-cusped deltoid hypocycloid, with animated tangent lines.
Nephroid Caustic
Parallel light rays reflecting inside a circle converge on a nephroid — the 2-cusped kidney curve lurking in caustic envelopes.
Talbot's Curve
A parametric family interpolating between ovals and figure-eights, controlled by three parameters with real-time sliders.
Kampyle of Eudoxus
Ancient Greek curve a²x⁴ = b⁴(x²+y²), used by Eudoxus around 350 BCE in attempts to duplicate the cube.
Lame Spirals
The family r = a·θ^(1/n) — morphing continuously from Archimedean to parabolic to hyperbolic spiral as n varies.
Archimedean Spiral Art
Multi-arm Archimedean spirals with color gradients — adjust arms, spacing, and hue to generate mandalas and interference patterns.
Logarithmic Spiral Growth
r = ae^(bθ) — the equiangular spiral. Every radius cuts the curve at the same angle. Found in nautilus shells and galaxy arms.
Cochleoid Curves
r = a·sin(θ)/θ — the snail curve. Winding loops shrink like sinc(θ/π), with a pole singularity that resolves to r→a.
Rhodonea Polar Roses
r = cos(kθ) — rational k gives intricate multi-petal roses. Animate k through the family to watch petals split and merge.
Involute of a Circle
Unwinding a string from a circle traces an involute. Parallel involutes never intersect — the geometry behind every gear tooth.
Pedal Curves
The locus of feet of perpendiculars from a point to the tangents of a curve — drag the pedal point and watch the traced form transform across ellipses, parabolas, and lemniscates.
Isogonal Conjugates
Reflect cevians through angle bisectors — the reflected lines concur at the isogonal conjugate. Orthocenter and circumcenter are conjugates of each other.
Descartes Circle Theorem
(k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²) — four mutually tangent circles and their curvatures satisfy this elegant quartic identity. Animated with live verification.
Inversive Geometry
Inversion in a circle maps P to P′ where OP·OP′ = r². Lines become circles, grids become fractal webs — drag the inversion center to reshape the world.
Pascal's Mystic Hexagram
Six points on a conic — the three intersections of opposite sides are always collinear. Drag the points and watch the Pascal line pivot through every configuration.
Nine-Point Circle
Every triangle hides a circle through nine special points: midpoints of sides, feet of altitudes, and Euler midpoints — all on a single circle. Drag the vertices.
Simson Line
Any point on the circumcircle of a triangle projects to three collinear feet of perpendiculars. The Simson line rotates at exactly half the angular speed of the point.
Radical Axis
The locus of equal power w.r.t. two circles is a line — the radical axis. Three circles share a single radical center where all three axes meet.
Coaxial Circles
Two orthogonal families of circles: Apollonius circles (constant distance ratio to two foci) and the pencil of circles through the foci — every pair meets at right angles.
Steiner Chains
A chain of circles tangent to two given circles, each tangent to the next — Steiner's porism: if the chain closes once, it closes for every starting angle. Animated.
Harmonic Range
Four collinear points with cross-ratio −1. Drag two points — the harmonic conjugate updates instantly, always keeping (AC,BD) = −1.
Cross-Ratio & Möbius
Four points on the Riemann sphere. Drag them freely, then apply a Möbius transformation — the cross-ratio is invariant, always preserved.
Continued Fraction Convergents
Watch convergents of √2, φ, π, e, and ∛2 close in on the true value. Fractions alternate above and below, narrowing exponentially.
Euclidean Algorithm
The geometric Euclidean algorithm: repeatedly subtract the smaller side from the rectangle. The GCD emerges as the final square's side length.
Chinese Remainder Theorem
Two clock faces sweep congruences. The unique solution mod m₁·m₂ lights up when both hands align — CRT made visual.
Gaussian Integers
The complex lattice ℤ[i] colored by norm. Units glow red, primes gold, composites blue. Click any point to see its factorization.
Eisenstein Integers
The triangular lattice ℤ[ω] where ω = e^(2πi/3). Six-fold hexagonal symmetry, Eisenstein primes highlighted, click to inspect any point.
Gaussian Primes Grid
A dense grid colored by Gaussian primality. Pan and zoom to explore the moat problem — how far can you walk stepping only on primes?
Ulam Spiral
Integers arranged in a spiral from the center — primes cluster mysteriously along diagonals. Adjustable size and coloring modes.
Sieve of Eratosthenes
Animated sieve: composites cross out in waves as each prime is found. Watch the primes light up one by one with adjustable speed.
Prime Gaps
Consecutive prime gaps up to N as a bar chart, with Cramér's conjecture bound (ln p)² overlaid. Twin primes and record gaps highlighted.
Collatz Trajectories
Multiple Collatz sequences overlaid in color, plus a stopping-time histogram. Delay records — numbers that set new highs — glow white.
Riemann Zeta Zeros
First 50 non-trivial zeros of ζ(s) on the critical line Re(s) = ½. Animate |ζ(½+it)| in real time — watch it vanish at each zero.
Möbius Function
μ(n) as ±1/0 bars and the Mertens function M(x) = Σμ(n). The Mertens conjecture |M(x)| < √x — bounded but ultimately false.
Euler's Totient φ(n)
φ(n) scatter, ratio φ(n)/n converging to 6/π², and the totient sum ~3n²/π². Primes glow blue; prime powers gold.
Divisor Sum σ(n)
σ(n) colored by perfect / abundant / deficient. Switch to d(n) bubble chart to spot highly composite numbers by their towering dot size.
Liouville Function λ(n)
λ(n) = (−1)^Ω(n) bars and the running sum L(x). The Pólya conjecture (L ≤ 0 always) was believed for decades — disproved in 1958.
Von Mangoldt Λ(n)
Λ(n) = log p at prime powers, 0 elsewhere. Primes spike tall in gold; squares and cubes of primes leave smaller echoes.
Chebyshev ψ(x)
ψ(x) = Σ Λ(n) vs the diagonal x. The Prime Number Theorem says ψ(x)/x → 1. Toggle to see the oscillating deviation ψ(x) − x.
Prime Counting π(x)
π(x) staircase vs Li(x) and x/ln x. Li(x) always leads — the Skewes number (~10^316) is where π finally overtakes it.
Bertrand's Postulate
For every n, watch an animated scan find the prime lurking in (n, 2n). Gap ratios shrink but never vanish — the postulate holds forever.
Legendre Symbol Grid
(a/p) = ±1 or 0 over an a×p grid. Color map of quadratic residues vs non-residues — quadratic reciprocity jumps off the screen.
Quadratic Residues
QR grid mod p with Gauss sum G(χ) visualized as a vector walk. Watch |G|² = ±p depending on p mod 4.
Primitive Roots
Powers of g mod p drawn as arcs on a cycle. Primitive roots sweep every residue; non-generators trace smaller orbits.
Baby-step Giant-step
Find log_g(h) mod p in O(√p) time. Watch baby steps and giant steps collide in the middle — the discrete log pops out.
Elliptic Curve Groups
y² = x³ + ax + b with draggable sliders. Click two points to see the chord-and-tangent group law construct P+Q geometrically.
Elliptic Curve Addition
Step through point addition, tangent doubling, and nP sequences on an elliptic curve. The group law made geometric and tactile.
Modular Forms
Domain coloring of E₄, E₆, j(τ), and Δ(τ) on the upper half-plane. Hue encodes argument, brightness encodes magnitude — zeros glow black.
j-invariant
j(τ) mapped across the upper half-plane with SL(2,Z) tessellation overlay. j(i)=1728, j(ρ)=0, and the cusp at i∞ blazes white.
Hecke Operators T_n
T_n decomposes a lattice into all its index-n sublattices. Visualize the σ₁(n) Hermite normal forms as colored sublattice overlays.
Partition Function p(n)
p(n) grows explosively — watch it against the Hardy-Ramanujan asymptotic formula. Hover to enumerate all partitions for small n.
Young Tableaux
Enter any partition to see hook lengths, the hook-length formula for counting standard fillings, and an RSK-style standard Young tableau.
Ferrers Diagram
Dot diagram of a partition with its conjugate (transpose) and Durfee square highlighted. Toggle layers and type any partition to explore.
Integer Partitions Tree
All p(n) partitions arranged as a tree by part removal. Color by number of parts, largest part, or count of distinct parts.
Pascal's Triangle mod k
C(n,r) mod k reveals fractal self-similarity. Sierpiński at k=2, intricate patterns at primes. Adjust k and row count live.
Catalan Numbers
C_n = (2n choose n)/(n+1) counts ballot paths, valid parenthesizations, and polygon triangulations. Flip between all three interpretations.
Fibonacci Identities
Visual proofs: Cassini's identity as signed area (det = ±1), Zeckendorf decompositions, golden ratio convergence, and the Fibonacci spiral.
Lucas Sequences
U_n(P,Q) and V_n(P,Q) unify Fibonacci, Lucas, Pell, and Jacobsthal. Tune P and Q live; the discriminant governs growth vs oscillation.
Bernoulli Numbers
B_n values with exact fractions, Faulhaber's power-sum formula, and Euler's ζ(2n) = rational × π^{2n} — the famous 691 included.
Stirling Numbers
S(n,k) counts set partitions into k blocks; c(n,k) counts permutations with k cycles. Color triangles for both kinds, Bell numbers, and their matrix inverse relationship.
Bell Numbers
Bell triangle construction animated, set partition visualization for small n, and the exponential generating function exp(exp(x)−1).
Eulerian Numbers
A(n,k) counts permutations with exactly k descents. Color triangle, bar chart by row, and the Worpitzky identity for powers of x.
Derangements
D(n)/n! converges to 1/e — animated. Inclusion-exclusion construction and the hat-check problem: ~36.79% chance nobody gets their own.
Necklace Counting
All distinct necklaces and bracelets of n beads in k colors drawn explicitly. Burnside/Pólya formula via Euler's totient.
Burnside's Lemma
Rotation group acting on a colored square — fixed colorings under each rotation, averaged to count distinct equivalence classes.
Pólya Enumeration
Cycle index of D4, D3, D5, D6 symmetry groups — cycle structure of each element, substitution giving distinct coloring counts.
Latin Squares
Interactive n×n Latin square editor with live conflict highlighting. Counts L(3)=12 to L(6)=812 million. Click cells to edit.
Magic Squares
Lo Shu (3×3), Dürer's 1514 Melancholia square, and animated Siamese method construction for odd n. Magic constant M=n(n²+1)/2.
Orthogonal Latin Squares
Graeco-Latin squares: overlay showing all n² distinct pairs. Euler's 36 Officers Problem — proved impossible for n=6 by Tarry in 1901.
Hadamard Matrices
Sylvester construction H_{2n} = H₂⊗H_n, Walsh functions as orthogonal basis, and Hadamard error-correcting codes achieving the Plotkin bound.
Murmur Field
Whispering particle field with subtle noise-driven drift and mouse guidance.
Vortex Spiral
Spinning spiral attractor with color trails orbiting a clickable center.
Neon Bloom
Glowing neon flowers that bloom, breathe, and shift with layered petals.
Echo Cascade
Ripples cascade and interfere, sparking light where wave fronts collide.
Fractal Moss
Organic fractal branches grow in real time with spore bursts and spreading tendrils.
Drift Engine
Particles navigate invisible drifting force fields, picking up color from each well.
Pulse Garden
Rhythmic pulsing flora breathe under a starlit sky — click to plant new life.
Aurora Weave
Aurora-like flowing ribbons of light ripple across a northern night sky.
Shadow Basin
Dark attractor wells orbit the canvas, pulling swirling shadows into their depths.
Tidal Rhythm
Layered wave harmonics rise and fall under a glowing moon with foam and reflection.
Plasma Web
Electric plasma arcs between charged nodes in a dynamic web.
Moth & Light
Moths spiral inward on transverse orientation paths toward a glowing light.
Deep Current
Bioluminescent particles drift on deep ocean thermohaline currents.
Spore Cloud
Spores drift on thermals, settle, and bloom into mycelium growth.
Magma Flow
Lava flows and cools into a glowing crust with cracked fissures.
Silk Thread
Invisible spinners weave luminous silk threads between anchor points.
Mirror Pond
A still night pond reflects stars and moon; ripples distort the mirror.
Wind Score
Gusts of wind leave visible sound waves written across musical staves.
Ember Drift
Glowing embers rise from an unseen fire, cooling as they ascend.
Hollow Star
A dying star cycles through stellar evolution, collapsing to a hollow void.
Tide Clock
A clock built from tidal wave rhythms — bioluminescent hands sweep through rippling ocean rings.
Moth Storm
A swarm of moths circles chaotically in a storm, drawn helplessly toward your light.
Iron Bloom
Metallic flowers grow from magnetic field lines between poles — click to plant new magnets.
Crystal Wave
A crystalline lattice ripples with wave energy — click to send pulses through the structure.
Fog Machine
Fog rolls in with diffusion physics, hiding objects that your lantern light slowly reveals.
Sand Clock
An hourglass with real sand physics — grains stack, slide, and tumble through the neck.
River Mind
A river that thinks — branching tributaries carry flowing thoughts toward the sea.
Comet Trail
Comets streak across deep space trailing rocky debris, gas tails, and glowing comas.
Nerve Web
A living neural network fires and propagates signals — click to stimulate and watch cascades.
Lunar Tide
Drag the moon and watch gravity pull and stretch the ocean beneath it in real time.
Storm Cell
A rotating supercell thunderstorm builds updrafts, spins a mesocyclone, and hurls lightning.
Brine Pool
A deep-sea brine pool glimmers at the halocline, hosting strange life forms in hypersaline stillness.
Velvet Dark
Dark matter web filaments traced with dim glow — collapse nodes, warp the cosmic web.
Amber Pulse
A heartbeat rhythm pulses outward through amber resin, inclusions glowing with each beat.
Ice Fracture
Apply stress to an ice sheet and watch fractures propagate, branch, and shatter crystal grain boundaries.
Quartz Mind
A crystal quartz lattice resonates with thought-like waves — excite nodes and send pulses through the grid.
Wire Garden
Plants grow from circuit board wire patterns, routing traces toward blooms of bioluminescent signal.
Deep Bloom
Bioluminescent deep-sea organisms bloom in cascade when pressure waves sweep through the dark water column.
Ghost Current
Invisible ocean currents made visible by drifting tracers — create vortices and watch gyres form.
Spark Colony
A colony of sparks with roles — scouts, workers, guards, and a queen — cooperate and self-organize.
Salt Flat
Shimmering heat mirages over a white salt flat — hexagonal crystal patterns and shimmering horizons.
Dune Wave
Sand ripple patterns evolving under shifting wind — watch crests migrate and dunes breathe.
Mycelium Net
Underground fungal threads grow toward nutrients and transport glowing pulses through the network.
Radio Bloom
Radio telescope signals bloom into visual flower patterns — each transmission a spectrum of petals.
Obsidian Flow
Molten lava cools from white-hot to glassy obsidian — pour heat and watch it solidify and gleam.
Petal Storm
Petals caught in competing vortices — complex spiral patterns emerge from layered whirlwinds.
Star Nursery
Gas clouds collapse under gravity, accrete, and ignite as new stars — supernovae scatter the cycle.
Cave Drip
Stalactites drip mineral-rich water into a cave pool — ripples spread and formations slowly build.
Sonic Web
Sound waves woven into a geometric web — pluck nodes to send vibrations through taut string networks.
Prism Rain
Raindrops act as prisms splitting light into spectra — splashes fan out into rainbow cascades.
Wave Collapse
Quantum probability clouds shimmer across a grid — click to collapse the wavefunction into definite states.
Quantum Lattice
A glowing grid of quantum nodes pulses with energy — hover to influence tunneling, click to energize.
Spiral Decay
Multi-armed spirals bloom, fade, and reform — fractal arms trace fiery paths through the dark.
Echo Field
Click to send rippling echoes across a resonant field — watch interference patterns form and dissolve.
Neon Pendulum
A Lissajous harmonograph traces neon curves — press R for new frequency ratios, Space to clear.
Crystal Bloom
Symmetric crystal structures bloom outward from seed points — geometric branches grow with glowing facets.
Void Ripple
Move through the dark void and watch light ripples trail your cursor — particles scatter in your wake.
Solar Tide
Solar wind particles stream past a magnetic planet — drag the field to bend particle trajectories.
Fractal Breath
A fractal tree expands and contracts in slow breathing rhythm — inhale to grow, exhale to retreat.
Liminal Static
Television static slowly resolves into spirals, waves, and moiré — signal emerging from pure noise.
Aurora Drift
Northern lights simulation — curtains of color drifting across a dark sky, stirred by your mouse.
Magnetic Bloom
Iron filings arrange around magnetic poles in real time. Drag poles to reshape the field.
Resonance Web
A spider-web of strings that vibrates when plucked. Click threads to send ripples through the lattice.
Entropy Rain
Ordered particles descend into chaos as entropy rises — watch structure dissolve in real time.
Bioluminescent
Deep-sea jellyfish drift in the dark, glowing as your cursor passes through the abyss.
Time Crystal
A geometric crystal lattice that breaks discrete time-translation symmetry — click to change the period.
Shadow Dance
Invisible objects cast shadows as they dance. Only the shadows are real — the forms remain unseen.
Nebula Forge
Gas clouds collapse under gravity, spinning into stars that erupt with bipolar jets. Click to seed new regions.
Plasma Thread
Birkeland current filaments connect charged nodes, writhing with plasma discharge energy.
Vortex Seed
Fluid vortices spawn from seeds, interact, merge, and split — click to plant new swirling structures.
Tide Engine
Gravitational tidal forces sculpt a fluid grid — add moons and watch the waves respond
Swarm Mind
Pheromone-guided agents form emergent networks connecting food sources
Glass Fracture
Click to shatter — beautiful fracture patterns propagate outward from each impact point
Orbit Decay
Atmospheric drag pulls satellites into spiraling death spirals toward a planet
Voltage Garden
Lichtenberg discharge figures bloom like electric flowers from each spark
Spore Network
Physarum-inspired slime mold builds efficient transport networks between food sources
Light Echo
A stellar explosion sends light pulses through interstellar dust — click to trigger a nova
Quantum Foam
Planck-scale spacetime topology flickers — virtual particle pairs wink in and out of existence
Desert Wind
Aeolian sand transport simulation — adjust wind speed and watch dunes form and migrate
Harmonic Well
Quantum harmonic oscillator wavefunctions and probability clouds in a parabolic potential
Acid Rain
Acid rain corrodes a city — chemical reactions eat through stone and steel as pH drops
Dark Matter Web
Cosmic web of dark matter filaments — invisible structure revealed by galaxy tracers
Nerve Pulse
Action potential propagating along an axon — Na⁺/K⁺ ion channels opening in sequence
Coral Growth
Coral reef simulation — polyps build calcium carbonate structures, colored by age
Glacier Flow
Glacial ice flowing under gravity — crevasse formation and calving at the terminus
Ice Crystal
Snowflake growth via diffusion-limited aggregation on a hexagonal lattice
Solar Flare
Solar flare eruption — magnetic field line reconnection and plasma ejection arcs
Thought Web
Neural network of concepts — click to spark ideas that cascade through associations
Dendrite Growth
Dendritic crystal electrodeposition — fractal metal trees grow via diffusion-limited aggregation
Synapse Storm
Neural firing cascade — excitatory and inhibitory synaptic storms ripple through a network
Pressure Wave
Acoustic pressure waves through different media — shock waves, reflections, and interference patterns
Magnetic Reconnection
Opposing magnetic field lines merge and snap — plasma jets and energy release in real time
Mitosis Engine
Cell division step by step — chromatin condensation, spindle formation, and cytokinesis
Rain Forest
Ecosystem simulation — rainfall feeds plants, plants feed herbivores, predators hunt prey
Smoke Signal
Buoyant smoke particles rise through turbulent air — heat, drag, and vortex dynamics
River Delta
Sediment transport and deposition builds a branching delta — erosion carves the landscape
Circuit Life
Electronic components as living creatures — electricity pulses like blood through a living board
Photon Scatter
Rayleigh and Mie scattering — why the sky is blue and sunsets are red
Topographic Flow
Water flows across a procedural landscape — watersheds form, rivers erode channels
Membrane Tension
Soap bubble physics — thin-film interference creates iridescent colors as the film drains
Stellar Wind
Charged particles stream from a star and deflect around planetary magnetospheres — Lorentz force in action
Fire Ant Raft
Fire ants link together to float on rising water — self-assembling living raft emerges from simple rules
Lava Flow
Lava descends a slope — crust forms, channels carve, pahoehoe and aa textures emerge from cooling dynamics
Binary Star
Two stars orbit their common center of mass — Roche lobe overflow and accretion disk form as mass transfers
Foam Bubble
2D soap foam physics — Plateau's laws govern topology, bubbles merge and T1 rearrangements restructure the network
Spectral Split
White light enters a glass prism — wavelength-dependent refraction disperses it into a full visible spectrum
Tectonic Shift
Tectonic plates collide and diverge — subduction zones, rift valleys, mountain ranges, and earthquakes emerge
Gravity Lens
A massive object bends spacetime — Einstein rings form as background stars are lensed into arcs of light
Pollen Drift
Pollen grains jostled by invisible water molecules execute a random walk — Brownian motion and diffusion visualized
Enzyme Dock
Substrates diffuse and bind to enzyme active sites — induced fit, catalysis, and Michaelis-Menten kinetics in motion
Superconductor
Cooper pair formation, Meissner effect, and Abrikosov vortex lattice — drag the temperature to cross the critical threshold.
Black Hole Jet
Relativistic jet from a spinning black hole — Blandford-Znajek mechanism, plasma beaming, and gravitational lensing.
Root System
Plant root growth simulation with gravitropism, obstacle avoidance, and nutrient tropism — click to place water and minerals.
Fever Dream
Hallucinogenic pattern generation — plasma waves, melting geometry, color cycling, and mouse-driven distortion fields.
Tide Pool
Intertidal ecosystem simulation — sea anemones, crabs, starfish, and algae responding to the rising and falling tide.
Wind Turbine
Wind turbine aerodynamics — blade vortex shedding, real-time power curve, and the Betz limit at 59.3% efficiency.
Lung Breath
Alveolar gas exchange simulation — branching airways, O₂/CO₂ diffusion, blood cell oxygenation, and breathing rhythm.
Graviton Field
Spacetime curvature as particle exchange — drag masses to distort the fabric of spacetime and watch gravitons fly.
Carbon Cycle
Global carbon flux through atmosphere, ocean, biosphere, and lithosphere — animated particles trace each reservoir exchange.
Aurora Engine
Aurora borealis generator — charged particle precipitation along magnetic field lines, colored by altitude and ion species.
Neural Plasticity
Hebbian learning visualized — synapses strengthening and weakening as neurons co-fire across a living network
Doppler Shift
Moving source compresses wavefronts — watch redshift and blueshift emerge for light and pitch change for sound
Immune Response
T cells hunt pathogens, B cells launch antibodies — a cellular battlefield inside your bloodstream
Comet Tail
Two tails diverge — curved dust tail follows the orbit while the ion tail points straight away from the sun
Sea-Floor Spreading
New crust rises at the mid-ocean ridge, recording magnetic polarity reversals as plates drift apart
Protein Motor
Kinesin walks along a microtubule one ATP-powered 8 nm step at a time, hauling molecular cargo
Hebb Synapse
Spike-timing dependent plasticity — pre before post strengthens, post before pre weakens; the STDP learning window live
Myelin Pulse
Action potential leaps between nodes of Ranvier in saltatory conduction — 10× faster than bare axon propagation
Chemotaxis
Bacteria run-and-tumble up chemical gradients, extending runs when attractant rises and tumbling when it falls
Kelvin Wave
A wave trapped by a coastline or the equator, decaying exponentially offshore — the ocean's guided mode
Photosynthesis
Light-harvesting complexes absorb photons, transfer energy to reaction centers, and drive the electron transport chain to produce ATP and NADPH
Meiosis Dance
Animated meiosis I and II — homologous chromosome pairing, crossing-over events, and separation into four haploid cells
Magnetar Pulse
A magnetar starquake fractures the crust and releases a gamma-ray burst — crustal fracture lines, magnetic field reconfiguration, and plasma shockwaves
Sediment Layers
Geological time unfolds as strata accumulate — sandstone, shale, limestone, unconformities, and embedded fossils building a rock column
Coriolis Spin
Deflection in the rotating frame — projectiles curve right in the northern hemisphere, left in the south, driving hurricane rotation and Foucault pendulums
Leaf Venation
Auxin canalization grows a leaf vein network via space colonization — transport optimization produces the branching fractal pattern of real leaves
Circuit Breaker
An electrical arc forms as contacts separate — plasma channel, magnetic quench field, and glowing particles as 20 kA breaks across the gap
Wormhole Transit
Morris-Thorne traversable wormhole — embedding diagram, geodesic paths through the throat, exotic matter, and a Penrose diagram of the spacetime structure
Kármán Vortex Street
Alternating vortex shedding behind a cylinder — Strouhal number, Reynolds-dependent frequency, and counter-rotating spirals drifting downstream
Wake Turbulence
Aircraft wingtip vortices form, descend by mutual induction, and spread downwash — a hazard for following aircraft that persists for minutes
Dark Energy Web
Cosmic web accelerating under Λ — galaxies recede along Hubble flow, filaments stretch, voids balloon as dark energy reshapes the large-scale structure of spacetime
Eigenvalue Dance
Matrix transformations made visual — eigenvectors hold their direction as space rotates and stretches, complex eigenvalues spiral, the unit circle deforms into an ellipse
Neutron Star Merger
Binary inspiral driven by gravitational wave emission — watch the chirp signal build, stars collide in a kilonova, and r-process nucleosynthesis scatter gold into the cosmos
Epigenetic Clock
CpG methylation accumulates on a DNA double helix as time passes — the Horvath clock ticks biological age, diverging from chronological time with a turn of the dial
Fourier Portrait
Draw any shape and watch it reconstructed by rotating epicycles — each Fourier component adds one circle, and together they trace your line with mathematical precision
Topological Defects
Click to plant vortices and anti-vortices in an XY order-parameter field — winding numbers tally, field lines swirl, and annihilation events flash as charges meet
Gauge Field
U(1) fiber bundle over a lattice — arrows show the connection, a Wilson loop accumulates holonomy, charged particles pick up geometric phase as they travel through the field
Holographic Principle
Bulk 3D geometry encoded on its 2D boundary — AdS/CFT correspondence, Bekenstein-Hawking entropy from event horizon area, and Rindler observers in Minkowski spacetime
Spiral Harmonics
Spirograph and Lissajous harmonic visualizer — tune two frequencies and a phase offset to trace intricate closed curves that reveal the hidden geometry of periodic motion
Topology Explorer
Interactive genus and Euler characteristic explorer — morph between sphere, torus, double torus, Klein bottle, and RP² while χ = V − E + F tracks every deformation
Fourier Symphony
Draw any wave and watch it instantly decompose into rotating epicycles — the DFT laid bare as spinning circles reconstruct your signal from pure frequency components
Strange Loop
A GEB homage — Penrose staircases, self-printing code, recursive frames, and Langton's ant converge in a visual meditation on self-reference and strange loops
Entropy Cascade
Watch the second law in action: ordered particles dissolve into chaos as Boltzmann entropy climbs — temperature, gravity, and collisions drive the irreversible cascade
Neural Field Theory
Excitation and inhibition waves ripple across a simulated 2D cortical sheet — tune lateral connectivity kernels and watch Turing-like patterns emerge from neural noise
Resonance Chamber
Standing waves bloom in a 2D closed cavity — select normal modes, place continuous or pulse sources, and observe the wave equation carve Chladni-like nodal patterns
Oscillator Chaos
The driven Duffing oscillator on the edge of chaos — tune drive amplitude and frequency to navigate period doubling, strange attractors, and the bifurcation landscape
Bifurcation Web
Zoomable bifurcation diagram for the logistic map — drag to explore the self-similar cascade of period doublings converging on chaos at the Feigenbaum constant δ ≈ 4.669
Vortex Field
Place and drag point vortices in a 2D inviscid fluid — thousands of tracer particles reveal the swirling streamlines as vortices waltz under their mutual induction
Reaction Diffusion
Gray-Scott Turing patterns — editable feed/kill rates, mitosis, coral, and worm presets
Lenia
Continuous cellular automaton with smooth, biological-looking lifeforms emerging from math
Drumhead Modes
Circular membrane standing waves — Bessel function modes, click to excite, tap and watch resonate
Galaxy Collision
Two galaxies in gravitational encounter — watch tidal tails form as stars fling and cores merge
Wavelet Piano
Play piano notes and see their real-time continuous wavelet decomposition — frequency meets time
Kaleidoscope Engine
Particle system reflected through n-fold symmetry — move your mouse to paint living mandalas
Voronoi Life
Conway's Game of Life on an irregular Voronoi grid — the rules are universal, the topology is not
Plasma Wave
Langmuir oscillations: electron density waves in a 1D plasma, with phase-space portrait
Pendulum Wave
N pendulums of tuned lengths create wave interference, ripples, and perfect re-synchronization
Surface Tension
Soap film relaxing to minimal surface — iridescent physics, click to perturb, watch area minimize
Tessellation Art
Penrose tiling generator with P2/P3 inflation rules — aperiodic quasicrystalline patterns
Schrödinger Box
Particle in a 1D box — quantum wavefunction time evolution with energy eigenstates superposed
Three-Body Dance
Choreographic solutions to the gravitational three-body problem — figure-8, butterfly, braid orbits
Riemannian Surface
Parametric surfaces (sphere, torus, pseudosphere, saddle) with geodesic curves drawn live
Swarm Emergence
Boids flocking — three simple rules (cohesion, separation, alignment) produce collective behavior
Sand Ripple
Aeolian sand ripple formation — saltation, creep, and avalanche dynamics producing periodic bedforms
Photon Gas
Blackbody radiation — Planck spectrum, Wien displacement law, Stefan-Boltzmann law visualized
Network Epidemic
SIR epidemic spreading on random networks — tune R₀ and watch epidemic threshold behavior
Metabolic Network
Animated metabolic pathway graph — glycolysis, TCA cycle, ETC with live flux visualization
Crowd Dynamics
Social force model (Helbing & Molnár 1995) — pedestrian corridor flow, bottleneck jamming, evacuation
Bessel Modes
J₀–J₅ Bessel functions plotted with zeros marked, plus animated circular drum vibrational mode shapes
Soliton Wave
KdV equation soliton solutions — two solitons collide and pass through each other completely unchanged
Hilbert Curve 3D
3D space-filling Hilbert path animated through orders 1–4; drag to rotate, watch locality preservation
Strange Attractors II
Rössler and Lorenz attractors side by side — tune parameters, watch chaos emerge, drag to rotate
N-Magnet Pendulum
Pendulum swinging above N magnets — fractal basins of attraction drawn live, click to launch
Voronoi Distance Field
Voronoi diagram with distance field overlay — drag sites and watch cells and gradients update live
Power Diagram
Weighted Voronoi diagram — circles with weights define power-distance cells, animate weights live
Medial Axis
Draw a polygon and watch its medial axis skeleton computed in real time via distance field ridge-finding
Substitution Tiling
Fibonacci, Thue-Morse, Tribonacci, and Rudin-Shapiro substitution sequences as animated aperiodic tilings
TDA Mapper
Topological Data Analysis Mapper — point cloud to Mapper graph with tunable cover intervals and clustering
3D Standing Wave
Membrane eigenmodes ψ(x,y,t) — tune mode numbers (m,n) and watch the 3D surface oscillate
Acoustic Cavity
Helmholtz eigenmodes in a 2D resonant cavity — pressure field visualized with nodal lines
Light Polarization
Linear, circular, elliptical polarization states — E-field ellipse, wave propagation, Poincaré sphere
Snell's Law Ray Tracer
Drag the ray angle and refractive indices — watch refraction and total internal reflection in action
Diffraction Simulator
Single and double slit Fraunhofer diffraction — tune slit width, separation, and wavelength
Atomic Spectroscopy
Hydrogen Balmer/Lyman/Paschen series — energy level diagram and emission spectrum, click lines for details
Morse Code Wave
Type text, encode it as Morse code, and watch the signal propagate as a carrier wave
2D Crystal Lattice
Bravais lattice explorer — square, hexagonal, triangular, rectangular, oblique with Brillouin zones
Fibonacci & Golden Ratio
Fibonacci spiral, golden angle phyllotaxis, ratio convergence to φ — four animated views
Euler's Identity
e^(iπ)+1=0 visualized — rotating phasor, Taylor series buildup, 3D helix in the complex plane
Stellar Nucleosynthesis
Particle simulation of nuclear fusion inside stars — H fuses to He, C, O, Fe with color-coded element trails and energy flashes
Membrane Vibration
Circular drum normal modes described by Bessel functions Jₙ(αₙₘr/R)cos(nθ) — select radial and angular mode numbers
Ising Spin Lattice
2D Ising model with Metropolis algorithm — watch ferromagnetic order emerge below the critical temperature Tc ≈ 2.269
Euler / Cornu Spiral
The Cornu spiral traced by Fresnel integrals ∫cos(πt²/2)dt — curvature grows linearly, used in road design for smooth transitions
Plasma Confinement
Charged particles spiraling along tokamak magnetic field lines — adjust field strength and watch Larmor radius and confinement change
Vortex Rings
Toroidal vortex rings (smoke rings) self-propelling via Biot-Savart induction — add rings and observe leapfrogging dynamics
Resonance Modes
Standing waves on a fixed string — superpose harmonics, observe nodes and antinodes, frequencies fₙ = n·f₁
Braids & Topology
Braid group Bₙ visualization — compose generators σᵢ and σᵢ⁻¹, watch strands cross over and under in scrolling animation
Sierpinski Cave
Sierpinski triangle as an explorable fractal cave — click to zoom into voids, Hausdorff dimension log(3)/log(2) ≈ 1.585
Coupled Oscillators
Spring-mass chain with normal mode excitation — energy transfers between modes, frequencies ωₙ = 2√(k/m)|sin(nπ/2(N+1))|
Bifurcation Map
Logistic map xₙ₊₁=rxₙ(1−xₙ) — period-doubling cascade to chaos, zoom into Feigenbaum's constant δ≈4.669
Rössler Attractor
Rössler strange attractor in 3D projection — fractal structure, sensitive dependence, rotating fading trails
3D Cellular Automaton
Game of Life extended to three dimensions — isometric view of emergent 3D structures from local rules
Lenia Creature
Lenia continuous cellular automaton — smooth Gaussian kernel convolution produces lifelike moving organisms
Double Pendulum Phase Space
Multiple double pendulums diverge exponentially from nearby initial conditions — real motion + phase portrait (θ₁,θ₂)
Gray-Scott Reaction-Diffusion
Two-chemical reaction-diffusion system — spots, stripes, solitons, and mitosis from du/dt = Du∇²u − uv² + F(1−u)
Chladni Figures
Sand particles migrate to nodal lines of a vibrating plate — select mode (m,n) to see different geometric patterns
Tsunami Wave
Shallow-water equations on a continental shelf — wave slows and amplifies as depth decreases, c=√(gH)
Fractal Terrain
Diamond-square algorithm generates fractal heightmaps — fly-through with perspective projection and height-based coloring
Sandbox Physics
Falling sand simulation — place sand, water, fire, stone, and wood; gravity, fluid flow, and fire spreading
WireWorld
WireWorld cellular automaton — draw circuits with electron heads and conductors, watch signals propagate through AND gates, diodes, and oscillators
Flocking Boids 3D
Three-dimensional boids flocking with perspective projection — separation, alignment, and cohesion rules create emergent aerial murmurations
Elastic Collision Gas
2D ideal gas with elastic ball collisions — Maxwell-Boltzmann speed distribution builds up in real time as temperature and pressure emerge from chaos
Crowd Simulation
Social force model crowd dynamics — agents navigate obstacles to exits, density heatmap reveals congestion, panic mode triggers stampede behavior
Ant Colony Optimization
Ants deposit pheromones to multiple food sources — evaporation and reinforcement cause shortest paths to emerge from collective stigmergy
Swarm Intelligence PSO
Particle Swarm Optimization on a fitness landscape — particles share memories of personal and global bests to converge on the global minimum
Galaxy N-Body Collision
Two spiral galaxies on a collision course — gravitational N-body dynamics form tidal tails and merger debris; drag to rotate the 3D view
Protein Folding 2D
HP lattice model with Monte Carlo optimization — hydrophobic residues attract in a 2D grid, minimizing free energy through simulated annealing
Automata Music
Cellular automata as a musical instrument — Game of Life, Rule 30, and Seeds patterns drive tones via Web Audio API in selectable scales
Turing Morphogenesis
Gray-Scott reaction-diffusion — activator-inhibitor dynamics spontaneously form jaguar spots, zebra stripes, and labyrinths from uniform noise
Flow Field Noise
Simplex noise vector field — thousands of particles follow swirling organic currents, leaving luminous trails in a living tapestry of fluid motion
Cyclone Formation
2D tropical cyclone dynamics — Coriolis deflection and central low pressure create a rotating inflow structure with spiral rainbands and a warm eye
Reaction Diffusion 3-Species
Belousov-Zhabotinsky-like oscillator with three competing species — spiral waves and target patterns emerge from Turing instability, color-coded RGB
Topological Sort
Visualize Kahn's algorithm on a directed acyclic graph — step through in-degree queuing, watch nodes resolve in dependency order with live animation
Markov Chain Walk
Random walk on a Markov chain — animated walker traverses state transition diagrams while empirical visit frequencies converge to the stationary distribution
Spiro Curves
Hypotrochoid and epitrochoid parametric curves — rolling circles trace intricate Spirograph patterns; layer multiple curves with R, r, d sliders
Wave Interference Color
Two-source light wave interference with visible-spectrum color — drag the sources to reshape constructive and destructive fringe patterns in real time
Lorenz-96 Model
Lorenz's toy atmospheric model on a periodic ring — adjust forcing F to cross the chaos threshold; view Hovmöller diagrams and Lyapunov exponent estimates
SPH Fluid 2D
Smoothed Particle Hydrodynamics — 200 fluid particles with pressure and viscosity forces slosh in a bounded domain; click to stir and push the fluid
Bell Curve / CLT
Galton board Plinko simulation — balls bounce through pegs accumulating a binomial distribution that converges to a normal curve as the Central Limit Theorem predicts
Tree Fractal Wind
L-system fractal tree swaying in simulated wind — spring physics animate each branch with gusts, damping, and seasonal color modes from spring green to winter bare
Polyhedra Viewer
3D Platonic and Archimedean solids with mouse-drag rotation — Euler characteristic V−E+F=2 displayed live for tetrahedron, icosahedron, truncated icosahedron and more
Harmonograph
Two pendulums at different frequencies draw decaying Lissajous curves — adjust frequency ratio, phase offset, and damping to explore the full family of harmonograph figures
Magnetic Field Visualizer
Field lines for a magnetic dipole or hypothetical monopole — color-coded by strength, with a test particle following the Lorentz force through the field
Minority Game
El Farol bar problem: N adaptive agents independently choose A or B each round and the minority wins — watch emergent anti-coordination beat random play over time
Complex Plane
Domain coloring for complex functions — hue encodes argument, brightness encodes log-magnitude; pan and zoom to explore z², sin(z), exp(z), and Möbius transforms
Iterated Function Systems
Chaos game builds fractals from random affine transformations — watch the Barnsley fern, Sierpinski triangle, dragon curve, and Lévy C curve emerge dot by dot
Newton Fractal
Newton's method basins of attraction — each pixel is colored by which root it converges to; brightness shows convergence speed; fractal boundaries emerge at root interfaces
Infinity Mirror
Recursive mirror tunnel with animated light orbs bouncing between shrinking reflections — adjust reflection depth, perspective angle, rotation speed, and tint color
Moiré Pattern
Two overlapping periodic grids at a slight angle produce large-scale interference fringes — choose parallel lines, concentric circles, dot grids, or radial lines
Optical Illusions
Rotating snakes, Hermann grid, Café wall, Kanizsa triangle, and scintillating grid — five classic illusions that exploit neural mechanisms, animated and cycling
Necker Cube
Bistable figures — Necker cube, Rubin vase, duck-rabbit — animate the spontaneous perceptual flip between two equally valid interpretations with adjustable switch rate
Terrain Erosion
Diamond-square heightmap eroded by hydraulic simulation: raindrops pick up sediment, flow downhill, deposit where they slow, carving rivers and valleys over time
DLA Fractal
Diffusion-limited aggregation: random walkers stick on contact, building dendritic crystal clusters with fractal dimension ≈ 1.71 — colored by attachment order
Delaunay Mesh
Live Delaunay triangulation of moving points with optional Voronoi dual overlay and circumcircles — click to add points, watch the mesh recompute in real time
Space-Filling Curves
Hilbert, Peano, Gosper flowsnake, and Sierpinski curves animated level by level — dimension 2, yet drawn as a 1D line threading every point in a square
Logistic Network
Small-world network of coupled logistic maps showing synchronization, chaos, and pattern formation — adjust growth rate r and coupling strength ε in real time
Seeds Automaton
Seeds (B2/S), Brian's Brain, Conway's Life, and Day & Night cellular automata — click to draw, watch explosive symmetric patterns erupt from tiny seeds
Hodgkin-Huxley Neuron
Nobel Prize 1963 model: four ODEs for membrane voltage, Na⁺ activation/inactivation, and K⁺ gating — inject current to fire action potentials and see refractory periods
Van der Pol Oscillator
Nonlinear oscillator with a global limit cycle attractor in phase space — all trajectories spiral in from outside or out from inside; μ controls relaxation oscillation shape
Cosmic Web
N-body gravity simulation with Gaussian density perturbations — watch dark-matter filaments, voids, and halos assemble into the large-scale structure of the universe
Fractal Canopy
Recursive fractal tree viewed from below — a full seasonal cycle animates from bare winter branches through spring buds and summer leaves to autumn gold, with wind sway
L-System Plant
L-system string rewriting and turtle graphics — five plant grammars (bush, fern, seaweed, dragon curve, twig) animate from axiom to full structure as iteration depth grows
Reaction Wave
Excitable medium wave propagation — click to fire circular pulses that spread outward as annihilating rings, exactly like trigger waves in the Belousov-Zhabotinsky reaction
Excitable Medium
FitzHugh-Nagumo equations on a 2D grid — spiral waves and action-potential propagation emerge spontaneously; click to plant a pacemaker and watch waves collide
Target Pattern
Lambda-omega reaction-diffusion system driven by central oscillators — concentric ring targets expand outward and interfere where wave trains from multiple centers overlap
Turing Ring
Activator-inhibitor pattern formation on a 1D ring — the Hovmöller diagram traces space-time as diffusion-driven instability selects a wavelength from random noise
Oregonator BZ
Oregonator model of the Belousov-Zhabotinsky reaction — ferroin indicator cycles blue-to-red through spiral waves on a 2D grid; ε, stoichiometry, and ratio are tunable
Snow Crystal
Snowflake growth on a hexagonal lattice via vapor diffusion and anisotropic attachment — temperature selects plates, columns, needles, or dendrites from the Nakaya morphology diagram
Mud Crack
Desiccation crack pattern formation — stress accumulates as a surface dries, new cracks nucleate at weak points and propagate perpendicular to existing ones, forming polygonal networks
Bone Trabecula
Wolff's Law in action — trabecular bone self-organizes along principal stress trajectories, forming dense lattices under compression and resorbing in unloaded zones
Coral Reef
Diffusion-limited growth produces branching staghorn, massive boulder, and encrusting coral forms competing for light and space — bleaching simulated by raising water temperature
Slime Mold
Physarum polycephalum agent model — thousands of agents deposit and follow chemoattractant trails, self-organizing into efficient transport networks mirroring Tokyo's rail system
Root Growth
Plant roots grow by chemotropism toward water sources, branch stochastically, and navigate around rock obstacles — click to place new water sources
Mycelium Network
Fungal hyphal tips explore space and fuse via anastomosis, creating redundant nutrient transport loops — resource pulses flow visibly through the growing network
Heart Rhythm
Cardiac action potential propagates from SA node through atria, delays at AV node, then sweeps the ventricles — switch between normal sinus, A-fib, AV block, and V-tach
Brain Waves
Eight-channel EEG oscilloscope showing delta, theta, alpha, beta, and gamma bands — morphs smoothly between focused, relaxed, drowsy, deep sleep, and REM brain states
Retinal Mosaic
Photoreceptor mosaic with cone density peaking at the fovea and falling to near-zero in the periphery — shows L/M/S cone distribution, the blind spot, and visual acuity gradient
Shell Spiral
Logarithmic spiral shell growth in 3D perspective — varying expansion rate, height ratio, and aperture angle generates the full range from flat nautilus to tall turritella
Phyllotaxis
Golden angle (137.5°) generates Fibonacci spiral counts in sunflower seed packing — drag the angle slightly off-golden to watch the optimal packing collapse dramatically
Fire Simulation
Cellular automaton fire and smoke dynamics
Wave Interference 3D
Three-dimensional wave interference, rotatable
Crystal Growth
DLA and Eden model crystal formation
Plasma Filaments
Magnetic plasma tendrils in simulation
Neural Harmonics
Synchronized oscillator networks
Fractal Coastline
Richardson plots and coastline dimension
Vortex Streets
Kármán vortex street behind a cylinder
Stellar Nursery
Gas cloud collapse into proto-stars
Quantum Lattice Walk
Quantum random walk probability spreading
Aurora Diffraction
Interference patterns in auroral colors
Geodesic Dome
Icosahedron subdivision into geodesic sphere
Topology Garden
Torus, Klein bottle, Möbius strip in 3D
Traffic Flow
Nagel-Schreckenberg jam formation model
Soap Bubble Physics
Minimal surface relaxation simulation
CA Explorer
All 256 elementary rules and 2D life-like automata
Boids 3D
Three-dimensional flocking with perspective projection
Langton's Ant Variants
Multiple ant rule variants running simultaneously
Sand Pile Model
Abelian sandpile avalanche dynamics
Ising Spin Glass
Disordered spin glass with Metropolis algorithm
Mandelbrot Navigator
Zoomable Mandelbrot set with smooth coloring
Fluid Pressure Field
Navier-Stokes pressure visualization with obstacles
Atmospheric Circulation
Hadley, Ferrel, and Polar cell dynamics
Soliton Waves
KdV solitons passing through each other intact
Voronoi 3D
Three-dimensional Voronoi cells, rotatable
Quaternion Julia
3D slice of quaternion Julia set
Spectral Graph Explorer
Laplacian eigenvectors and spectral embedding
Airy Stress Function
Elastic stress fields from Airy potential
Cellular Neural Network
CNN-style cellular automaton with learned kernels
Electric Field Lines
Place charges, visualize fields and equipotentials
Phase Portrait Explorer
Dynamical systems nullclines and streamlines
Kolmogorov Turbulence
2D turbulence and the k⁻⁵/³ energy cascade
Order Parameter Field
Phase transitions, domain walls, and defects
Morse Theory
Critical points and sublevel set topology
Random Matrix Spectrum
GUE/GOE eigenvalue spacing and level repulsion
Fiber Bundle
Base space, fiber, and connection visualization
Minimal Surfaces
Catenoid, helicoid, Enneper, Costa surfaces
Diffusion Map
Manifold embedding via diffusion coordinates
Spin Network
Loop quantum gravity spin network visualization
Perlin Terrain 3D
Procedural terrain with noise octaves
Lenia
Continuous cellular automaton with life-like creatures
Kolmogorov Complexity
Algorithmic complexity and compression ratios
Coriolis Effect
Rotating frame trajectories and weather patterns
Density Functional
Electron density in a 1D quantum well
Heat Kernel
Heat diffusion on graphs and manifolds
Vicsek Model
Collective motion and order-disorder transition
Quantum Measurement
Bloch sphere, collapse, and Born rule statistics
Catastrophe Theory
Fold, cusp, and swallowtail bifurcations
Lie Algebra Roots
Root systems for A2, B2, G2 algebras
Neural Turing Machine
Attention-based memory read/write heads
Persistence Diagram
TDA filtration, birth-death pairs, Betti numbers
Monstrous Moonshine
j-function coefficients and Monster group dimensions
Noncommutative Torus
Rotation algebra and Rieffel projections
Mirror Symmetry
CY moduli spaces and SYZ T-duality
Floer Homology
Gradient flow lines and Morse boundary operator
Toric Variety
Fan, polytope, and moment map image
Spectral Sequence
Pages E₁, E₂… converging with differentials
Brauer Group
Central simple algebras and Brauer equivalence
Mean Field Theory
Self-consistency equation and phase diagram
Isoperimetric Problem
Curve deformation approaching the circle optimum
Ergodic Theory
Irrational rotation, cat map, Birkhoff averages
Codon Usage Bias
Codon frequencies, GC content, and genetic code
Enzyme Kinetics
Michaelis-Menten, inhibition, Lineweaver-Burk
Membrane Potential
Hodgkin-Huxley model and action potentials
Allosteric Regulation
MWC model, cooperativity, hemoglobin binding
Cell Division Cycle
CDK/cyclin oscillator and checkpoint control
Circadian Clock
Goodwin oscillator and light entrainment
Synaptic Plasticity
STDP spike timing and Hebbian learning
Ecological Network
Food web trophic levels and extinction cascades
Gene Regulatory Network
Boolean gene dynamics and steady states
Protein Contact Map
Residue contacts and backbone folding
River Delta Formation
Sediment bifurcation and delta morphologies
Slime Mold Network
Physarum optimal transport network formation
Murmuration
Starling topological interaction flocking
Airfoil Flow
Joukowski mapping, lift, and streamlines
Bubble Packing
2D foam with T1 topological transitions
Soap Film Network
Steiner tree and Plateau 120° junctions
Lightning Discharge
Dielectric breakdown fractal branching
Sand Dune Formation
Wind transport and barchan dune emergence
Tensegrity Structures
Cable-strut equilibrium and force distribution
Origami Mathematics
Kawasaki theorem, crease patterns, Miura-ori
Protein Folding Energy
Energy landscape of protein folding — watch a chain find its native state through a funnel of configurations
Calcium Waves
Calcium-induced calcium release propagating as waves through a cellular grid — click to trigger
Stochastic Predator-Prey
Individual wolves and rabbits on a spatial grid — emergent population cycles from local interactions
Vesicle Fusion
SNARE-mediated vesicle fusion step by step — from docking to neurotransmitter release
Immune Response
Pathogen invasion, B-cell activation, antibody surge, and immunological memory formation
Photosynthesis Light Reactions
Light-dependent reactions in the thylakoid — photons driving the electron transport chain
ATP Synthase Motor
The molecular motor spinning from proton flow — watch F0 rotate and F1 synthesize ATP
Signal Transduction Cascade
GPCR activation amplifying through G-proteins, cAMP, and PKA — click receptor to trigger
Heat Shock Response
Protein misfolding under thermal stress, HSP chaperones rescuing aggregates — adjust temperature
Autophagy Dynamics
Cellular self-eating — phagophore formation, cargo capture, lysosomal degradation
Cytoskeleton Dynamics
Actin treadmilling and microtubule dynamic instability — the cell's ever-changing scaffold
Mitochondrial Membrane
Electron transport chain and ATP synthase — the inner membrane as a proton-driven power plant
Nuclear Pore Transport
Importin-cargo threading through FG-nucleoporin hydrogel — selective transport in and out of the nucleus
Epigenetic Landscape
Waddington's landscape of cell fate — a ball rolling toward attractors of differentiation
CRISPR Mechanism
Cas9 scanning DNA, PAM recognition, R-loop formation, and targeted gene editing step by step
Chromosome Segregation
Spindle assembly and mitotic chromosome separation — from kinetochore capture to anaphase
Lateral Inhibition
Notch-Delta signaling on a hex grid — uniform tissue spontaneously resolving into patterned fates
Stem Cell Differentiation
Lineage decisions driven by transcription factor signals — navigating the branching tree of cell identity
Synapse Development
Axon growth cone navigation and synaptogenesis — the molecular handshake that builds a connection
Reaction-Diffusion in Cells
Turing activator-inhibitor producing biological patterns — stripes and spots from noise
Molecular Crowding
Excluded volume effects in a packed cytoplasm — how crowding slows diffusion and drives reactions
Phase Separation & IDRs
Liquid-liquid phase separation of disordered proteins forming condensates from homogeneous solution
Membrane Curvature
BAR domain proteins sensing and amplifying membrane bends — click to indent and watch curvature cascade
Clathrin Endocytosis
Clathrin lattice assembly, membrane invagination, and dynamin-mediated vesicle pinching
Actin Polymerization Force
Brownian ratchet mechanism generating pushing force from actin monomer addition against a load
Cell Polarity
Symmetry breaking in GTPase activation — uniform membrane spontaneously resolving into a single pole
Gap Junction Coupling
Signal propagation through gap junctions — click a cell, watch activation ripple through the network
Osmoregulation
Cell volume dynamics under osmotic stress — water flux, swelling, shrinking, and regulatory recovery
Mechanosensing
Focal adhesion assembly on stiff vs soft substrates — how cells feel and respond to their physical environment
Lipid Bilayer Dynamics
Lateral diffusion, flip-flop, and lipid raft formation — cholesterol and temperature driving membrane phases
Stochastic Gene Expression
Gillespie simulation of bursty transcription and translation — noise in a single cell vs population
mRNA Splicing
Spliceosome assembly and intron lariat removal — the molecular choreography of pre-mRNA processing
Ribosome Translation
Codon-by-codon protein synthesis — tRNA entry, peptide bond formation, and translocation
Proteasome Degradation
26S proteasome recognizing ubiquitin tags, unfolding substrates, and cleaving them into peptides
Ubiquitin-Proteasome System
E1-E2-E3 cascade building a polyubiquitin chain — tagging proteins for proteasomal destruction
DNA Repair Mechanisms
BER, NER, and DSB repair pathways responding to different lesions — click to damage and watch repair
Telomere Dynamics
Telomerase extending chromosome ends, t-loop protection, and the erosion counting toward senescence
Histone Modifications
Click to add and remove histone marks on H3/H4 tails — writing and reading the histone code
Chromatin Remodeling
ATP-dependent nucleosome repositioning by SWI/SNF — opening and closing chromatin access
Transcription Factor Binding
TF searching by 3D diffusion and 1D sliding — finding its binding site in a sea of DNA
Population Genetics & Drift
Wright-Fisher allele frequency trajectories — watching fixation and loss events from random sampling
Natural Selection
Heritable traits under predation pressure — watch the population's trait distribution climb the fitness peak
Speciation Model
Allopatric speciation — two populations diverging in isolation until reproductive isolation is complete
Horizontal Gene Transfer
Bacterial transformation, transduction, and conjugation spreading resistance genes through a population
Biofilm Formation
Bacterial biofilm lifecycle — attachment, microcolony, EPS maturation, and quorum-triggered dispersal
Antibiotic Resistance Evolution
Stepwise resistance evolution across an antibiotic gradient — mutation, selection, and clonal expansion
Adaptive Radiation
One colonizing species diversifying to fill island niches — watch the phylogeny branch in real time
Coevolution Dynamics
Host-parasite Red Queen cycling — allele frequencies locked in perpetual frequency-dependent arms race
Genetic Drift & Bottleneck
Population crash and recovery — heterozygosity collapse through a bottleneck and the founder effect
Sexual Selection Runaway
Fisherian runaway — female preference and male trait becoming genetically correlated through sexual selection
Kin Selection & Altruism
Hamilton's rule in action — altruism spreading through kin networks when r·b exceeds the cost
Multilevel Selection
Within-group selfishness vs. between-group altruism — the Price equation decomposing selection's two faces
Niche Construction
Organisms reshaping their environment which reshapes selection — the feedback loop of niche construction
Cultural Evolution
Cultural variants spreading via unbiased, conformist, or success-biased transmission across a population
Language Evolution
Naming games on networks — how shared lexicons emerge from pairwise communication attempts
Cooperation & Punishment
Public goods game with costly punishment — the rock-paper-scissors dynamics that sustain cooperation
Social Network Contagion
Simple vs complex contagion spreading on scale-free and random networks — topology shapes spread
Opinion Dynamics
Voter model coarsening and bounded confidence clustering — how opinions separate or converge
Conformity Thresholds
Granovetter cascade model — small triggers igniting population-wide adoption when thresholds align
Norm Emergence
Social conventions crystallizing from noise — symmetry breaking as one norm dominates all others
Quantum Entanglement
Bell states, correlated measurement outcomes, and violation of classical inequalities
Bloch Sphere Dynamics
Qubit state evolution under quantum gates — watch the state vector precess on the Bloch sphere
Double-Slit Experiment
Wave interference building up particle by particle — and collapsing when you look
Quantum Decoherence
Coherence decaying as a qubit couples to its environment — the quantum-to-classical transition
Quantum Gate Circuit
Build quantum circuits from H, X, Y, Z, CNOT gates — watch the statevector transform
Grover's Search
Amplitude amplification finding a marked item in √N steps — quantum speedup made visible
Quantum Error Correction
3-qubit bit-flip code detecting and correcting errors without measuring the logical qubit
Density Matrix Evolution
Watching coherences and populations shift under unitary and noisy quantum channels
Quantum Phase Transition
Transverse-field Ising model — quantum fluctuations destroying order at the critical point
Aharonov-Bohm Effect
Interference fringe shifting from vector potential alone — topology affecting physics with no local force
Percolation Clusters
Giant cluster emergence at the critical threshold — connectivity phase transition on a 2D lattice
XY Model Vortices
Topological defects in the 2D XY model — vortex-antivortex pairs near the BKT transition
Traffic Flow
Nagel-Schreckenberg cellular automaton — phantom jams emerging from simple local driving rules
Metropolis Algorithm
MCMC random walk sampling a 2D energy landscape — watching the Boltzmann distribution build up
Simulated Annealing
Traveling salesman solved by cooling — thermal fluctuations escaping local minima to find global ones
Spin Glass
Frustrated random bonds creating a rugged energy landscape — the physics of disorder and memory
Kuramoto Synchronization
Coupled oscillators locking their phases — order parameter rising as coupling crosses the critical threshold
Chimera States
Synchronized and chaotic oscillators coexisting in the same network — a counterintuitive partial order
Noise-Induced Transitions
Stochastic escape from potential wells — noise not just adding error but creating qualitatively new behavior
Anomalous Diffusion
Normal vs sub vs superdiffusion — MSD power laws from heavy-tailed waiting times and Lévy flights
Integrate-and-Fire Network
Excitatory-inhibitory neural network dynamics — tuning E/I balance between synchrony, oscillation, and chaos
Wilson-Cowan Model
Mean-field neural population dynamics — phase plane showing rest, oscillations, and bistability
Neural Field Theory
Mexican hat connectivity producing traveling waves and memory bumps in a 1D cortical sheet
Spike-Timing Plasticity
STDP learning rule — LTP or LTD depending on whether pre fires before or after post
Attractor Network Memory
Hopfield network retrieving corrupted patterns — energy landscape and convergence to stored memories
Neural Avalanches
Cascading activity at the critical branching ratio — power-law size distributions at the edge of instability
Cortical Oscillations
Delta, theta, alpha, beta, gamma bands from coupled neural populations — tuning the brain's rhythms
Theta-Gamma Coupling
Slow theta phase modulating fast gamma amplitude — neural multiplexing across frequency bands
Neural Criticality
Brain dynamics at the edge of chaos — maximum dynamic range and information transmission at criticality
Connectome Dynamics
Signal propagation through a small connectome — sensory input cascading through interneurons to motor output
Crystal Annealing
Atoms finding minimum energy configurations as temperature slowly cools — Metropolis criterion and crystallization
Strange Attractor Zoo
Lorenz, Rössler, Thomas, Dadras, Halvorsen side by side — chaotic orbits and phase-space butterfly wings
Geodesic Paths
Shortest paths on curved surfaces — great circles on sphere, wrap-around on torus, arcs in hyperbolic disk
Tessellation Engine
Regular, semi-regular, and aperiodic Penrose tilings — interactive colors, rotation, and wave animation
Quantum Decoherence
Bloch sphere coherence decay, density matrix off-diagonals vanishing, von Neumann entropy rising
Fraunhofer Diffraction
Single slit, double slit, circular aperture — vary wavelength and geometry to sculpt far-field patterns
Hologram Simulator
Interference patterns from point sources, Fresnel zone plates, wavefront reconstruction visualized
Earth's Magnetosphere
Solar wind bowing field lines, Van Allen belts, bow shock, magnetopause, and magnetic reconnection
Plasma Physics
Particle-in-cell simulation — plasma oscillations, Debye shielding, cyclotron motion, Z-pinch confinement
Seebeck Effect
Temperature gradient drives electron drift — thermoelectric voltage, efficiency, ZT figure of merit
Anderson Localization
Electron wave functions in disordered lattices — localized vs extended states, IPR, 1D tight-binding
Bose-Einstein Condensate
Atoms cooling into macroscopic ground state — condensate fraction, momentum distribution, phase transition
Fermi Surface Explorer
2D Brillouin zone with Fermi contours — van Hove singularities, nesting, tight-binding hopping
Exciton-Polariton Condensate
Strong light-matter coupling — polariton branches, avoided crossing, Hopfield coefficients
Cooper Pairs & BCS
Phonon-mediated pairing — energy gap, quasiparticle spectrum, BCS density of states
Mott Insulator Transition
Hubbard model bandwidth vs Coulomb repulsion — metal-insulator transition, double occupancy
Kondo Effect
Magnetic impurity spin screening — resistivity minimum, Kondo temperature, screening cloud
Fractional Quantum Hall
Landau levels and filling fractions — Laughlin wavefunction, anyonic excitations, Hall plateaus
Weyl Semimetal
Weyl points in k-space — topological Fermi arcs, Berry curvature monopoles, chiral anomaly
Quantum Dot
Confinement energy levels, size-dependent spectrum — Coulomb blockade, charging diagram
Black Hole Accretion
Shakura-Sunyaev thin disk — temperature profile, emitted spectrum, relativistic ISCO, Eddington luminosity
Neutron Star Crust
Nuclear pasta phases — gnocchi→spaghetti→lasagna→Swiss cheese, density layers, phase transitions
Hawking Radiation
Virtual pair creation near horizon — T=ℏc³/8πGMk, evaporation timescale, blackbody spectrum
Baryon Acoustic Oscillations
Sound horizon in early universe — photon-baryon plasma waves, BAO peak in galaxy power spectrum
Dark Matter Halo
NFW profile density — flat vs Keplerian rotation curves, subhalo distribution, mass function
Cosmic Reionization
HII bubbles from first stars/galaxies — Gunn-Peterson trough, overlap epoch, neutral fraction evolution
CMB Anisotropy
Angular power spectrum Cl vs ℓ — acoustic peaks, Sachs-Wolfe plateau, Silk diffusion damping tail
Nucleosynthesis Path
Chart of nuclides — s-process and r-process paths, beta-decay chains, nuclear magic numbers
Supernova Light Curve
Type Ia Phillips relation — 56Ni decay chain powering peak, SN type comparison
Exoplanet Transit
Star+planet simulation — Mandel-Agol transit model, limb darkening, live light curve, transit timing
Möbius Transformations
Compose rotations, inversions, translations on the Riemann sphere — circles map to circles, six types, fixed points
Minimal Surfaces
Enneper, catenoid, helicoid, Costa surface — zero mean curvature H=0, curvature coloring, saddle geometry
Willmore Flow
Surfaces evolving to minimize bending energy ∫H²dA — tori flowing toward Clifford torus, Marques-Neves theorem
Mean Curvature Flow
Curves shrinking to circles, dumbbells pinching, Gage-Hamilton-Grayson theorem — topology change in finite time
Alexandrov Polyhedra
Unfold convex polyhedra to nets, fold back — any intrinsic metric on S² = unique convex polyhedron (1941)
Discrete Differential Geometry
Angle defect curvature, cotangent Laplacian, mesh smoothing — Gauss-Bonnet on triangle meshes
Spinor Visualization
Dirac belt trick, plate trick, SU(2)→SO(3) double cover — π₁(SO(3))=ℤ/2 made interactive
Clifford Algebra
Geometric product, rotors, bivectors in Cl(3,0) — rotate vectors via R·v·R† with live blade decomposition
Octonion Multiplication
Fano plane mnemonic, non-associativity (ab)c≠a(bc), G₂ symmetry — the largest normed division algebra
Dynkin Diagrams
ADE classification: A_n, B_n, C_n, D_n, E₆, E₇, E₈, F₄, G₂ — root systems, Weyl groups, Cartan matrices
Axelrod Culture Model
F cultural features, K traits each — agents influence similar neighbors, watch polarization or consensus emerge
El Farol Bar Problem
Minority game dynamics — agents use competing predictors, attendance self-organizes near capacity threshold
Auction Dynamics
First-price, second-price, Dutch, English auctions — bid strategies, revenue equivalence theorem, winner's curse
Vicsek Model Flocking
N particles align with neighbors plus noise η — order parameter φ reveals collective motion phase transition
Termite Mound Building
Stigmergic construction — pheromone-driven piling creates emergent arches and pillars from simple local rules
Bee Swarm Intelligence
Scouts advertise nest sites via waggle dance — quorum sensing drives collective decision toward best site
Traffic Jam Formation
Nagel-Schreckenberg CA — phantom jams emerge from stop-and-go waves, density-flow fundamental diagram
Watts-Strogatz Small World
Rewire a lattice with prob p — clustering stays high while path length drops: six degrees of separation
Granovetter Threshold Model
Agents adopt when neighbor fraction exceeds threshold — cascade dynamics, bimodal outcomes, tipping points
Spatial Prisoner's Dilemma
Cooperators and defectors on a lattice — clusters of cooperation survive against defectors, Nowak & May 1992
Entropy Visualization
Shannon entropy explorer: H(p) curve, joint entropy, mutual information, KL divergence, Markov chain entropy rates
Lempel-Ziv Compression
LZ77/LZ78 algorithm builds dictionary on live text — sliding window, phrase tree, compression ratio vs source entropy
Channel Capacity
Shannon capacity for BSC, BEC, AWGN channels — C=max I(X;Y), water-filling for parallel channels
LDPC Codes
Bipartite Tanner graph, belief propagation decoding animation, BER vs Eb/N0 approaching Shannon limit
Polar Codes
Channel polarization: Bhattacharyya parameters bifurcating toward 0 and 1, frozen vs information bit selection
Compressed Sensing
Sparse signal recovery from few measurements — l1 vs l2 geometry, RIP, OMP algorithm step-by-step
Matrix Completion
Netflix problem: low-rank matrix recovery via SVT — revealed entries reconstruct the full matrix
Factor Graph Inference
Sum-product and max-product message passing on bipartite factor graphs — marginals, MAP, loopy vs tree
Viterbi Algorithm
HMM trellis decoding: dynamic programming path tracking and animated backtrace to most-likely state sequence
Particle Filter
Sequential Monte Carlo robot localization — particles represent belief, resampling, motion and measurement updates
Leaky Integrate-and-Fire
Membrane potential trace, F-I curve, refractory period — explore spike generation and neural excitability
Cable Equation
Voltage spreading along passive dendrite — space constant λ, electrotonic distance, signal attenuation
Dendritic Computation
Compartmental model, local dendritic spikes, coincidence detection — neurons as more than threshold devices
Spike Timing Plasticity
STDP: Hebbian learning with temporal asymmetry, LTP/LTD windows, synaptic weight evolution
Neural Coding
Rate vs temporal coding, tuning curves, population vector decoding, Fisher information, sparse coding
Hippocampal Place Cells
Rat navigating 2D environment, place fields, theta sequences, phase precession, cognitive map
Grid Cells
Hexagonal firing fields in entorhinal cortex — grid scale, orientation, path integration, population code
Drift-Diffusion Model
Evidence accumulation to threshold, reaction time distributions, speed-accuracy tradeoff, SPRT
Dopamine & Reward
Temporal difference learning, Schultz RPE experiment, δ = R + γV(s') − V(s), habit vs goal-directed
Predictive Coding
Hierarchical generative model — prediction errors up, predictions down, precision-weighted inference
Rubber Elasticity
Freely-jointed chain model, entropic spring force, Mooney-Rivlin stress-strain, crosslink density
Glass Transition
VTF viscosity divergence, fragility index, Kauzmann paradox, dynamic heterogeneity
Liquid Crystal Phases
Nematic, smectic, cholesteric director fields, Frederiks transition, ±½ disclination defects
Colloidal Crystal
Hard sphere phase diagram, entropy-driven crystallization, Bragg diffraction, Asakura-Oosawa depletion
Emulsion Stability
DLVO theory, van der Waals vs electrostatics, energy barrier vs salt, Ostwald ripening
Surfactant Self-Assembly
CMC, packing parameter, sphere/cylinder/bilayer transitions, micelle dynamics
Catalytic Cycle
Turnover mechanism, energy landscape, TON/TOF, substrate binding, Sabatier volcano plot
Enzyme Kinetics & Inhibition
Michaelis-Menten, competitive/uncompetitive/mixed inhibition, Lineweaver-Burk, Hill cooperativity
Michaelis-Menten: Single Molecule
Stochastic single-molecule kinetics, burst statistics, FRET efficiency histogram, trajectory vs ensemble
Protein Aggregation
Nucleation-elongation kinetics, Avrami equation, fibril growth, seeding, inhibitor screening, ThT fluorescence
Prospect Theory
Kahneman & Tversky's S-shaped value function — loss aversion, diminishing sensitivity, and the reflection effect.
Hyperbolic Discounting
Exponential vs hyperbolic discount curves — preference reversals and why we procrastinate.
Cournot Duopoly
Two firms choose quantities simultaneously — reaction functions, Nash equilibrium, and efficiency benchmarks.
Cobweb Model
Agricultural price cycles with lagged supply — convergent, divergent, and neutral oscillations.
Edgeworth Box
Two-person exchange economy with indifference curves, contract curve, and draggable endowment point.
Price Discrimination
First, second, and third degree pricing — consumer surplus, producer surplus, and deadweight loss.
Rent-Seeking (Tullock Contest)
N firms spending to win a prize — socially wasteful equilibrium and full rent dissipation as N grows.
Giffen Good
Upward-sloping demand decomposed via Slutsky equation — substitution vs income effect for inferior goods.
Nash Bargaining
Maximize the product of gains from disagreement — geometric construction on the Pareto frontier.
Vickrey Auction
Second-price sealed-bid — dominant strategy truthfulness and revenue equivalence with first-price.
Phantom Traffic Jams
Stop-and-go waves emerge spontaneously on a ring road above critical car density — no accident required.
Forest Succession
Cellular automaton: bare ground → pioneer grasses → shrubs → mature forest driven by shade tolerance.
Population Viability
Stochastic simulations reveal the minimum viable population size needed to avoid extinction over 100 years.
Island Biogeography
MacArthur-Wilson theory: species richness as equilibrium between immigration and extinction on islands.
Optimal Foraging
Marginal Value Theorem: find the optimal patch residence time via the graphical tangent-line solution.
Evolutionary Arms Race
Red Queen dynamics: predator and prey coevolve on fitness landscapes with escalating trait values.
Mimicry Evolution
Batesian vs Müllerian mimicry: how warning-color evolution depends on predator learning and mimic frequency.
Niche Partitioning
Tilman's ZNGI phase plane: see exactly when two species coexist versus when competitive exclusion occurs.
Metapopulation Dynamics
Levins model: patch occupancy as balance of colonization and extinction, with rescue effect and mainland variants.
Trophic Cascade
Remove the apex predator and watch mesopredator release drive herbivore explosion and vegetation collapse.
Jet Stream Rossby Waves
Planetary-scale meanders in the jet stream driven by Coriolis and PV conservation, with blocking pattern formation.
Orographic Mountain Waves
Airflow over topography creates lee waves, rotor zones, and hydraulic jumps controlled by the Froude number.
ENSO — El Niño / La Niña
Walker circulation, thermocline depth oscillation, and coupled ocean-atmosphere Bjerknes feedback cycle.
Hadley Cell Circulation
Equatorial convection drives subtropical subsidence; cell width depends on rotation rate via the Held-Hou model.
Ocean Gyres — Sverdrup Balance
Wind-driven Sverdrup interior meets intensified western boundary currents in the Stommel gyre model.
Thermohaline Circulation
Density-driven AMOC with salinity-temperature feedbacks and the freshwater-forcing collapse bifurcation.
South Asian Monsoon
Seasonal land-sea heating contrast drives wind reversal; Himalayan orography focuses extreme rainfall.
Arctic Amplification
Ice-albedo and lapse-rate feedbacks make the Arctic warm 3–4× faster than the global mean.
Cloud Radiative Feedback
Low clouds cool by reflecting SW; high clouds warm by trapping LW — the sign of net CRE determines feedback.
Radiative Forcing
Earth's energy budget with interactive greenhouse gas sliders showing logarithmic CO₂ forcing and equilibrium warming.
Crystal Growth PDE
Phase-field model of dendritic solidification: anisotropic surface energy creates snowflake-like dendrites from a supercooled melt.
Dislocation Dynamics
Edge and screw dislocations glide under applied stress, with Frank-Read source multiplication and Hall-Petch strengthening.
Grain Boundary Migration
Ostwald ripening in polycrystals: larger grains consume smaller ones via curvature-driven boundary migration and coarsening kinetics.
Spinodal Decomposition
Cahn-Hilliard equation drives spontaneous phase separation in unstable mixtures, forming bicontinuous microstructure without nucleation.
Martensitic Transformation
Diffusionless austenite-to-martensite shear transformation with Bain correspondence, habit plane, and Koistinen-Marburger kinetics.
Work Hardening
Taylor hardening law: dislocation density increases with plastic strain, raising flow stress via σ = σ₀ + αμb√ρ.
Creep Deformation
Time-dependent plastic flow at high temperature: primary, secondary, and tertiary creep regimes with Arrhenius temperature dependence.
Fatigue Crack Growth
Paris law da/dN = C(ΔK)^m governs crack growth rate; S-N curve, threshold, stable Paris regime, and fracture toughness limit.
Shape Memory Alloy
NiTi superelastic and shape-memory effects: reversible stress- or temperature-driven martensitic transformation with hysteresis loop.
Superelasticity
Stress-induced martensitic transformation enables up to 8% recoverable strain with a characteristic loading-unloading hysteresis loop.
Seismic Wave Propagation
P-waves and S-waves travel through Earth's layered interior, refracting at the Moho and core-mantle boundary to create shadow zones and travel-time curves.
Mantle Convection
Thermal Rayleigh-Bénard convection in Earth's mantle: hot material rises, cools, and sinks in cells driven by temperature and viscosity contrast.
Plate Tectonics Forces
Ridge push, slab pull, and basal drag drive plate motion; explore which forces dominate and how plate length affects velocity.
Geomagnetic Dynamo
Convection in the liquid outer core generates Earth's dipole field; trigger polarity reversals and explore the geomagnetic timescale.
Geoid Anomaly
Earth's equipotential gravity surface deviates from the ellipsoid due to mountains, dense mantle bodies, and ice sheets; explore spherical harmonic decomposition.
Volcanic Eruption Dynamics
Magma chamber pressurization and volatile exsolution control eruption style from effusive Hawaiian flows to catastrophic Plinian columns on the VEI scale.
Earthquake Rupture Dynamics
Static friction gives way to dynamic weakening as a rupture front propagates along a fault; visualize stress drop, radiation pattern, and Omori aftershock decay.
Mineral Phase Diagram
Pressure-temperature phase diagram for silicate minerals traces olivine through wadsleyite and ringwoodite to bridgmanite, explaining mantle seismic discontinuities.
Subduction Zone
Oceanic plate descends, releases water that triggers arc volcanism above; explore Wadati-Benioff seismicity, slab dehydration, and back-arc spreading.
Earth Rotation Wobble
The Chandler wobble (~433 days) and annual oscillation combine to trace Earth's pole in a beating spiral; length-of-day varies by milliseconds with seasonal atmosphere exchange.
Opponent Process Color Theory
Red-green and blue-yellow opponent channels explain color afterimages, unique hues, and the structure of the CIE chromaticity diagram.
Retinal Ganglion Receptive Fields
Center-surround DoG receptive fields create bandpass spatial frequency tuning and lateral inhibition that sharpens edges into Mach bands.
Motion Aftereffect
Adapt to downward motion and a static pattern appears to drift upward — the waterfall illusion modeled as opponent MT neuron fatigue.
Multistable Perception
Necker cube, Rubin vase, and binocular rivalry modeled as competing neural populations with mutual inhibition and slow adaptation.
Depth from Stereo
Julesz random-dot stereograms, binocular disparity geometry, the Vieth-Müller horopter, and Panum's fusional area explored interactively.
Auditory Scene Analysis
Bregman's stream segregation: interleaved tone sequences split into separate auditory objects above a frequency-separation and tempo threshold.
Speech Formants & Vowel Space
F1 and F2 resonances of the vocal tract map tongue height and advancement to the IPA vowel quadrilateral with live spectrum synthesis.
Tactile Two-Point Discrimination
Receptive field size and cortical magnification explain why fingertips resolve 2mm separations while the back requires 65mm.
Pain Gate Control Theory
Melzack-Wall 1965: large Aβ fibers excite SG interneurons that inhibit nociceptive T-cells, closing the gate — the basis of TENS therapy.
Vestibulo-Ocular Reflex
Head rotation drives compensatory eye movement via a 3-neuron arc in 7ms; explore semicircular canal dynamics, VOR gain, nystagmus, and adaptation.
Quantum Chromodynamics
Running coupling αₛ(Q²), color string confinement, and the QCD phase diagram from hadronic matter to quark-gluon plasma.
Feynman Path Integral
A particle takes all paths simultaneously — animate quantum amplitudes, watch phases interfere, and see the classical trajectory emerge from stationary phase.
Spontaneous Symmetry Breaking
Mexican hat potential: symmetric Lagrangian, asymmetric vacuum — animate field rolling to minimum and the massless Goldstone boson along the valley.
Higgs Mechanism
Gauge symmetry breaking gives mass to W/Z bosons by absorbing the Goldstone mode; Yukawa coupling generates fermion masses from the Higgs VEV.
Wilson Renormalization Group
Integrate out high-momentum shells, flow through theory space, and watch the Wilson-Fisher fixed point attract all 3D Ising universality class theories.
Anomalous Dimension
Quantum loop corrections shift field scaling dimensions away from classical values, controlling critical exponents and the running of couplings.
Conformal Bootstrap
Crossing symmetry of the 4-point function carves out an allowed region in (Δ_σ, Δ_ε) space with the 3D Ising CFT pinned at the kink.
Topological Field Theory
Chern-Simons theory: Wilson loop expectation values compute knot invariants, linking numbers, and the Jones polynomial from quantum field theory.
Instanton Tunneling
Instantons are Euclidean classical solutions tunneling between degenerate vacua — explore the double-well, instanton gas, and exponential tunneling rate.
Supersymmetry Breaking
SUSY algebra, superpartner spectrum, and soft terms from hidden-sector F-term breaking — see mass splittings and the Goldstino from broken SUSY.
Fourier Synthesis
Build square, sawtooth, and triangle waves from sine harmonics — watch the Gibbs overshoot and rotating phasor circles in real time.
Wolfram Rule 110
The Turing-complete elementary cellular automaton — explore all 256 Wolfram rules and watch Class IV complexity emerge from three-cell neighborhoods.
Mandala Generator
Rotationally symmetric patterns from rose curves and Lissajous figures — tune symmetry order, petal count, and layers to reveal dihedral group Dₙ structure.
Attractor Basin
Newton fractal basins of attraction for zⁿ=1 — the fractal boundaries reveal where tiny perturbations flip which root Newton's method converges to.
Voltage Clamp
Live Hodgkin-Huxley action potential simulation — tune sodium/potassium conductances and watch the Nobel Prize-winning equations fire in real time.
Boolean Lattice
The power set 2ⁿ as a Hasse diagram — click nodes to highlight upsets and explore Sperner's theorem about maximum antichains.
Torus Knot Flow
T(p,q) torus knots winding around an invisible torus — gcd(p,q) determines whether you get a knot or multi-component link.
Phase Space Portrait
Pendulum trajectories in (θ, ω) space — see libration orbits, separatrices, and how a periodic drive pushes the system toward chaos.
Fibonacci Spiral 3D
Phyllotactic seed placement on a 3D paraboloid using the golden angle 137.5° — the least-rational angle that maximizes packing efficiency.
Magnon Dispersion
Spin waves in a ferromagnetic lattice — the quadratic dispersion ω∝k² and anisotropy gap that underlies the Bloch T^(3/2) magnetization law.
Hyperboloid of Ruled Lines
A doubly-ruled quadric surface built from two families of straight lines — drag parameters to watch the waist radius and ruling geometry transform.
SSH Model: 1D Topological Insulator
Su-Schrieffer-Heeger chain: tune intra- and inter-cell hopping to cross the topological phase transition and see protected edge states appear.
Oseen-Burgers Vortex
An exact Navier-Stokes solution balancing axial strain and viscous diffusion — the vortex tube model underlying turbulent intermittency.
Bravais Lattice Explorer
All five 2D Bravais lattices with live Wigner-Seitz cell construction — the symmetry classification underlying every crystal structure.
Quantum Scattering Matrix
S-matrix transmission and reflection amplitudes for a 1D potential barrier — see quantum tunneling, resonances, and exact unitarity.
Liouville's Theorem
Phase-space volume is conserved under Hamiltonian flow — watch a blob of trajectories shear and stretch while its area stays exactly constant.
Adiabatic Invariant J = ∮p dq
The action integral preserved under slow parameter changes — sweep the oscillator frequency and watch E/ω remain constant while energy grows.
Hodge Star Operator
Duality between k-forms and (n−k)-forms — visualize how ★ maps vectors to planes, bivectors to normals, and underpins Maxwell's equations.
Clebsch-Gordan Coefficients
Angular momentum coupling: see how product states |j₁,m₁⟩ decompose into total angular momentum eigenstates |J,M⟩.
Wigner-Seitz Cell
Voronoi tessellation of crystal lattices — the primitive unit cell with full point-group symmetry, dual to the first Brillouin zone in k-space.
Bloch Oscillation
Electrons in a crystal lattice under a DC field oscillate instead of accelerating — Bragg reflection at the Brillouin zone boundary.
Legendre Transform
Geometric duality converting a convex function into its slope-space dual — the bridge between Lagrangians and Hamiltonians.
Noether Current
Every continuous symmetry of a physical action yields a conserved current ∂_μj^μ = 0 and a globally conserved charge.
Ward Identity
Gauge symmetry forces q_μM^μ = 0 in QED, decoupling longitudinal photons and constraining renormalization at all orders.
Polaron Cloud
An electron in a polar crystal drags a cloud of virtual phonons, increasing its effective mass — from weak to self-trapped regimes.
Debye Model
Heat capacity of solids from quantized lattice vibrations: T³ law at low temperatures, Dulong-Petit 3Nk_B at high temperatures.
Fermi's Golden Rule
Transition rate Γ = (2π/ℏ)|V|²ρ(E) from time-dependent perturbation theory — the sinc² energy window sharpens to a delta function.
Path Integral QM
Feynman's sum over all histories: every path contributes e^(iS/ℏ), with classical trajectories dominating via stationary phase.
Retarded Green Function
Causal impulse response G^R(t,t') — zero before the source, oscillatory decay after; poles below the real axis encode causality.
Spontaneous Emission
Excited atoms decay by emitting photons into the quantum vacuum — rate Γ ∝ ω³, ensemble follows N(t) = N₀e^(−Γt).
Jaynes-Cummings Model
Quantum atom-photon coupling in a cavity. Watch Rabi oscillations and collapse-revival as excitation coherently swaps between atom and field.
Berry Curvature
Momentum-space topology and geometric phase. Visualize the Berry curvature field and Chern number across the Brillouin zone.
Chern-Simons Theory
Anyonic braiding and topological gauge theory. See how anyons acquire fractional statistical phase as they wind around each other.
WKB Approximation
Semiclassical quantum tunneling through potential barriers. Watch the wavefunction decay exponentially in forbidden regions.
Born-Oppenheimer
Electronic and nuclear motion separation in molecules. Explore Morse potential vibrational levels and nuclear wavepacket dynamics.
Franck-Condon Principle
Vertical electronic transitions and vibrational overlap integrals. See how wavefunction overlap shapes vibronic absorption spectra.
Jahn-Teller Effect
Spontaneous symmetry breaking in degenerate electronic states. Explore the Mexican hat potential and pseudo-rotational dynamics.
Stark Effect
Electric field splitting of atomic energy levels. Visualize linear and quadratic Stark shifts and polarization of hydrogen wavefunctions.
Anomalous Zeeman Effect
Magnetic splitting with electron spin and Landé g-factor. See how spin doubles the magnetic moment and multiplies spectral lines.
Cooper Pairs
Phonon-mediated electron pairing and the BCS superconducting gap. Watch lattice distortions bind electrons across the Fermi sea.
Einstein Solid
Quantum harmonic oscillators model a crystal. Visualize Bose-Einstein occupancy, energy levels, and the heat capacity crossover from quantum to classical.
Van Hove Singularity
Where ∇_k E(k)=0, the density of states diverges. Explore the tight-binding Brillouin zone and the logarithmic DOS peak at the saddle point.
Density Matrix
Visualize quantum states on the Bloch sphere. Explore coherence, purity Tr(ρ²), and how off-diagonal elements encode quantum interference.
Lindblad Master Equation
Open quantum system dynamics with amplitude decay and dephasing. Watch the Bloch vector spiral toward steady state under Markovian dissipation.
Quantum Discord
Quantum correlations beyond entanglement: D = I(A:B) − J(A:B). Nonzero even for separable states, discord captures fundamental measurement disturbance.
Topological Order
Kitaev toric code anyons on a lattice. Place e-anyons and m-anyons, watch membrane operators, and explore topologically protected ground state degeneracy.
Fracton Model
Immobile fracton excitations and mobility-restricted lineons in the X-cube model. Sub-extensive GSD and ultra-slow thermalization on an interactive lattice.
Sachdev-Ye-Kitaev Model
N Majorana fermions with random all-to-all couplings. The SYK model saturates the chaos bound λ_L=2π/β and is dual to a 2D black hole via holography.
Holographic Entanglement
Ryu-Takayanagi formula: S(A) = Area(γ_A)/4G_N. Minimal geodesics in the Poincaré disk compute CFT entanglement entropy from AdS bulk geometry.
Replica Trick
Entanglement entropy via analytic continuation: S = −lim_{n→1} ∂_n Tr(ρ^n). Visualize Rényi entropies on n-sheeted Riemann surfaces with twist operators.
Zeta Regularization
Analytic continuation tames divergent sums: ζ(−1) = −1/12. Watch partial sum spirals in the complex plane as Re(s) crosses the convergence boundary.
Casimir Piston
Vacuum fluctuations between conducting plates produce a measurable attractive force F = −π²ℏcA/240d⁴. Visualize confined modes vs. the external continuum.
Penrose Process
Extract rotational energy from a Kerr black hole's ergosphere. A particle splits inside: one fragment falls with negative energy, the other escapes with more than it came in with.
Hawking Temperature
Black holes radiate thermally at T = ℏc³/8πGMk_B. As mass falls, temperature rises, accelerating evaporation in a runaway cascade ending in a gamma-ray burst.
Unruh Effect
An accelerating observer sees the quantum vacuum as a thermal bath at T = ℏa/2πck_B. "Particle" is observer-dependent — a vacuum for one is a furnace for another.
Gravitational Wave Chirp
Binary black hole inspiral sweeps upward in frequency as energy radiates away. The chirp mass M_c determines the entire waveform evolution f(t) ∝ (t_c−t)^(−3/8).
Baryon Asymmetry
Sakharov's three conditions explain why matter won over antimatter. One extra particle per billion survived annihilation — everything we see is that tiny residue.
Lensing Convergence
Gravitational lensing convergence κ maps the projected dark matter density. Watch Einstein rings form and arcs stretch as mass bends light around foreground galaxies.
CMB Power Spectrum
Acoustic oscillations in the primordial plasma froze at recombination. The spectrum of peaks encodes baryon density, dark matter, and the geometry of the universe.
Reionization Bubble
First stars carved Strömgren spheres through neutral hydrogen fog at z≈6–12. Sweep through the Epoch of Reionization and watch ionized bubbles percolate across the IGM.
Geodesic Deviation
Visualize how spacetime curvature separates nearby free-falling paths via the Jacobi equation and Riemann tensor.
Petrov Classification
Explore the six algebraic types of the Weyl curvature tensor and their principal null directions — from generic Type I to flat Type O.
Penrose Diagram
Conformal diagrams compress all of spacetime into a finite picture: explore Minkowski, Schwarzschild, and de Sitter causal structure.
Singularity Theorem
Watch the Penrose-Hawking theorem unfold: once a trapped surface forms, the Raychaudhuri equation guarantees geodesic incompleteness.
ADC Sampling
See Nyquist-Shannon sampling in action — tune signal frequency past the Nyquist limit to trigger aliasing, and explore quantization noise.
Signal Envelope
Visualize AM modulation, ADSR envelopes, and instantaneous amplitude as the smooth curve outlining an oscillating signal.
Hilbert Transform
The 90° phase-shift operator creates the analytic signal, enabling instantaneous frequency and amplitude from any real waveform.
Matched Filter
The optimal SNR detector: cross-correlate the received signal with a chirp template to find hidden pulses in noise.
Viterbi Decoder
Dynamic programming finds the most likely hidden state sequence in a Markov model — watch the trellis build and traceback.
EM Algorithm
Expectation-Maximization fits Gaussian mixture models by alternating E-step responsibilities and M-step parameter updates.
Variational Bayes
Approximate Bayesian inference by minimizing KL divergence. Watch the ELBO climb as a variational distribution fits a complex posterior.
Gaussian Process
Click to add observations and watch the GP posterior update in real time. Uncertainty collapses near data and expands in unexplored regions.
Kernel PCA
Nonlinear dimensionality reduction via the kernel trick. RBF, polynomial, and linear kernels transform inseparable data into separable projections.
Manifold Learning
Swiss rolls, tori, and spheres unrolled into their intrinsic 2D parameterizations. Geodesic distances reveal structure hidden from Euclidean metrics.
Boltzmann Machine
Energy-based RBM with Gibbs sampling and contrastive divergence. Watch the free energy descend as visible and hidden units reach equilibrium.
Diffusion Score Models
Forward noising and reverse SDE denoising visualized. Particles follow the learned score field from Gaussian noise back to data manifold.
Score Matching
Learning the gradient of the log-density without the partition function. Langevin dynamics particles navigate the score field to sample from the distribution.
Contrastive Learning
Self-supervised representation learning via NT-Xent loss. Watch embeddings cluster by class as positives attract and negatives repel on the hypersphere.
Attention Mechanism
Scaled dot-product attention with queries, keys, and values. Interactive multi-head attention matrix shows how each token attends to every other token.
Transformer Circuit
Mechanistic interpretability: residual stream, induction heads, and neuron superposition. Trace information flow through transformer layers at the circuit level.
Graph Neural Networks
Watch activation waves propagate through graph nodes as GNN message-passing layers build richer neighborhood representations.
Message Passing
Belief propagation on factor graphs: variable and factor nodes exchange messages until marginal beliefs converge.
Spectral Graph Convolution
Filter signals on graphs in the Laplacian eigenvector domain — visualize low-pass, high-pass, and band-pass graph filters.
Flow Matching
Transport particles from Gaussian noise to complex target distributions along straight conditional flow trajectories.
Consistency Models
Self-consistent denoising: every noisy version of a sample maps directly to its clean endpoint in a single network call.
Rectified Flow
Watch crossing transport paths straighten through reflow iterations — the mechanism powering Stable Diffusion 3 and FLUX.
Latent Diffusion
The full Stable Diffusion pipeline: VAE encoding, latent denoising with classifier-free guidance, and pixel-space decoding.
VAE Disentanglement
Traverse independent latent dimensions in a β-VAE: each axis controls one factor of variation — shape, hue, or size.
GAN Training Dynamics
Live minimax competition between generator and discriminator, with real-time loss curves and Nash equilibrium detection.
Mode Collapse
Visualize how GANs drop modes of the data distribution and compare mode coverage between real and generated samples.
Wasserstein GAN
Watch generator distribution converge to real data using Earth Mover's distance and the Lipschitz critic.
Policy Gradient
REINFORCE algorithm trains an agent on a grid world by increasing probabilities of high-return actions.
Q-Learning
Temporal difference learning builds optimal Q-values via Bellman updates — watch the value heatmap emerge.
Actor-Critic
Joint actor-critic training balances a pendulum — critic's value function visualized as an angle-velocity heatmap.
MCTS Search
Monte Carlo Tree Search grows a game tree via UCB1-guided selection, expansion, simulation, and backpropagation.
Minimax Tree
Animated minimax search backs up leaf values through alternating MAX and MIN layers to find optimal play.
Alpha-Beta Pruning
See alpha-beta cutoffs eliminate irrelevant game tree branches in real time, slashing nodes evaluated.
Evolutionary Strategy
Population of solutions mutates and selects across multimodal landscapes with adaptive step size.
Particle Swarm
Swarm particles navigate a fitness landscape guided by personal memory and collective global best.
Tabu Search
Memory-guided local search escapes local optima by forbidding recently visited solutions via a tabu list.
Branch & Bound
Watch a search tree grow and prune as B&B solves integer programs exactly, bounding subtrees to avoid exhaustive enumeration.
Simplex Method
Dantzig's 1947 algorithm walks the vertices of the feasible polytope, following improving edges until the global optimum is reached.
Interior Point Method
Karmarkar's polynomial-time LP algorithm stays strictly inside the feasible region, following the central path via logarithmic barriers.
Duality Gap
Visualize primal and dual bounds converging — strong duality holds for convex programs, while non-convexity leaves a persistent gap.
Semidefinite Programming
Optimize over the cone of positive semidefinite matrices — the foundation of MAX-CUT relaxations, quantum info, and sum-of-squares proofs.
Sum of Squares
Certify polynomial nonnegativity by decomposing into squared components — Hilbert's 17th problem, answered by Artin 1927, now an SDP.
Ellipsoid Method
The first polynomial-time LP algorithm: shrink an ellipsoid containing the feasible set by cutting through its center with violated constraints.
Cutting Plane Method
Gomory's 1958 method adds valid inequalities that exclude fractional LP optima, progressively tightening the relaxation toward integer optimality.
Benders Decomposition
Split large MIPs into a master problem (integer) and subproblem (continuous), exchanging Benders cuts until upper and lower bounds converge.
Lagrangian Relaxation
Move hard constraints into the objective with multiplier λ, then maximize the concave dual function via subgradient ascent for tight lower bounds.
Column Generation
Solve massive LPs by iteratively generating only profitable columns via pricing subproblems. Powers crew scheduling and cutting stock.
Stable Matching
Gale-Shapley algorithm finds a matching with no blocking pairs in O(n²). Nobel Prize-winning theory behind medical residency assignment.
Vehicle Routing
Capacitated VRP with nearest-neighbor construction and 2-opt improvement. Generalizes TSP to fleets serving customers from a depot.
Bin Packing
Compare First Fit vs First Fit Decreasing heuristics. FFD uses at most (11/9)OPT+1 bins — tight bound proven by Johnson 1973.
Maximum Clique
Find the largest fully-connected subgraph using Bron-Kerbosch with pivoting. NP-complete and hard to approximate within n^(1-ε).
Vertex Cover
Minimum node set touching every edge via the classic 2-approximation. Tight inapproximability under the Unique Games Conjecture.
Feedback Arc Set
Remove the fewest edges to make a directed graph acyclic. Topological ordering yields a 2-approximation. Used in deadlock prevention.
Facility Location
Open facilities to minimize total opening cost plus client assignment distances. Best approximation ratio: 1.488 (Li 2013).
Set Cover
Greedy algorithm achieves optimal ln(n) approximation for covering a universe with minimum subsets. Feige 1998 proves no better is possible.
PageRank
Random surfer Markov chain converges via power iteration to the principal eigenvector. Visualize damping factor and convergence rate live.
HITS Algorithm
Kleinberg's hub-authority duality: hubs link to good authorities, authorities are linked by good hubs. Converges to eigenvectors of AᵀA and AAᵀ.
Girvan-Newman
Community detection by iteratively removing edges with highest betweenness centrality. Bridge edges score highest, revealing natural cluster structure.
Louvain Method
Greedy modularity maximization in two phases: local node reassignment then community aggregation. Scales to billions of nodes.
Label Propagation
Near-linear O(m) community detection: each node adopts its neighbors' most frequent label. Watch community identity spread from seed nodes outward.
Knowledge Graph
Semantic triples (Subject, Predicate, Object) encode structured world knowledge. Explore multi-hop subgraph queries over typed entities and relations.
Embedding Walk
Random walks generate node sequences for Word2Vec skip-gram training. Visit frequency shapes the embedding space — hubs cluster at the center.
node2vec
Biased walks interpolate BFS (homophily) and DFS (structural roles) via return bias p and in-out bias q. Switch modes and watch the explorer shift strategy.
Graph Attention
GAT learns per-edge attention weights via softmax over neighbor compatibility scores. Edge brightness reveals which neighbors the network trusts most.
Dynamic Graph
Edges are born and die over time; preferential attachment drives scale-free degree distributions. Edge color tracks age from gold (fresh) to blue (fading).
Temporal Network
Timestamped contact sequences reveal bursty human interaction patterns. Time-respecting paths and contact windows control information spreading speed.
Cascade Model
Watch information spread through a social network. Each activated node attempts to convert neighbors with probability p — the independent cascade model.
Hypergraph Neural Networks
Message passing beyond pairwise edges. Hyperedges connect arbitrary groups of nodes, capturing higher-order interactions invisible to standard GNNs.
Motif Counting
Triangles, paths, stars, and squares — the recurring subgraph patterns that define network identity. Compare motif frequencies across random graph densities.
Graphlet Kernel
Measure graph similarity via normalized graphlet degree distributions. The cosine similarity of two graphs' subgraph fingerprints defines their kernel value.
Weisfeiler-Leman
Iterative color refinement for graph isomorphism — and the exact expressive limit of Graph Neural Networks. Watch color histograms stabilize across rounds.
Persistent Homology
Track topological features as filtration ε grows. The persistence diagram records when components merge and loops form, robust to noise by the stability theorem.
TDA Mapper
Compress high-dimensional point clouds into graph summaries via overlapping covers and clustering. Loops and branches reveal global shape invisible to PCA.
Cellular Sheaf
Attach vector stalks to graph cells and watch sheaf diffusion minimize the coherence energy — data flowing toward globally consistent sections.
Discrete Morse Theory
Critical cells, gradient vector fields, and topological invariants on simplicial grids. The Euler characteristic emerges from counting minima, saddles, and maxima.
Combinatorial Topology
Build simplicial complexes and compute Betti numbers live. Vertices, edges, triangles, and tetrahedra combine to reveal connected components, loops, and voids.
Random Simplicial Complexes
Watch topology emerge from random geometry — Betti numbers, phase transitions, and the mathematical foundation of TDA.
Matroid Theory
The abstract structure of independence unifying linear algebra and graph theory — spanning trees, greedy algorithms, and the Rado-Edmonds theorem.
Newton Polytope
The convex hull of polynomial exponents encodes geometric structure — BKK theorem, mixed volume, and tropical geometry.
Ehrhart Polynomial
Count lattice points in dilated polytopes — a polynomial in the dilation factor connecting combinatorics, volume, and number theory.
Gröbner Basis
The polynomial analog of Gaussian elimination — Buchberger's algorithm, monomial orderings, and solving polynomial systems geometrically.
Resultant Theory
The Sylvester matrix determinant detects shared roots — classical elimination theory and the bridge to polynomial system solving.
Algebraic Variety
Zero sets of polynomial systems — the Nullstellensatz, singularities, elliptic curves, and Hilbert's foundational theorem.
Moduli Space
The space parametrizing all geometric shapes — tori, the modular group SL(2,Z), and the j-invariant classifying elliptic curves.
Derived Category
Chain complexes with quasi-isomorphisms inverted — Verdier's construction unifying cohomology and Kontsevich's mirror symmetry.
Intersection Theory
Counting curve crossings with sign and multiplicity — Bezout's theorem, the Chow ring, and enumerative geometry.
Spin Glass Energy Landscape
Frustrated magnetic systems with random bonds, metastable states, and slow relaxation below T_g.
Quasicrystal Diffraction
Penrose tiling with sharp 5-fold diffraction spots — long-range order without a periodic lattice.
Amphiphile Bilayer
Lipid molecules self-assembling into a bilayer membrane driven by the hydrophobic effect.
Topological Phonons
SSH phonon chain with alternating springs — topological edge modes inside the mechanical bandgap.
Geodetic Precession
Gyroscope spin axis drifts per orbit in curved spacetime — de Sitter precession from Schwarzschild geometry.
Spin Wave Magnon
Collective spin precession in a ferromagnet — quadratic dispersion ω∝k² and anisotropy gap.
Soap Film Minimal Surface
Plateau's problem solved numerically — zero mean curvature surfaces spanning wire-frame boundaries.
Granular Jamming
2D disk packing near the jamming transition — force chains and stress transmission at φ_J ≈ 0.84.
Phononic Bandgap
1D diatomic mass-spring chain — complete bandgap from Bragg scattering with transmission spectrum.
Photoelectric Threshold
Einstein's KE = hν − φ: vary photon frequency and metal work function to find the emission threshold.
Rayleigh-Bénard Convection
Convection rolls above the critical Rayleigh number
Bragg X-ray Reflection
Crystal plane interference and the 2d·sin(θ)=nλ condition
Polariton Condensate
Exciton-polariton anticrossing and non-equilibrium BEC
Faraday Wave Patterns
Parametrically driven standing waves — stripes, hexagons, squares
Brillouin Zone Explorer
First Brillouin zone for SC, BCC, FCC cubic lattices in 3D
Thermal Diffusivity
2D heat equation solver — draw heat and watch it diffuse
Van der Waals Gas
P-V isotherms, Maxwell construction, and the phase diagram
Andreev Reflection
Electron-hole retroreflection at a normal-superconductor interface
Ising Critical Point
2D Ising model with exact critical exponents β=1/8, γ=7/4
Adiabatic Demagnetization
Magnetic cooling cycle reaching millikelvin temperatures
Piezoelectric Crystal
Electric polarization from mechanical deformation
Mode-Locked Laser
Random vs locked phases — ultrashort pulse trains
Bose-Einstein Statistics
MB, FD, and BE distributions with BEC onset
Fano Resonance
Asymmetric lineshape from discrete-continuum interference
Microorganism Swimmer
Flagellar locomotion and the scallop theorem at Re≪1
Levy Flight Search
Levy flights vs Brownian motion for sparse foraging
Dislocation Glide
Edge dislocation motion and Peierls-Nabarro barrier
Nematic Liquid Crystal
Director field with +/-1/2 topological defects
Weak Quantum Measurement
Aharonov-Albert-Vaidman weak values beyond eigenrange
Ekman Spiral
Wind-driven ocean currents rotating with depth via Coriolis
Protein Folding Funnel
Energy landscape guiding chains to native state
Kirchhoff Migration
Seismic wavefield imaging via hyperbolic moveout
Topological Insulator Edge States
SSH chain bulk-boundary correspondence and winding number
Solar Wind Bow Shock
Magnetosphere deflecting supersonic solar wind plasma
Reaction Network Flux
Metabolic flux balance analysis on glycolysis network
Entanglement Entropy Block
Area law vs volume law in 1D quantum chains
Polymer Reptation
Tube-constrained snake diffusion of entangled chains
Magneto-Optical Kerr Effect
Polarization rotation and MOKE hysteresis loop
Bacterial Chemotaxis
E. coli run-and-tumble biased random walk up gradient
Quantum Random Walk
Ballistic quantum spreading vs diffusive classical walk
Acoustic Phonon Scattering
Electron-phonon coupling and scattering rates in crystals
Zak Phase Topology
Berry phase and topological edge states in 1D chains
Taylor-Couette Flow
Laminar to vortex to turbulent flow between rotating cylinders
Spatial SIR Epidemic
Wave fronts and R₀ on a 2D infection lattice
Hawking Radiation Analogue
Acoustic black hole sonic horizon and thermal phonon emission
Arrhenius Transition State
Reaction rates, energy barriers, and catalysis
Magnetic Frustration
Triangular antiferromagnet degenerate ground states
Wigner-Seitz Cell
Voronoi construction of Bravais lattice primitive cells
Genetic Toggle Switch
Bistable gene circuit nullclines and stochastic switching
Membrane Action Potential
Hodgkin-Huxley neuron Na⁺/K⁺ conductance dynamics
Coulomb Blockade
Single-electron tunneling and Coulomb diamonds
Kelvin-Helmholtz Instability
Shear flow vortex roll-up at fluid interfaces
Stochastic Differential Equations
Langevin dynamics, Fokker-Planck, double-well escape
Nuclear Shell Model
Magic numbers from spin-orbit splitting
Spin-Orbit Coupling
Fine structure splitting and Rashba-Dresselhaus bands
Autocatalytic Reaction
A+B→2B sigmoid growth and logistic equivalence
Newtonian vs Non-Newtonian
Viscosity models and Poiseuille flow profiles
Ion Trap (Paul Trap)
Mathieu stability diagram and ion trajectory
Cavity QED
Jaynes-Cummings Rabi oscillations and revivals
Rabi Splitting
Avoided crossing and dressed state energies E±
Surface Tension & Capillary
Liquid rise and contact angle in capillary tubes
Flux Quantization
Φ₀ = h/2e in superconducting rings and SQUID interference
Delay Differential Equations
Delayed feedback: stable → oscillation → chaos via Mackey-Glass
Virial Theorem
⟨KE⟩ = −½⟨PE⟩ in live N-body gravitational simulations
Seismic Wave Modes
P, S, Love, and Rayleigh waves; S-wave shadow zone
Photoionization Cross Section
σ(ν) ∝ ν⁻⁸/³ above threshold for hydrogen and multi-electron atoms
Kelvin Probe Force Microscopy
Contact potential difference via AC bias nulling and work function mapping
Molecular Vibration & IR
Normal modes of H₂O and CO₂; IR-active modes and spectra
Quantum Dot Spectrum
Discrete levels E ∝ 1/R² and size-tunable blueshift in 3D spherical box
Polaron Formation
Fröhlich coupling α: large vs small polaron, effective mass enhancement
Meissner Effect
Magnetic field expulsion in Type I & II superconductors
Logistic Map Bifurcation
Period-doubling cascade and Feigenbaum constant δ≈4.669
Gravitational Microlensing
Einstein ring, dual images & magnification light curve
Electron Spin Resonance
Bloch equations, Larmor precession & Rabi oscillations
Topological Defect Winding
2D vector field defects with winding numbers ±1, +2
Renormalization Group Flow
RG flow, Wilson-Fisher fixed point & universality classes
Quantum Phase Estimation
QPE circuit, QFT measurement & 1/2ⁿ precision scaling
Fiber Optic Dispersion
GVD pulse broadening, chirp & dispersion-shifted fiber
Swift-Hohenberg Patterns
Stripes, hexagons & squares from the SH PDE with ε control
Kinesin Molecular Motor
8 nm hand-over-hand steps, ATP cycle & force-velocity curve
Tidal Locking
Tidal torque synchronizing moon's spin to orbit
Bell Inequality
CHSH quantum violation S=2√2 vs classical S≤2
Elastic Wave Anisotropy
Slowness surfaces for anisotropic crystal qP/qS modes
Free Electron Laser
Undulator wiggling, FEL resonance, SASE exponential gain
Birefringence
Ordinary and extraordinary rays in a uniaxial crystal
Solar Convection Zone
Granulation, mixing-length theory, Schwarzschild criterion
Hysteresis in Elasticity
Stress-strain loop area equals energy dissipated per cycle
Optical Vortex Beam
Laguerre-Gaussian OAM modes with helical phase singularity
Critical Branching Process
Avalanche size P(s)~s^{-3/2} at σ=1 criticality
Nonequilibrium Entropy
Entropy production σ=J·X, Onsager reciprocal relations
Gravitational Wave Inspiral
Binary merger chirp and orbital decay
Fermi Surface Topology
Square lattice tight-binding and van Hove singularities
Reaction-Diffusion Spirals
FitzHugh-Nagumo spiral waves and cardiac arrhythmias
Cosmic Inflation Scalar Field
Inflaton slow-roll, de Sitter expansion, CMB tilt
Anharmonic Oscillator
Quartic potential, thermal expansion, Grüneisen parameter
Quantum Tunneling Barrier
Wave packet tunneling, WKB, resonant double barrier
Active Matter Flocking
Vicsek model order-disorder transition and density waves
Langmuir Probe
Plasma I-V characteristic, T_e, ion saturation current
Helfrich Membrane
Lipid bilayer fluctuations and bending rigidity spectrum
Graph Laplacian Spectrum
Fiedler vector, spectral gap, and graph partitioning
Topological Band Inversion
Band gap inversion and topological surface states
Ocean Internal Waves
Internal gravity waves in stratified fluid — ω²=N²sin²θ
Muon Anomalous Moment
g-2 precession, Feynman loops, and the 4.2σ discrepancy
Swarmalator Sync
Oscillators that synchronize and aggregate in space
Dendritic Crystal Growth
Diffusion-limited solidification with anisotropy and sidebranching
2D Inverse Energy Cascade
Energy flows upscale, enstrophy downscale — dual cascade
Neutrino Oscillation
Two-flavor mixing, MSW matter effect, and mass eigenstates
r-Process Nucleosynthesis
Rapid neutron capture in neutron star mergers — forging gold
Surface Code QEC
Topological quantum error correction and syndrome detection
Jamming Transition
Soft spheres at point J — coordination jump and force chains
Plasma Wakefield Acceleration
Laser-driven plasma waves accelerating particles to GeV/cm
Resonance Fluorescence
Mollow triplet and photon antibunching in a driven two-level atom
Topological Semimetal
Weyl nodes, Fermi arcs, and chiral anomaly in k-space
Atmospheric Gravity Waves
Mountain waves, critical level absorption, momentum flux in stratosphere
Quantum Dot Cellular Automata
Electrostatic binary encoding, wire propagation, and majority gate logic
Spin Squeezing
One-axis twisting squeezes quantum noise below the standard quantum limit
Phonon Polariton
IR photon–phonon strong coupling, Reststrahlen band, hyperbolic dispersion
BTW Sandpile (SOC)
Bak-Tang-Wiesenfeld sandpile spontaneously reaching criticality with power-law avalanches
Parity Violation in β-Decay
Wu experiment: asymmetric electron emission reveals V−A chiral weak force
Rayleigh-Plateau Jet Instability
Surface tension breaking a liquid jet into droplets at kR ≈ 0.697
Wigner Function
Quantum phase-space quasi-probability distribution
Tectonic Subduction
Oceanic plate diving beneath continental crust — Wadati-Benioff zone
Exciton Energy Transfer
Förster R⁻⁶ and Dexter exponential mechanisms for excitation transfer
Transmon Qubit
Superconducting anharmonic oscillator with Josephson junction
Stochastic Resonance
Noise-enhanced signal detection in a leaky integrate-and-fire neuron
Abrikosov Vortex Lattice
Triangular flux-tube lattice in Type II superconductors
Classical Nucleation Theory
Free energy barrier and critical nucleus in droplet formation
QCD Flux Tube
Linear confinement and string breaking between quark-antiquark pairs
Elastica: Buckled Beam
Euler buckling loads and post-buckling elastica curve from ODE
Poincaré Disk
Hyperbolic geometry with circular arc geodesics and {7,3} tiling
MHD Dynamo
Self-sustaining magnetic field from conducting fluid flow
Color Center (F-Center)
Electron at anion vacancy colors alkali halide crystals
Quantum Hall Plateaus
σ_xy = νe²/h plateaus as Landau levels fill in a 2DEG
Soft Mode Ferroelectric
Phonon frequency softening at the ferroelectric transition
Hawkes Process
Self-exciting point process with branching and avalanches
Geodesic Dome
Icosahedron subdivided into geodesic polyhedron in 3D
Ising Duality
Kramers-Wannier duality maps high-T to low-T Ising model
2D Phononic Crystal
Periodic inclusions create acoustic band gaps in 2D
AdS/CFT Holography
Ryu-Takayanagi entropy from geodesics in the AdS bulk
Dielectric Breakdown
Townsend avalanche, streamer formation, and Paschen curve
Smectic Liquid Crystal
Layered phase with focal conic defect domains and Sm-A→Sm-C tilt transition
Quantum Zeno Dynamics
Frequent measurement freezes quantum evolution; anti-Zeno effect at intermediate rates
Cosmic String Network
Topological defects from Kibble mechanism; scaling solution with loop reconnection
Spin Wave Spectrum
Magnon dispersion E(k) = D·k²; Bloch T^(3/2) law for magnetization
Topological Fluid Knot
Knotted vortex filaments; helicity H = ∫v·ω dV is a topological invariant
Entropic Spring Polymer
Freely-jointed chain entropy elasticity; WLC model for DNA stiffness
Superfluid Helium Roton
Phonon-roton dispersion of ⁴He; two-fluid superfluid fraction vs temperature
Amplitude Death Oscillators
Stuart-Landau oscillators: coupling kills oscillations; death island in parameter space
Optical Soliton (NLSE)
GVD and SPM balance in optical fiber; soliton maintains shape while linear pulse disperses
Conformal Mapping Flow
Joukowski transform maps cylinder to airfoil; lift from Kutta-Joukowski L = ρUΓ
Chandrasekhar Limit
White dwarf stability and stellar collapse at the quantum boundary
Dark Matter Halos
NFW density profiles and the invisible scaffolding of galaxies
Baryon Acoustic Oscillations
Sound waves frozen in the cosmic microwave background
Primordial Nucleosynthesis
Big Bang element formation in the first three minutes
Reionization Epoch
How the first stars lit up the dark cosmic ages
Magnetar Bursts
Catastrophic magnetic field rearrangements on neutron stars
Neutron Star Merger
Gravitational waves and r-process nucleosynthesis in kilonovae
Monopole Problem
Why inflation solved the overproduction of magnetic monopoles
Flatness Problem
Fine-tuning of cosmic density parameter and inflationary solution
Horizon Problem
CMB uniformity across causally disconnected regions
Frustration Networks
Competing interactions on graphs and the geometry of spin glass order
Replica Method
Analytic continuation over replicas to compute disordered system averages
Random Energy Model
Derrida's exactly solvable glass with frozen low-energy states
Cavity Method
Bethe-Peierls iteration on sparse factor graphs
Satisfiability Threshold
Sharp phase transition in random k-SAT at the satisfiability cliff
Random SAT
Random clause-variable graphs and the emergence of hard instances
Constraint Satisfaction
Arc consistency, backtracking, and the structure of solution spaces
SAT Phase Transition
From easy-SAT through hard-SAT to UNSAT as clause density grows
Algorithmic Threshold
The gap between statistical and algorithmic phase transitions
Turbo Codes
Iterative decoding near the Shannon limit via parallel concatenation
Hebbian Learning
Neurons that fire together wire together — synaptic weight dynamics
Neural Avalanches
Power-law distributed cascades at the edge of criticality in cortex
Brain Criticality
Self-organized criticality and optimal information processing in neural networks
Hippocampal Replay
Memory consolidation via sharp-wave ripples and sequence reactivation
Oscillatory Binding
Phase synchrony as a mechanism for binding distributed neural representations
Gamma Oscillations
40 Hz rhythms from excitatory-inhibitory balance in cortical circuits
Working Memory
Persistent activity in prefrontal cortex and the capacity limit of 7±2
Attractor Network
Fixed-point attractors, limit cycles, and chaotic attractors in neural dynamics
Energy-Based Model
Learning probability distributions via energy landscapes and contrastive terms
Restricted Boltzmann Machine
Bipartite stochastic network for unsupervised feature learning
Chemical Oscillator
Limit-cycle dynamics in coupled reaction networks
Turing Instability
Diffusion-driven pattern formation from homogeneous states
Reaction-Diffusion Waves
Traveling chemical fronts and pulse propagation in excitable media
FitzHugh-Nagumo Waves
Excitation and inhibition waves in the reduced nerve-impulse model
Spiral Waves
Rotating spiral patterns in reaction-diffusion systems and cardiac tissue
Target Patterns
Concentric chemical waves emanating from pacemaker centers
Scroll Waves
Three-dimensional spiral waves rotating around a filament
Autocatalysis
Self-amplifying reactions and the origin of chemical complexity
Bistable Switch
Hysteresis and snap-through in bistable chemical and genetic circuits
Oscillating Reactions
Periodic color changes and concentration oscillations in batch reactors
Modular Forms
Symmetric functions on the upper half-plane and the modularity theorem
L-Functions
Dirichlet series encoding arithmetic information in analytic form
Goldbach Conjecture
Every even integer greater than 2 as a sum of two primes
Twin Primes
Pairs of primes differing by 2 and the Hardy-Littlewood conjecture
Perfect Numbers
Numbers equal to the sum of their proper divisors and Mersenne connections
Diophantine Equations
Integer solutions to polynomial equations and Fermat's Last Theorem
Primitive Roots
Generators of multiplicative groups modulo n and discrete logarithms
Chinese Remainder Theorem
Simultaneous congruences and isomorphisms of residue rings
Algebraic Integers
Roots of monic integer polynomials and rings of integers in number fields
Class Field Theory
Abelian extensions of number fields and Artin reciprocity
Double-Diffusive Convection
Salt fingers and diffusive layering from competing heat and solute gradients
Saffman-Taylor Instability
Viscous fingering when a less viscous fluid displaces a more viscous one
Jamming Transition
Rigidity onset in dense granular packings and soft matter
Force Chains
Stress transmission through heterogeneous contact networks in granular media
Aeolian Transport
Wind-driven saltation, creep, and suspension of sediment grains
Fluvial Erosion
Stream power, incision, and landscape evolution by running water
River Meanders
Secondary flow, bank erosion, and oxbow lake formation in alluvial rivers
Delta Formation
Avulsion, lobe switching, and distributary network growth at river mouths
Braided Rivers
Multiple shifting channels and bar migration in high-energy fluvial systems
Turbidity Current
Density-driven submarine avalanches carving deep-sea canyons
Selection Pressure
How selection strength shapes allele frequency change over generations
Fitness Landscape
Adaptive peaks, valleys, and the topology of evolutionary search
Epistasis
Gene interactions that make fitness non-additive across loci
Sexual Selection
Runaway selection, honest signals, and the peacock's tail paradox
Kin Selection
Hamilton's rule rb > c and the evolution of altruism among relatives
Group Selection
Between-group competition and the evolution of cooperative traits
Inclusive Fitness
Total effect on reproduction weighted by genetic relatedness
Evolutionary Game Theory
Frequency-dependent selection and evolutionarily stable strategies
Hawk-Dove Game
Mixed-strategy ESS from conflict over resources with injury costs
Evolving Cooperation
Tit-for-tat, spatial structure, and the evolution of cooperation
Dirac Semimetal
Linear band crossings protected by crystal symmetry in 3D materials
Topological Surface States
Gapless Dirac cone surface states protected by time-reversal symmetry
Quantum Spin Hall
Spin-momentum locked edge states in 2D topological insulators
Anomalous Hall Effect
Berry curvature contribution to transverse conductivity in magnets
Rashba Effect
Momentum-dependent spin splitting from structural inversion asymmetry
Dresselhaus Effect
Bulk inversion asymmetry spin splitting in zinc-blende semiconductors
Spin Texture
Momentum-space spin orientation patterns on Fermi surfaces
Fermi Arcs
Topological surface states connecting Weyl point projections
Chern Insulator
Integer quantum Hall effect without Landau levels from Berry phase
SSH Model
Su-Schrieffer-Heeger chain with topological edge states and winding number
VC Dimension
Vapnik-Chervonenkis dimension and the capacity of hypothesis classes
PAC Learning
Probably approximately correct learning and generalization bounds
Sample Complexity
How many examples are needed to learn a concept to given accuracy?
Bias-Variance Tradeoff
Decomposing generalization error into bias, variance, and irreducible noise
Double Descent
Test error peaks at interpolation threshold then descends again in overparameterized models
Neural Scaling Laws
Power-law relationships between model size, data, compute, and loss
Lottery Ticket Hypothesis
Sparse subnetworks that can train to full accuracy from initialization
Neural Tangent Kernel
Infinite-width neural networks as kernel machines with a fixed NTK
Mean Field Neural Network
Signal propagation, chaos edge, and depth limits in random neural networks
Feature Learning
How neural networks learn structured representations beyond kernel methods
Endemic SIR
Disease persistence and endemic equilibria in SIR models with vital dynamics
Superspreading
Overdispersed transmission and the 80/20 rule of epidemic dynamics
Herd Immunity
Threshold vaccination coverage and indirect protection dynamics
Source-Sink Dynamics
Population persistence through dispersal from productive to sink habitats
Species-Area Relationship
Power-law S = cA^z and the fragmentation cost to biodiversity
Neutral Theory
Hubbell's unified neutral theory of biodiversity and biogeography
Niche Theory
Hutchinsonian hypervolumes and trait-based community assembly
Competitive Exclusion
Gause's principle: two species competing for one resource cannot coexist
Coexistence Mechanisms
Storage effect, relative nonlinearity, and modern coexistence theory
Keystone Species
Disproportionate community impact and the Paine starfish experiment
Preferential Attachment
Barabási-Albert scale-free networks from rich-get-richer growth
Small-World Rewiring
Watts-Strogatz interpolation between regular lattice and random graph
Network Resilience
Giant component collapse under random failures and targeted attacks
Network Synchronization
Kuramoto oscillators coupled on networks and the master stability function
Deffuant Model
Continuous opinion dynamics with bounded confidence and pairwise averaging
Bounded Confidence
Opinion clusters emerging from tolerance thresholds in social dynamics
Echo Chambers
Homophily, selective exposure, and radicalization in social networks
Social Contagion
Simple contagion spreading through social networks via weak and strong ties
Threshold Models
Granovetter cascade model and tipping points in collective behavior
Complex Contagion
Adoption requiring multiple exposures and the role of network structure
Stabilizer Codes
Pauli group stabilizers and the structure of quantum error correcting codes
Surface Code
Topological quantum error correction on a 2D lattice of qubits
Quantum Teleportation
Transmitting quantum states via entanglement and classical communication
Quantum Dense Coding
Sending two classical bits through a single qubit using entanglement
Quantum Key Distribution
Unconditionally secure key exchange via quantum no-cloning theorem
BB84 Protocol
Bennett-Brassard 1984 QKD using four polarization states in two bases
VQE Algorithm
Hybrid quantum-classical ground state energy estimation via variational circuits
QAOA
Quantum approximate optimization algorithm for combinatorial problems
Grover's Algorithm
Quadratic quantum speedup for unstructured search via amplitude amplification
Shor's Algorithm
Quantum period finding and polynomial-time integer factorization
Molecular Motors
ATP-driven protein machines converting chemical energy to mechanical work
Kinesin Walk
Hand-over-hand processive stepping of kinesin along microtubules
Myosin Power Stroke
Lever arm rotation and force generation in muscle contraction
Flagellar Motor
Bacterial rotary motor driven by proton motive force through the membrane
Cell Membrane Dynamics
Lateral diffusion, rafts, and the fluid mosaic model of the plasma membrane
Lipid Bilayer Phase
Gel-to-fluid transition and phase separation in lipid membranes
Membrane Undulations
Helfrich bending elasticity and thermal fluctuations of lipid membranes
Actin Polymerization
Treadmilling, critical concentration, and force generation by actin filaments
Dynamic Instability
GTP cap model and catastrophe-rescue cycles in microtubule dynamics
Cell Division Mechanics
Spindle assembly, chromosome segregation, and cytokinetic ring constriction
Szilard Engine
One-molecule heat engine, information, and the thermodynamics of measurement
Jarzynski Equality
Free energy differences from nonequilibrium work distributions
Fluctuation Theorem
Probability ratio for entropy-consuming vs entropy-producing trajectories
Landauer Principle
Minimum energy cost kT ln 2 to erase one bit of information
Irreversibility
How macroscopic time asymmetry emerges from time-reversible microscopic laws
Arrow of Time
Entropy increase, the past hypothesis, and the thermodynamic direction of time
Boltzmann H-Theorem
Monotonic decrease of the H-functional and approach to equilibrium
Loschmidt Paradox
Time-reversal symmetry vs irreversibility and the role of initial conditions
Poincaré Recurrence
Phase space volume conservation and eventual return to near-initial states
Ergodic Hypothesis
Time averages equal ensemble averages in ergodic dynamical systems
Wilson-Fisher Fixed Point
Epsilon expansion and the interacting fixed point of scalar field theory
Scaling Hypothesis
Widom scaling and the collapse of thermodynamic functions near criticality
Universality Classes
Why disparate systems share identical critical exponents near phase transitions
Lower Critical Dimension
Mermin-Wagner theorem and the destruction of order by fluctuations in low d
Upper Critical Dimension
Mean field theory validity and Ginzburg criterion above d_uc
Dangerously Irrelevant
RG operators that are irrelevant but still affect scaling in singular ways
CFT Basics
Conformal group, primary operators, and the state-operator correspondence
Operator Product Expansion
Short-distance singularities and the algebraic structure of local operators
Crossing Symmetry
Equality of s- and t-channel OPE decompositions constraining CFT data
Conformal Bootstrap
Constraining CFT spectra via crossing symmetry and unitarity bounds
Turing Pattern
Gray-Scott reaction-diffusion morphogenesis — spots and stripes self-organizing from chemical noise
Flocking Rules
Reynolds' Boids: separation, alignment, cohesion producing emergent murmurations from 200+ agents
Hyperbolic Space
Poincaré disk tessellations — infinitely many congruent polygons tiling negatively curved space
Automaton Music
Wolfram elementary automata sonified — Rule 30, 90, 110 mapped to musical scales beat by beat
Langton's Loops
Self-replicating loops in a cellular automaton — each loop encodes its own construction blueprint
Growing Network
Barabási–Albert preferential attachment — watch scale-free hubs emerge with P(k)~k⁻³ degree distribution
Quasicrystal Tiling
Penrose-style aperiodic tiling via de Bruijn pentagrid — five-fold symmetry that never repeats
Emergence Game
Conway's Game of Life and variants — paint cells, discover gliders, guns, pulsars, and alien rule universes
Spin Dynamics
2D XY model Metropolis simulation — watch vortex-antivortex pairs emerge near the Kosterlitz-Thouless transition
Network Cascade
Watts threshold contagion model — click nodes to seed cascades of failure, infection, or belief spreading
Strange Basin
Newton fractal basins of attraction on complex polynomials — fractal boundaries between roots, zoomable and pannable.
Sandpile Avalanche
Abelian sandpile model — add grains, watch cascading avalanches self-organize into a critical state with power-law sizes.
Topological Phase
SSH chain topology — tune hopping parameters to cross the topological phase transition and watch edge states appear.
Magnetic Vortex
Abrikosov vortex lattice in a type-II superconductor — vortices repel and pin to form the famous triangular lattice.
Evolutionary Arms Race
Predator-prey coevolution on a grid — watch speed and camouflage traits shift under Red Queen selection pressure.
Crowd Wisdom
Wisdom of crowds aggregation — add noisy estimators and watch the crowd mean converge to truth as 1/√N.
Self-Organized Critical
Forest fire model hovering at criticality — trees grow, lightning strikes, fires cascade with power-law size distribution.
Synchronization
Kuramoto model — N coupled oscillators with random frequencies spontaneously synchronize as coupling K crosses K_c.
Chimera State
Non-locally coupled oscillators sustaining simultaneous synchronized and chaotic regions — a counterintuitive symmetry break.
Entropy Landscape
Shannon entropy visualization across distribution shapes — watch entropy respond as you reshape probability landscapes.
Protein Energy Landscape
Funnel-shaped folding landscape with a Monte Carlo walker escaping misfolding traps — temperature and frustration controls.
Economic Cycles
Goodwin model: labor share and employment rate cycle endogenously without external shocks — limit cycles in the phase plane.
Drunkard's Walk
Random walks in diffusive, ballistic, and Lévy regimes — beautiful trails showing the √t displacement law and superdiffusion.
Adaptive Landscape
Wright's fitness landscape — populations drift up peaks under selection and genetic drift, splitting across valleys into species.
Genotype Space
NK model fitness landscape — ruggedness slider from smooth single-peak to multi-peaked chaos; population climbs by mutation and selection.
SEIR Epidemic Model
Full SEIR with seasonal forcing, births, deaths, and vaccination — endemic equilibria, epidemic waves, and herd immunity threshold.
Viscous Fingering
Saffman-Taylor instability — less viscous fluid invades more viscous fluid in a Hele-Shaw cell, growing beautiful fractal fingers.
Osmotic Pressure
Van't Hoff osmosis — solute and water molecules animate across a semipermeable membrane until pressure equilibrium is reached.
Membrane Bilayer
Lipid bilayer self-assembly — amphiphilic molecules spontaneously organize from random scatter into an ordered membrane bilayer.
Circadian Rhythm
Goodwin oscillator — negative feedback of mRNA, protein, and repressor generates a ~24-hour biological clock with light entrainment.
Reaction-Diffusion Patterns
Gray-Scott two-species model generating spots, stripes, labyrinths, and spirals with adjustable kinetics.
Belousov-Zhabotinsky Reaction
Oregonator model of the BZ reaction — spiral waves and target patterns propagating across a chemical grid.
Glycolytic Oscillator
Selkov model: limit cycle in ATP/ADP phase space reveals how metabolism generates sustained biochemical oscillations.
Repressilator
Elowitz & Leibler's synthetic gene oscillator — three mutually repressing genes producing rhythmic protein waves.
Genetic Toggle Switch
Gardner et al. bistable gene circuit with two attractors, bifurcation diagram, and noise-driven state switching.
Stochastic Gene Expression
Gillespie SSA exact simulation of mRNA and protein birth-death — bursty transcription, Fano factor, distributions.
Intrinsic vs Extrinsic Noise
Dual-reporter decomposition: CFP/YFP scatter plot reveals how noise sources differ inside living cells.
Cell Cycle Oscillator
Goldbeter CDK model: cyclin waves driving G1/S/G2/M phase transitions with temperature and checkpoint controls.
Biofilm Growth
Individual-based simulation of bacterial biofilm formation — cells grow, divide, and organize around nutrient channels.
Somitogenesis
Clock-and-wavefront mechanism: oscillating gene expression meets a receding FGF gradient to stamp out vertebrae.
Cell Polarity Establishment
Turing mechanism in a round cell — spontaneous symmetry breaking concentrates the activator at one pole via reaction-diffusion.
Actin Dynamics
Treadmilling and Arp2/3-mediated dendritic nucleation at the leading edge — watch the retrograde flow in real time.
Microtubule Assembly
Dynamic instability: filaments switch stochastically between growth and catastrophe. GTP cap, rescue events, length histogram.
Motor Protein Walking
Kinesin stepping hand-over-hand along a microtubule — 8 nm steps, cargo transport, force-velocity curve, multiple motors.
MAPK Signal Transduction
Three-tier kinase cascade converts a graded Ras signal into a switch-like ERK output via zero-order ultrasensitivity.
Receptor Kinetics
Watch ligands diffuse, bind, and unbind at membrane receptors. Tune kon, koff, competition between two ligands.
Cooperative Binding
Hemoglobin vs myoglobin oxygen binding. Bohr effect, Hill plot linearization, T/R state transitions animated.
Hill Equation
How the Hill coefficient n shapes response from hyperbolic to switch-like. Sensitivity, log dose-response, side-by-side comparison.
Feedback Amplifier
Positive feedback creates bistability and hysteresis; negative feedback drives adaptation. Phase plane, bifurcation diagram.
Feedforward Loop
C1-FFL network motif: sign-sensitive delay, pulse filtering. Animate X, Y, Z concentrations; compare with simple regulation.
Robustness Landscape
Boolean gene network robustness to perturbations. Many genotypes share one phenotype — explore neutral paths via random walks on the robustness landscape.
Evolvability
NK fitness landscapes: compare smooth (low K) vs rugged (high K) epistasis. Watch populations climb hills and see how modularity accelerates adaptation.
Neutral Network
RNA sequences folding to the same structure form a connected neutral network in sequence space. Animate random walks preserving phenotype across genotype space.
Sequence Logo
Interactive position weight matrix. Letter height encodes information content in bits. Load Kozak, TATA box, splice sites, or enter your own sequences.
Molecular Clock
Neutral mutations accumulate at constant rate μ. Watch lineages diverge, measure JC-corrected substitutions per site, calibrate with fossil dates.
Selection vs Drift
Wright-Fisher model: allele frequency trajectories with selection and drift. Fixation probability phase diagram — when does Ns matter?
Neutral Theory
Kimura's neutral theory: most fixed mutations are neutral. Distribution of fitness effects, site frequency spectrum, fixation accumulation by class.
Coalescent Theory
Kingman's coalescent animated: gene copies trace back to a common ancestor. Coalescence rate depends on N. Watch lineages merge backwards in time.
Founder Effect
Small founding population colonizes a new habitat. Track rapid allele frequency drift, reduced heterozygosity, and divergence from the mainland source.
Bottleneck Effect
Population crash then recovery. Heterozygosity permanently reduced even after population rebounds — a genetic scar that persists through recovery.
Linkage Disequilibrium
Interactive r² heatmap showing how recombination breaks down LD between SNP pairs over generations. Watch correlation structure decay in real time.
Recombination Map
Genetic recombination rate along a chromosome with hotspots and cold regions. Overlay gene density and simulate meiotic crossover events.
Haplotype Block Structure
LD-defined haplotype blocks in a simulated population. Detect block boundaries, select tag SNPs, and visualize colored block segments.
GWAS Manhattan Plot
Genome-wide association study visualization. Simulated p-values across chromosomes with significance thresholds and zoom into hit regions.
Genomic PCA
Principal component analysis separating simulated populations by ancestry. Add admixed individuals and watch them fall between parental clusters.
Admixture Plot
STRUCTURE-style ancestry barplot. Each individual shown as stacked colored bars. Adjust K populations and admixture level to see migration signatures.
Demographic History
PSMC-style Ne(t) visualization. Simulate bottlenecks, expansions, and migrations. Compare human OOA, island, and post-Ice-Age expansion scenarios.
Selective Sweep Detection
Positive selection creates a hitchhiking valley. Visualize reduced diversity, extended haplotype homozygosity (EHH), and the iHS sweep statistic.
Balancing Selection
Overdominance maintains polymorphism at equilibrium. Sickle cell anemia example, fitness landscapes, phase portraits, and multi-replicate drift.
Meme Dynamics
Replicator dynamics for cultural variants competing for mindshare. Introduce viral memes, kill dominant ones, and watch selection and drift in memespace.
Extended Phenotype
Dawkins' extended phenotype: genes selected for effects beyond the body. Watch spider web-building genes evolve under predator selection pressure.
Major Transitions
Szathmáry & Maynard Smith's eight major transitions in evolution. Interactive timeline exploring each transition's core challenge and evolutionary solution.
Endosymbiosis
Margulis' endosymbiosis theory animated. Watch an alpha-proteobacterium become a mitochondrion as genes transfer to the nucleus over evolutionary time.
Plasmid Dynamics
Plasmid spread through bacterial conjugation. Toggle antibiotic pressure and watch resistance plasmids sweep through the population via horizontal gene transfer.
Phage-Bacteria Coevolution
Lytic phage hunt bacteria that evolve CRISPR resistance. Phage escape via mutation. Red Queen dynamics drive oscillating population crashes and recoveries.
Arms Race Coevolution
Lotka-Volterra trait matching: predator attack and prey defense traits chase each other in endless cycles. Escalation vs stasis phase portraits.
Camouflage Evolution
Predators hunt the most visible individuals. Watch population color evolve toward the background. Switch environments to see polymorphism maintained by heterogeneity.
Warning Coloration
Aposematism paradox: rare warning colors are disfavored until predators learn. Müllerian mimicry drives multiple unpalatable species to converge on one signal.
Mutualism Network
Nested plant-pollinator network with co-extinction cascades. Remove a species and watch dependent partners go extinct. Quantify network robustness.
Pollination Network
Bipartite plant-pollinator network with phenological timing. Simulate climate change shifting pollinator emergence relative to flowering — watch interactions collapse.
Metapopulation Dynamics
Levins model — patches colonize and go extinct across a fragmented landscape with connectivity networks.
Habitat Fragmentation
Watch species vanish as continuous habitat is cut into isolated fragments with edge effects and corridors.
Species-Area Relationship
S = cA^z — the power law governing species richness on islands. MacArthur-Wilson equilibrium theory.
Ecological Succession
Pioneer to climax — grid simulation of succession stages with disturbance, fire, and recovery dynamics.
Nutrient Cycling
Carbon, nitrogen, phosphorus — pools and fluxes flowing between atmosphere, biomass, soil, and ocean.
Nitrogen Cycle
Fixation, nitrification, denitrification — toggle Haber-Bosch and see how humans doubled reactive nitrogen.
Greenhouse Effect
Solar in, infrared trapped — visualize CO2 forcing, temperature anomalies, and future climate scenarios.
Ocean Acidification
CO2 dissolves into seawater, forming carbonic acid — watch pH fall and shell-forming organisms struggle.
Sixth Mass Extinction
Species vanishing 1000x above background rates — tune habitat loss, climate change, and invasive species.
Extinction Vortex
Gilpin and Soule — inbreeding depression, demographic stochasticity, and genetic erosion spiral small populations to extinction.
Minimum Viable Population
Population viability analysis — how small populations spiral to extinction via demographic stochasticity, inbreeding, and the 50/500 rule.
Rewilding Dynamics
Trophic cascades from wolf reintroduction — elk behavior, riparian vegetation, river morphology, and the "rivers of life" effect.
Invasive Species
Invasion spread on a landscape grid — competitive exclusion, connectivity effects, long-range dispersal, and control strategies.
Biological Control
Lotka-Volterra predator-prey cycles as pest management — success, failure, and the risk of the predator becoming invasive itself.
Pesticide Resistance
Hardy-Weinberg selection in action — resistance alleles sweep through a pest population, fitness costs, dominance, and refuge strategies.
Superorganism
Ant colony as superorganism — response-threshold division of labor, emergent role allocation, and colony response to perturbation.
Colony Collapse Disorder
Forager lifespan threshold dynamics — how pesticides, pathogens, and habitat loss trigger sudden colony collapse in honeybees.
Bee Waggle Dance
Honeybee dance language — encoding distance and direction to food sources in the waggle run, sun compass, and recruit dispatch.
Ant Foraging
Pheromone trail networks with stigmergy — random walkers converge on shortest paths, competing food sources, and evaporation dynamics.
Collective Navigation
Couzin 2005 — how an informed minority leads an uninformed majority without explicit communication, split-target conflicts.
Parasite Manipulation
Toxoplasma rewires mouse fear, hairworms drive crickets to water — host behavior hijacked to complete the parasite lifecycle. Fitness landscapes.
Zombie Ant Fungus
Ophiocordyceps infects ants, forces them to climb exactly 25cm high, then erupts spores. Watch colony infection dynamics unfold in real time.
Hyperparasitism
Parasites of parasites — 3-species Lotka-Volterra with chaotic dynamics. How hyperparasitoids undermine biological pest control.
Social Parasitism
Cuckoo bumblebees invade host colonies, kill the queen, and trick workers into raising parasite brood. Colony takeover simulation.
Cuckoo Arms Race
Cuckoos evolve egg mimicry per host gente; hosts evolve finer discrimination. Coevolutionary arms race across multiple host species.
Brood Parasitism
Parasite eggs in host nests — acceptance vs rejection, chick eviction, host discrimination evolution. Optimal rejection threshold model.
Cleaner Fish
Cleaner wrasse cleaning station economics — cheating vs honest service, reputation effects, and why predators don't eat their cleaners.
Symbiosis Evolution
From parasitism to mutualism — microbe virulence evolves along the symbiosis continuum depending on transmission mode and host resources.
Mycorrhizal Network
Fungal networks trading carbon for phosphorus among forest trees. Carbon flows to shaded seedlings. Disruption cascades through the network.
Wood Wide Web
Mother trees preferentially send carbon to kin seedlings via mycorrhizal networks. Hub removal, kin recognition, and forest resilience.
Plant Volatile Signaling
Caterpillar herbivory triggers jasmonate cascade — plants release HIPVs that warn and prime neighboring plants' defenses.
Allelopathy
Chemical warfare between plants. Allelopathic toxins diffuse through soil creating zones of inhibition around the aggressor.
Root Competition
Two plants' root systems race through soil, sensing nutrient gradients and competing for finite phosphorus and water patches.
Arbuscular Mycorrhiza
Fungal hyphae colonize root cortex forming arbuscules — the carbon-for-phosphorus exchange interface of 80% of land plants.
Nitrogen Fixation
Nitrogenase splits N≡N using 16 ATP. Leghemoglobin buffers oxygen in the nodule, protecting the O₂-sensitive enzyme.
Legume–Rhizobium Mutualism
Root hair curling, infection thread formation, and nodule organogenesis — Nod factor signaling from soil to symbiosis.
Lichen Symbiosis
Fungus + photobiont composite: carbon flows from alga to fungus, fungal cortex retains water, desiccation tolerance on bare rock.
Coral Bleaching
Zooxanthellae photosynthesize inside coral tissue. Heat stress expels them — watch bleaching and recovery as temperature shifts.
Squid Bioluminescence
Vibrio fischeri use quorum sensing — autoinducer accumulates until the entire colony switches on bioluminescence simultaneously.
Gut Microbiome Ecology
Firmicutes, Bacteroidetes, Lactobacillus compete in the intestine. Watch diversity collapse under antibiotics and recover with fiber.
Immune-Microbiome Crosstalk
IgA, Tregs, and effector T cells interact with gut commensals and pathogens. Watch dysbiosis tip the immune balance toward autoimmunity.
Horizontal Gene Transfer Network
Bacteria swap resistance genes across species via plasmids, phage transduction, and transformation. Live network of spreading antibiotic resistance.
The Mobilome
Plasmids, transposons, phages, and integrons form the mobilome. Watch resistance and virulence genes hitchhike across bacterial genomes.
Transposon Dynamics
Transposable elements jump and duplicate through genomes. piRNA silencing suppresses TE activity — until stress triggers a burst.
Transgenerational Epigenetic Inheritance
Epigenetic marks survive fertilization and pass to offspring. Some escape erasure, carrying environmental memory across generations.
Histone Code
Combinatorial histone modifications — H3K4me3, H3K27me3, H3K9ac — define chromatin states. Writers, erasers, and readers decode the code.
DNA Methylation Dynamics
DNMT1 maintains, DNMT3a/b writes de novo, TET demethylates. Watch CpG methylation silence a gene at its promoter in real time.
Genomic Imprinting
IGF2, H19, and CTCF loops control parent-of-origin expression. The genomic conflict theory explains why fathers and mothers want different outcomes.
X Chromosome Inactivation
XIST RNA coats one X chromosome, forming the Barr body. Random inactivation creates mosaics — including the calico cat's color patches.
Alternative Splicing Code
Spliceosome assembles on pre-mRNA; ESEs and RBPs guide exon choice. One gene, many protein isoforms — tissue-specific splicing decoded.
RNA World
Animate the RNA world hypothesis — molecules that both store information and catalyze reactions, self-replicating on early Earth.
Ribozyme Evolution
Explore sequence-space fitness landscapes for catalytic RNA. Watch populations evolve toward hammerhead and HDV ribozyme peaks.
Protocell Origin
Fatty acid vesicles self-assemble, grow, divide, and compete. RNA inside replicates — the birth of cellular individuality.
Hypercycle
Eigen's mutually catalytic RNA cycle — each species helps replicate the next. Cooperate or collapse to parasites.
Error Threshold
When mutation rate exceeds 1/L, the master sequence dissolves into noise. Eigen's threshold governs all replicating information.
Quasispecies
Viral populations exist as clouds of mutants around a fitness peak. High mutation enables immune escape — the HIV strategy.
RNA Folding
Enter any RNA sequence and watch it fold into stem-loops and hairpins via minimum free energy. Mountain plot included.
Aptamer Evolution (SELEX)
Run in vitro evolution of RNA aptamers — select, amplify, mutate, repeat. Watch binding populations enrich over rounds.
Ribosome Evolution
The ribosome as a ribozyme: concentric evolutionary rings from the 3.8-billion-year-old peptidyl transferase center outward.
Genetic Code Origin
Explore the codon table, error robustness, and three theories — frozen accident, stereochemical, coevolution — for how the code arose.
Codon Table Explorer
Interactive codon wheel and table — click any of the 64 codons to explore amino acid properties, synonymous codons, and usage bias across human, E. coli, and yeast.
Wobble Base Pairing
Animated codon-anticodon pairing in the ribosome A-site — see Watson-Crick vs wobble G:U pairs and inosine triple-decoding at the molecular level.
tRNA Structure Explorer
Click through the cloverleaf and L-shaped 3D structure of tRNA — acceptor stem, D-loop, anticodon loop, TΨC loop, and aminoacyl-tRNA synthetase recognition.
Riboswitch Mechanism
Drag the ligand concentration slider to watch RNA change shape — aptamer binding switches gene expression ON or OFF. TPP, SAM, and adenine riboswitches.
CRISPR-Cas9 Editing
Step through guide RNA targeting, PAM recognition, R-loop formation, and double-strand break — then choose NHEJ or HDR repair pathways.
Base Editing
Watch cytosine deaminase convert C→U (CBE) or TadA convert A→I (ABE) in the editing window — precise base changes without double-strand breaks or templates.
Prime Editing
Step through the pegRNA search-and-replace mechanism — PBS hybridization, reverse transcription of new sequence, and 3'-flap resolution into any desired edit.
Gene Drive Simulation
Simulate CRISPR homing drives spreading through a wild population — compare standard, crash, threshold, and daisy chain drives with interactive allele frequency graphs.
Synthetic Biology Designer
Assemble genetic parts — promoters, RBS, CDS, terminators — into circuits. Build AND, OR, NOT, NAND, and NOR logic gates from transcription factors.
Synthetic Genetic Circuits
Simulate repressilator oscillations, bistable toggle switches, band-pass filters, and pulse generators — tune Hill coefficients and watch ODE dynamics in real time.
Mycelial Networks
Underground fungal communication webs
Coral Bleaching
Thermal stress and reef ecosystem collapse
Black Hole Entropy
Hawking radiation and information paradox
Metamorphosis
Holometabolous transformation dynamics
River Deltas
Sediment transport and branching patterns
Protein Misfolding
Aggregation cascades in neurodegenerative disease
Cephalopod Camouflage
Chromatophore control and pattern generation
Tidal Forcing
Gravitational harmonics and ocean resonance
Magnetoreception
Cryptochrome compass and radical pair mechanism
Ice Core Records
Paleoclimate proxies and orbital forcing
Fluid Instability
Kelvin-Helmholtz and Rayleigh-Taylor mixing
Bird Migration
Magnetic compass and star navigation strategies
Deep-Sea Vents
Chemosynthetic ecosystems at hydrothermal vents
Gravitational Waves
Spacetime ripples from merging black holes
Immune Memory
B-cell clonal selection and antibody affinity maturation
Kelp Forest
Canopy dynamics and sea urchin trophic cascades
Neutron Star
Pulsar timing, glitches, and dense matter physics
Soil Microbiome
Decomposition networks and nutrient cycling
Wildfire Spread
Percolation thresholds and wind-driven fire fronts
Mangrove Roots
Coastal carbon sinks and pneumatophore architecture
Hormone Signaling
Receptor binding, second messengers, and feedback loops
Snowball Earth
Ice-albedo runaway and Neoproterozoic glaciation
Atmospheric Rivers
Pineapple Express and filamentary water vapor transport
DNA Repair
Base excision, nucleotide excision, and double-strand break repair
Tectonic Stress
Strain accumulation and elastic rebound at fault zones
Animal Coloration
Structural color, iridescence, and Turing pattern pigmentation
Bubble Dynamics
Rayleigh collapse, sonoluminescence, and cavitation
Lymphatic System
Fluid drainage, lymph node filtering, and immune surveillance
Cave Ecosystems
Trophic webs in perpetual darkness
Solar Flares
Magnetic reconnection and coronal mass ejections
Stress Response
HPA axis, cortisol cascade, and allostatic load
Moon Formation
Giant impact hypothesis and accretion disk dynamics
Predator-Prey Cycles
Lotka-Volterra oscillations and population dynamics
Geomagnetic Reversal
Polarity flips and paleomagnetic field reconstruction
Social Insects
Stigmergy, trail pheromones, and superorganism intelligence
Earthquake Aftershocks
Omori's law and spatiotemporal clustering
Whale Migration
Ocean basin navigation and acoustic signaling
Supercell Thunderstorm
Mesocyclone rotation and tornado genesis
Speciation on Islands
Allopatric divergence and adaptive radiation
Thermohaline Circulation
Global ocean conveyor belt and AMOC dynamics
Planetary Rings
Resonance gaps, shepherd moons, and ring particle dynamics
Fire Ecology
Burn cycles, pyrophytes, and post-fire succession
Neurotransmission
Synaptic vesicle release and receptor binding kinetics
Symbiosis
Mutualism, commensalism, and parasitism coevolution
Coral Triangle
Biodiversity hotspot and larval dispersal currents
Thermocline
Ocean stratification, nutrient upwelling, and mixing
Salt Marsh
Tidal carbon sequestration and halophyte zonation
Planetary Formation
Planetesimal accretion and protoplanetary disk evolution
Sleep Cycles
REM/NREM architecture and memory consolidation
Chemiluminescence
Excited-state chemistry and bioluminescent reaction cascades
Stem Cells
Pluripotency, differentiation landscapes, and organoid growth
Lightning Physics
Stepped leader propagation and return stroke dynamics
Flowering Plants
Pollinator coevolution and angiosperm radiation
Permafrost Thaw
Arctic carbon release and thermokarst formation
Blood Coagulation
Clotting cascade, fibrin mesh formation, and platelet aggregation
Galaxy Mergers
Tidal tails, starburst triggers, and elliptical remnants
Sensory Adaptation
Receptor desensitization and contrast enhancement
Quasicrystals
Penrose tiling, forbidden symmetry, and Shechtman's discovery
Desert Ecology
Water harvesting strategies and boom-bust population cycles
Mitochondria
Electron transport chain and ATP synthase rotary motor
Ocean Plastic
Gyre accumulation, microplastic fragmentation, and food web entry
Quantum Computing
Qubit superposition, entanglement, and gate operations
Teeth Evolution
Cusp patterns, diet transitions, and dental morphospace
Algal Bloom
Nutrient loading, eutrophication, and dead zone dynamics
Wolf Reintroduction
Trophic cascade and landscape of fear in Yellowstone
Neutron Capture
s-process and r-process nucleosynthesis in stellar environments
Embryo Development
Cleavage, gastrulation, and morphogen gradients
Soil Erosion
Raindrop impact, rill formation, and conservation tillage
Vesicle Trafficking
Golgi sorting, SNARE proteins, and endocytic pathways
Dust Storms
Saltation thresholds, haboob formation, and global transport
Leaf Litter
Decomposer food webs and nutrient mineralization
Coral Spawning
Synchronized mass spawning and larval settlement
Dark Energy
Accelerating expansion and the cosmological constant
Gut-Brain Axis
Vagal signaling, microbiome metabolites, and mood regulation
River Flooding
Hydrograph dynamics, floodplain inundation, and flood memory
Ribosome
mRNA decoding, peptide bond formation, and translocation
Impact Cratering
Shock metamorphism, ejecta curtains, and crater scaling laws
Bird Song
Syrinx mechanics, dialect learning, and dawn chorus dynamics
Fungal Spores
Ballistospore discharge, atmospheric dispersal, and germination
Sea Level Rise
Ice sheet contributions, thermal expansion, and coastal impacts
Photosynthetic Antenna
Quantum coherence in light-harvesting complexes — excitons navigating the FMO complex
Mid-Ocean Ridge
Seafloor spreading, magnetic stripe chronology, and divergent plate boundaries
Hydrothermal Vent
Black smoker chemistry and chemosynthetic ecosystems at the seafloor
Deep Sea Bioluminescence
Light production in the eternal darkness — anglerfish, comb jellies, and more
Chemosynthesis
Energy from chemical redox reactions — life powered by H₂S oxidation, not sunlight
Abyssal Zone
Life at crushing pressure in the deep ocean trenches — marine snow and extremophiles
Kelp Forest
Giant kelp trophic cascade — sea otters, urchins, and canopy ecosystem dynamics
Sea Floor Spreading
Magnetic anomaly striping and the Vine-Matthews-Morley evidence for plate tectonics
Volcanic Plume
Hotspot volcanism and island chain formation — Hawaii's 80-million-year tape recorder
Ice Core Record
800,000 years of paleoclimate from δ¹⁸O isotopes, CO₂ bubbles, and dust layers
Cave Painting Pigments
Mineral chemistry of Paleolithic art — ochre, manganese, charcoal pigments
Acoustic Archaeology
Sound resonance in ancient caves and stone circles — RT60, standing waves
Ancient DNA Analysis
aDNA damage patterns, C→T deamination, population genetics from bone
Dendrochronology
Tree-ring dating and climate reconstruction — crossdating, drought years
Radiocarbon Dating
¹⁴C decay curve, IntCal calibration, age probability distributions
Archaeoastronomy
Solstice alignments at Stonehenge, Newgrange, Karnak, and Chichen Itza
Obsidian Hydration
Volcanic glass dating by water diffusion rind — Fickian diffusion, Arrhenius
Stratigraphy
Layer sequencing, unconformities, and superposition in geological sections
Pollen Analysis
Palynology — pollen grain morphology, stratigraphic diagrams, vegetation zones
Zooarchaeology
Animal bones in archaeology — NISP, MNI, butchery marks, diet reconstruction
Spindle Assembly Checkpoint
Mitotic checkpoint and APC/C activation via kinetochore tension sensing
Kinetochore Dynamics
Chromosome-microtubule attachment, Aurora B error correction, and tension
Centrosome Duplication
Centriole cycle, PLK4 licensing, and bipolar spindle formation
Cilia Motility
Axonemal dynein, 9+2 structure, beat cycle and fluid flow
Glycolysis Regulation
Allosteric control of PFK-1 and energy sensing via ATP/F2,6BP/citrate
Citric Acid Cycle
Krebs cycle intermediates, NADH generation, and anaplerosis
Oxidative Phosphorylation
Electron transport chain, proton gradient, and ATP synthase rotation
Fatty Acid Synthesis
FAS complex, malonyl-CoA elongation cycles, and NADPH consumption
Gluconeogenesis
Glucose synthesis from lactate and amino acids, PEPCK and bypass enzymes
Pentose Phosphate Pathway
NADPH production, ribose synthesis, and oxidative vs non-oxidative branches
Muon g−2
Anomalous magnetic moment, virtual particle loops, tension with the Standard Model
W and Z Bosons
Weak force carriers, charged vs neutral currents, decay modes and branching ratios
Color Confinement
Quark color charge, gluon flux tubes, and string breaking in QCD
Asymptotic Freedom
QCD running coupling, short-distance quark freedom, Nobel Prize 2004
Hadronization
Lund string fragmentation: quark-gluon strings breaking into mesons and baryons
Jet Fragmentation
Parton showers, QCD jets, and collider detector signatures
Parton Distribution Functions
Quarks and gluons inside the proton, Bjorken scaling, DGLAP evolution
Deep Inelastic Scattering
Electron-proton scattering revealing pointlike quark structure at SLAC 1968
Baryogenesis
Sakharov's three conditions for generating the matter-antimatter asymmetry
Matter-Antimatter Asymmetry
CP violation, kaon oscillations, CKM matrix, and why the universe exists
Spiking Neural Network
Leaky integrate-and-fire neurons, membrane potential, spike propagation
STDP
Spike-timing dependent plasticity, Hebbian learning rule, temporal windows
Grid Cells
Hexagonal firing fields, entorhinal cortex, path integration
Place Cell Remapping
Hippocampal place fields, global vs partial remapping, context switching
Theta Oscillations
Hippocampal theta rhythm, phase precession, temporal coding
Cortical Oscillations
Gamma, beta, alpha rhythms, cross-frequency coupling, neural synchrony
Working Memory
Persistent activity, attractor networks, capacity limits
Winner-Take-All Network
Competitive inhibition, decision making, neural selection
Gain Modulation
Multiplicative neural responses, attention and context effects
Attention Spotlight
Spatial attention, feature-based attention, biased competition model
Positional Information
Wolpert's French flag model — morphogen gradients specifying cell identity across a tissue field
Limb Patterning
ZPA, Sonic Hedgehog gradient, and digit specification in the developing vertebrate limb bud
Neural Tube Closure
Neurulation, multiple closure initiation points, and neural crest cell emigration
Gastrulation
Cell movements, germ layer formation, archenteron invagination and primary mesenchyme ingression
Convergent Extension
Tissue narrowing and elongation via planar polarized mediolateral cell intercalation
Cell Fate Commitment
Waddington landscape, bistable gene circuits, and irreversible commitment to cell identity
Lateral Inhibition
Notch-Delta signaling generates salt-and-pepper patterning on a hexagonal epithelium
Planar Cell Polarity
Tissue-level alignment of cells via the Frizzled-Dishevelled-Vangl2 core PCP complex
Wnt Signaling
Beta-catenin destruction complex, TCF/LEF activation, and stem cell regulation
Hedgehog Pathway
Patched-Smoothened-Gli cascade interprets SHH morphogen gradient into discrete cell fates
Superconductor Vortices
Abrikosov vortex lattice in type-II superconductors with flux quantization and GL theory
Heavy Fermion Systems
f-electron Kondo lattice hybridization and effective mass enhancement up to 1000×mₑ
Frustrated Magnetism
Geometric frustration on triangular lattice with competing exchange and residual entropy
Quantum Spin Liquid
Resonating valence bonds, fractionalized spinons and visons, Z₂ topological order
Kagome Lattice
Flat band physics with macroscopic degeneracy from destructive interference
BEC Vortices
Superfluid vortex lattice, quantized circulation, and Feynman's rule in rotating BECs
Quantum Criticality
Quantum critical point, diverging correlation length, and non-Fermi liquid fan
Strange Metal
T-linear resistivity, Planckian dissipation, and the breakdown of Fermi liquid theory
Cuprate Superconductors
High-Tc phase diagram, pseudogap, d-wave pairing symmetry, and Fermi arcs
Many-Body Localization
ETH breakdown, area-law entanglement, Poisson level statistics in disordered quantum matter
Noise-Induced Transitions
Stochastic dynamics and Kramers escape in double-well potentials
Coherence Resonance
Optimal noise drives periodic firing in excitable FitzHugh-Nagumo systems
Free Energy Principle
Friston's FEP — variational inference and prediction error minimization
Active Inference
Action as inference — policy selection via expected free energy
Predictive Processing
Hierarchical prediction, precision-weighted errors, perception as inference
Bayesian Brain
Prior-likelihood integration and systematic perceptual biases
Neural Binding Problem
Gamma-frequency synchrony binds distributed features into unified percepts
Integrated Information Theory
Phi (Φ) as a measure of consciousness — cause-effect structure
Global Workspace Theory
Ignition and broadcasting — how access consciousness arises
Φ Computation
Calculating integrated information — partitions and minimum information bipartition
River Delta Dynamics
Sediment deposition, distributary channels, and delta lobe switching
Braided River
Multiple channels, bar migration, and high sediment load dynamics
Meandering River
Meander migration, oxbow lake formation, and neck cutoff events
Alluvial Fan
Fan deposition at mountain fronts, avulsion, and lobe switching
Debris Flow
Viscoplastic Bingham fluid, yield stress, surge and deposition dynamics
Landslide Mechanics
Factor of safety, slope stability, rainfall and seismic triggering
Coastal Dune Migration
Aeolian transport, barchan dunes, and vegetation stabilization
Beach Profile Evolution
Wave swash and backwash, berm building, longshore bar formation
Barrier Island Dynamics
Storm overwash, rollover migration, and sea-level rise response
Estuary Mixing
Salt wedge, tidal mixing, stratification and salinity intrusion
Ocean Thermal Energy Conversion
OTEC depth-temperature profile, Carnot efficiency, and heat engine power output
El Niño / Southern Oscillation
Pacific SST anomalies, Walker circulation, and global teleconnections
Arctic Oscillation
Polar vortex strength, AO index time series, and cold air outbreak patterns
Madden-Julian Oscillation
30–60 day tropical oscillation, RMM phase diagram, and convective envelope propagation
Hurricane Eyewall Dynamics
Radial wind profile, eyewall replacement cycles, and warm core structure
Tornado Vortex Simulator
EF-scale winds, angular momentum conservation, and pressure deficit visualization
Atmospheric Rivers
IVT moisture corridors, Pineapple Express, and global AR visualization
Polar Vortex Dynamics
Stratospheric temperature cross-section, SSW events, and downward coupling
Jet Stream Meandering
Rossby wave dispersion, blocking patterns, and zonal vs. wavy jet states
Sudden Stratospheric Warming
Rapid Arctic warming, vortex disruption timeline, and surface cold air coupling
Coastal Upwelling Dynamics
Ekman transport, wind-driven offshore flow, cold nutrient-rich water rising to surface
Internal Ocean Gravity Waves
Pycnocline interface waves, stratification, density-dependent phase speed and wave breaking
Tidal Bore Formation
Bay of Fundy-scale tidal ranges funneling into estuaries, creating upstream-propagating bores
Rogue Wave Formation
Benjamin-Feir instability, nonlinear wave focusing, and extreme wave height statistics
Sea Ice Dynamics
Stefan problem ice growth, albedo feedback, brine rejection, and Arctic seasonal extent
Ocean Carbon Cycle
CO₂ solubility pump, biological pump, carbonate chemistry, pH and aragonite saturation
Halocline & Double Diffusion
Freshwater lens, density stratification, salt fingers vs. diffusive convection staircases
Climate Feedback Mechanisms
Planck, water vapor, ice-albedo, lapse rate, and cloud feedbacks — equilibrium climate sensitivity
Abyssal Circulation
Thermohaline driving, NADW and AABW formation, deep ocean conveyor and ~1000-year residence time
Hydrothermal Vent Chemistry
Black smokers, metal sulfide precipitation, chemosynthesis, and extreme deep-sea geochemistry
Subduction Zone Dynamics
Oceanic plate diving under continental crust — slab rollback, accretionary wedge, megathrust earthquakes, arc volcanism
Mid-Ocean Ridge Spreading
Seafloor spreading, magnetic striping, hydrothermal circulation, and axial valley — interactive spreading rate slider
Mantle Convection Cells
Rayleigh-Bénard convection in Earth's mantle — Ra = αgΔTd³/νκ, whole-mantle vs layered convection debate
Hot Spot Volcanism
Mantle plumes, Hawaiian island chain age progression, fixed vs moving plume debate — interactive age-distance diagram
Volcanic Eruption Types
Effusive to explosive: Hawaiian, Strombolian, Vulcanian, Plinian — VEI scale and silica content vs viscosity slider
Earthquake Seismic Waves
P-waves, S-waves, Love and Rayleigh surface waves — animated propagation through Earth's layers and shadow zones
Fault Slip Mechanics
Coulomb failure criterion τ = μσₙ + c, normal/thrust/strike-slip faults, Mohr circle, and Omori aftershock law
Isostasy & Glacial Rebound
Airy vs Pratt isostasy, glacial isostatic adjustment, Scandinavia +8 mm/yr uplift — animated crustal response
Rock Cycle Processes
Igneous, sedimentary, metamorphic transformations — interactive P-T phase diagram with metamorphic facies fields
Geothermal Gradient
Heat flow from Earth's interior — continental vs oceanic crust, conductive vs advective transport, borehole profiles
DNA Replication Fork
Leading/lagging strand synthesis, Okazaki fragments, helicase unwinding — animated replication fork with polymerase fidelity
Michaelis-Menten Kinetics
Substrate binding, V_max, K_m, competitive/noncompetitive inhibition, Hill cooperativity — interactive kinetics curves
Cell Membrane Transport
Passive diffusion, facilitated diffusion, active transport, osmosis — membrane cross-section with animated molecule movement
Action Potential Propagation
Hodgkin-Huxley model, Na⁺/K⁺ channel dynamics, threshold, myelination and saltatory conduction — animated voltage trace
Krebs Cycle Animation
8-step citric acid cycle, NADH/FADH₂ production, ATP yield accounting, acetyl-CoA entry — rotating animated pathway
Natural Selection & Fitness
Fitness landscapes, selection coefficient, heterozygote advantage, frequency-dependent selection — animated allele frequency evolution
Epigenetic Modification
DNA methylation, histone acetylation/methylation, chromatin states, gene silencing — interactive nucleosome model
RNA Splicing Mechanism
Pre-mRNA to mRNA, spliceosome assembly, branch point, lariat intermediate, exon joining — animated splicing diagram
CRISPR-Cas9 Mechanism
Guide RNA targeting, PAM sequence, double-strand break, HDR vs NHEJ repair pathways — interactive gene editing simulation
Synaptic Transmission
Vesicle release, receptor binding, EPSP/IPSP, reuptake, signal integration — animated synapse with Ca²⁺ influx
Neutron Star Structure
Interior layers from crust to quark-gluon core, TOV mass limit (~2 M☉), equation of state, and interactive mass-radius diagram
Black Hole Accretion Disk
ISCO at 3r_s for Schwarzschild BH, T∝r^(−3/4) temperature profile, relativistic jets, color-temperature mapping
Cosmic Ray Air Shower
Primary cosmic ray striking atmosphere, hadronic and electromagnetic cascade, Heitler model, muon component — animated particle shower
Galaxy Rotation Curves
Keplerian prediction vs observed flat curves, NFW dark matter halo profile, interactive mass fraction slider
Hubble's Law & Redshift
v = H₀d, cosmic expansion, peculiar velocities, Hubble tension (67 vs 73 km/s/Mpc) — interactive Hubble diagram
Wolf-Rayet Stellar Wind
Extreme radiation-driven mass loss (10⁻⁵ M☉/yr), wind-blown bubble animation, terminal velocity ~2000 km/s
Binary Star Mass Transfer
Roche lobe overflow, L₁ Lagrange point, accretion stream, cataclysmic variables, Type Ia supernova progenitor
Supernova Light Curves
Type Ia standard candle (Ni-56 decay), Type IIp hydrogen plateau, rise/peak/decline — interactive comparison
Gamma-Ray Burst Jet
Collapsar vs NS merger, relativistic beaming (Γ~100–1000), Doppler boost, afterglow — animated jet animation
Exoplanet Transit Method
Flux dip depth = (R_p/R_*)², limb darkening, transit duration, Kepler's third law — animated transit with live light curve
Spin Glass Frustration
Frustrated magnetic interactions, Parisi replica symmetry breaking, exponentially many metastable states — interactive frustrated lattice
Ising Model at Criticality
T_c = 2J/ln(1+√2), Onsager's exact solution, fractal spin clusters, order parameter — animated Monte Carlo near the critical point
Vicsek Flocking Model
Local alignment rule, noise-driven order-disorder transition, true long-range order in 2D — animated agent swarm simulation
Random-Field Ising Model
Quenched disorder, Imry-Ma argument, domain formation below d=4, dilute antiferromagnet realization — domain visualization
Bak-Sneppen Co-evolution
Punctuated equilibrium, avalanches of co-extinction, self-organized critical fitness threshold f*≈0.667 — power-law avalanche distribution
Traffic Jam Formation
Nagel-Schreckenberg model, phantom jams without bottlenecks, backward wave at ~15 km/h, fundamental diagram — ring road simulation
Conway's Game of Life
Gliders, oscillators, Gosper gun, Turing completeness, density phase transitions — interactive grid with pattern presets
Sugarscape Agent Model
Heterogeneous agents gathering resources, wealth inequality emergence, Gini coefficient evolution — animated landscape simulation
Anomalous Diffusion
MSD∝t^α: normal, subdiffusion (CTRW), superdiffusion / Lévy flights, fractional Brownian motion — interactive exponent comparison
Zipf's Law Emergence
Rank-frequency power law in language and cities, Simon preferential-attachment derivation — log-log plot with real data
Fourier Series Animation
Epicycles building arbitrary periodic functions from sinusoids — Gibbs phenomenon at discontinuities, live frequency spectrum, interactive harmonic editor
Laplace Transform Visualizer
s-plane poles and zeros, magnitude landscape, frequency response along the imaginary axis — probe the complex plane interactively
Complex Analysis & Residues
Domain coloring of complex functions, contour integration, residue theorem ∮f dz = 2πi·ΣRes — animated contour paths
Riemann Surface Explorer
√z, log z and ∛z as single-valued functions on branched sheets — 3D surface with monodromy around branch points, drag to rotate
Differential Forms Visualizer
1-forms as oriented measurement devices, 2-forms as area elements, exterior derivative d, Stokes theorem — flux tubes and arrows
Tensor Visualization
Rank-0/1/2 tensors, coordinate transformation rules, metric tensor, index raising and lowering, Einstein summation — interactive basis change
Lie Groups & SO(3)
3D rotation group, Lie algebra so(3), Rodrigues' formula exp(θ·n̂×), composition and non-commutativity — interactive rotation visualization
Topological Genus & Deformation
Surfaces with g=0,1,2,3 handles, Euler characteristic χ=2−2g, coffee mug equals torus — interactive 3D deformation animations
Knot Invariants & Jones Polynomial
Trefoil, figure-eight, unknot, Hopf link — Reidemeister moves, writhe, Jones polynomial skein relation, crossing diagrams
Category Theory & Functors
Objects, morphisms, functors, natural transformations, adjunctions L⊣R — animated commuting diagram checker
Decoherence & Pointer States
Einselection, environment-induced superselection, Zurek's pointer states, decoherence timescales — Bloch sphere purity decay
Adiabatic Quantum Computing
Slow evolution from easy to hard Hamiltonian, minimum spectral gap, quantum annealing, D-Wave — energy spectrum animation
Topological Qubits & Majorana Modes
Majorana zero modes, Kitaev wire, non-Abelian anyons, braiding operations for fault-tolerant gates — interactive braid diagram
Quantum Teleportation Circuit
Bell pair, Alice's CNOT + Hadamard, measurement, classical channel, Bob's correction — step-by-step circuit with state evolution
Quantum Annealing Landscape
Tunneling vs thermal hopping, transverse field Ising model, WKB approximation — 2D energy landscape with QA vs SA comparison
Grover's Search Algorithm
Amplitude amplification, oracle phase flip, diffusion operator, √N speedup — animated amplitude bars iteration by iteration
Shor's Factoring Algorithm
Period finding via QFT, modular exponentiation, GCD, factoring N=15 — circuit with register state visualization
Quantum Fourier Transform
QFT circuit, Hadamard + controlled-phase gates, phase kickback, comparison with classical FFT — interactive n-qubit register
Quantum Noise Channels
Bit flip, phase flip, depolarizing, amplitude damping — Bloch sphere distortion for each channel with error probability slider
Lindblad Master Equation
Open quantum dynamics, jump operators, cavity decay, spontaneous emission, thermal bath — density matrix evolution in real time
Gradient Descent
Loss landscape, learning rate effects, momentum, Adam optimizer, saddle points — animated 2D optimization paths
Neural Network Backpropagation
Forward pass, chain rule gradient flow, weight updates — animated network diagram with gradient magnitudes per layer
CNN Convolutional Filters
Sobel, Laplacian, Gaussian kernels, feature maps, receptive field — interactive image convolution visualizer
Transformer Attention
Q/K/V matrices, scaled dot-product attention, softmax, multi-head attention — interactive token attention heatmap
GAN Training Dynamics
Generator vs discriminator loss, mode collapse, Nash equilibrium — animated loss curves with distribution comparison
Q-Learning
Bellman equation, ε-greedy exploration, Q-table convergence — interactive gridworld with live Q-value arrow updates
Support Vector Machine
Maximum margin hyperplane, support vectors, RBF kernel, soft margin C — interactive 2D point classifier
Principal Component Analysis
Covariance eigenvectors, variance explained, dimensionality reduction — interactive data cloud with PC arrows
Gaussian Process Regression
Prior over functions, posterior conditioning, RBF/Matérn kernels, uncertainty bands — click to place observations
Bayesian Neural Network
Weight uncertainty, variational inference, epistemic vs aleatoric uncertainty, MC dropout — posterior weight distributions
Cellular Automaton Zoo
Explore all 256 rules of 1D elementary CA
Protein Folding Sim
HP lattice model with Monte Carlo optimization
Fluid Vortex Dance
Click to add vortices in Euler fluid flow
Fourier Transform Art
Draw a curve, watch it built from epicycles
Fibonacci Spirals
Golden angle phyllotaxis and packing patterns
Mandelbrot Deep Zoom
Click-to-zoom fractal with smooth coloring
Ising Model Sim
Ferromagnetic phase transition at critical T
Thermodynamic Cycle
PV diagrams for Carnot, Otto, Diesel cycles
Galaxy Formation
N-body gravity forms spiral galaxy structure
Solar System Orbits
Accurate Keplerian orbits with live planet data
2D Standing Waves
Membrane vibration modes with modal superposition
Tessellation Explorer
Regular and semi-regular tilings with color controls
Flow Field Particles
Thousands of particles trace Perlin noise streamlines
Hypercube Rotation
4D tesseract projected and rotated in all 6 planes
Diffusion-Limited Aggregation
Fractal dendrites grow from random-walking particles
Knot Visualizer
3D torus knots with tube rendering — drag to rotate
Harmonograph Sim
Dual-pendulum Lissajous curves with damping
Epicycloid Curves
Rolling circles trace epicycloids and hypocycloids
Atmospheric Pressure
Pressure, temperature, density across all layers
Sound Wave Synthesis
Additive synthesis: build sounds from harmonics
Complex Function Plotter
Domain coloring of f(z) — phase, magnitude, poles
Torus Rotation
Rotating torus with adjustable radii and shading
Geodesic Sphere
Subdivided icosahedron dome — Euler's V−E+F=2
Spring-Mass System
Coupled oscillators with normal modes and phase space
Escape Velocity Sim
Launch from planets — escape or fall back to gravity
Tree of Life
L-system fractal tree with seasons and wind
Voronoi Animation
Animated cells with Lloyd relaxation to centroids
Billiard Table
Chaotic vs integrable trajectories in six table shapes
Fourier Series Builder
Add harmonics — watch Gibbs phenomenon emerge
Butterfly Effect
Two Lorenz attractors diverge from near-identical start
Wave Equation 1D
Pluck a string — reflections and standing modes
Topology Explorer
Deform sphere to torus — genus and Euler characteristic
Prime Number Spiral
Ulam and Sacks spirals reveal hidden prime patterns
Barycentric Coordinates
Drag a point — weighted mix of triangle corners
Nonlinear Pendulum
Full ODE pendulum with phase portrait and period curve
Taylor Series Explorer
Watch polynomial approximations converge term by term
Doppler Effect
Moving source compresses wavefronts — blue and red shift
Lens Optics
Ray tracing through lenses — real and virtual images
Quantum Spin
Bloch sphere qubit — gates, measurement, precession
Matrix Transform
2D linear maps — eigenvectors, determinant, shear
Graph Isomorphism
Drag nodes, find isomorphism mapping between two graphs
Information Entropy
Shannon entropy, mutual information, channel capacity
Boolean Circuit Sim
Wire AND/OR/NOT/XOR gates, truth table output
Principle of Least Action
Drag paths, watch action S = ∫(T−V)dt minimize
Geometric Phases
Berry phase on Bloch sphere — holonomy from loops
Finite Automaton
Build DFAs and NFAs, test string acceptance
Turing Machine Sim
Visual tape, head, transitions — run example programs
Spline Interpolation
Compare cubic, Catmull-Rom, B-spline with drag points
Newton Interpolation
Divided difference polynomial grows term by term
Runge-Kutta Methods
Compare Euler, RK2, RK4 — step error and convergence
Lambda Calculus
Step through beta reduction — Church numerals, Y combinator
Type Inference
Hindley-Milner unification solves type constraints live
Numerical Integration
Riemann, trapezoid, Simpson's, Gauss — error convergence
Signal Filtering
Build harmonics, apply filters, see frequency domain live
Market Dynamics
Supply, demand, elasticity, price controls, deadweight loss
Ant Colony Sim
Pheromone stigmergy finds shortest paths, emergent trails
Immune System Sim
Pathogens, antibodies, clonal selection, memory cells
Backpropagation
Forward pass, gradients flow backward through layers
PCA Visualization
Principal components via power iteration on scatter data
K-Means Clustering
Click to add points, Voronoi boundaries, centroids converge
Decision Tree
Split data by thresholds, gini/entropy, grow and prune
Gradient Flow
SGD, Momentum, Adam racing down loss landscapes
Transformer Blocks
Step through Q/K/V attention and softmax weights
Latent Space Explorer
Clustered embeddings, interpolation, nearest neighbors
Attention Patterns
Heatmap of head attention across token pairs
Reinforcement Learning
Q-learning agent in a grid world, policy and values evolving
Diffusion Model
Signal dissolves into noise and reconstructs step by step
Autoencoder
Compress 2D shapes through a bottleneck, watch reconstruction
Word Embeddings
Semantic clusters and analogy arrows in 2D vector space
Unrolled RNN
Hidden state memory through time, vanishing gradient demo
Darcy Flow
Porous media seepage, permeability, pressure-driven flow
Hele-Shaw Cell
Viscous flow between plates, potential flow analogy
Buoyancy-Driven Flow
Natural convection, Rayleigh number, thermal plumes
Soap Film Dynamics
Minimal surfaces, drainage, thin-film interference
Capillary Rise
Jurin's law, meniscus, surface tension vs gravity
Meniscus Formation
Concave vs convex, adhesion, Young equation
Wetting & Dewetting
Thin film stability, spinodal dewetting, contact line
Contact Angle Hysteresis
Advancing/receding angles, roughness, lotus effect
Fluid Chaos
Sensitive dependence, turbulent cascade, Lyapunov exponents
Reynolds Number
Laminar vs turbulent, dimensionless flow, Re=UD/ν
Laminar-Turbulent Transition
Subcritical bifurcation, turbulent puffs, directed percolation
Plateau-Rayleigh Instability
Jet breakup, droplet formation, surface tension wins
Oscillating Pendulum
Phase portrait, nonlinear period, separatrix
Parametric Resonance
Mathieu equation, stability tongues, Floquet theory
Nonlinear Spring
Duffing oscillator, hardening/softening, jump phenomena
Pendulum Phase Space
Separatrix, libration vs rotation, Hamiltonian orbits
Single-Slit Diffraction
Fraunhofer pattern, Huygens-Fresnel, first minimum
Polarization of Light
Malus's law, Brewster angle, birefringence
Wave Packet Dispersion
Group vs phase velocity, spreading, de Broglie
Group vs Phase Velocity
Dispersive medium, anomalous dispersion, energy transport
Mach Cone
Supersonic shock waves, Mach angle, sonic boom
Acoustic Resonance
Standing waves in pipes, open/closed ends, harmonics
Room Modes
3D standing waves, Schroeder frequency, bass buildup
Psychoacoustics
Fletcher-Munson curves, loudness, frequency sensitivity
Musical Intervals
Frequency ratios, consonance, beating
Harmonic Series
Overtones, spectral content, Fourier building blocks
Just Intonation
Pure ratios, beating, wolf intervals
Equal Temperament
12-TET, cents, Werckmeister compromise
Circle of Fifths
Key relationships, modulation, enharmonic equivalence
Pythagorean Tuning
Stacked fifths, Pythagorean comma, ancient ratios
Overtone Series
Natural harmonics, brass instruments, missing fundamental
Timbre Synthesis
Spectral envelope, ADSR, instrument color
Additive Synthesis
Sine sum, Hammond organ, Fourier decomposition
Spectral Analysis
DFT, windowing, spectral leakage
Wavelet Analysis
Time-frequency, Morlet wavelet, multiresolution
Pitch Perception
Missing fundamental, residue pitch, mel scale
Rhythm Patterns
Metric hierarchy, syncopation, cross-rhythm
Polyrhythm
3 against 2, African drumming, LCM alignment
Euclidean Rhythm
Bjorklund algorithm, maximally even, tresillo
Music Theory Lattice
5-limit tuning, just intonation grid, Tonnetz
Chord Progressions
Harmonic function, voice leading, ii-V-I
Voice Leading
Contrary motion, parallel fifths, Bach chorales
Counterpoint
Species counterpoint, dissonance treatment, Fux
Fugue Structure
Subject, answer, stretto, Bach WTC
Canon Construction
Strict imitation, crab canon, Pachelbel
Species Counterpoint
Consonance rules, passing tones, suspensions
Modal Harmony
Church modes, Dorian-Phrygian-Lydian colors
Tonnetz Space
Neo-Riemannian theory, PLR operations, torus
Neo-Riemannian Theory
Chromatic mediant, hexatonic cycle, parsimonious voice leading
Pitch-Class Sets
Normal form, interval vector, Forte numbers
Twelve-Tone Row
Schoenberg, P/I/R/RI matrix, combinatoriality
Serialism
Total serialism, Webern symmetry, Boulez
Microtonality
31-TET, just lattice, xenharmonic tunings
Spectral Composition
Grisey, overtone tuning, acoustic phenomena
Granular Synthesis
Microsound, grain clouds, time-stretching
Convolution Reverb
Impulse response, RT60, Sabine formula
Physical Modeling
Karplus-Strong, digital waveguide, plucked string
Waveguide Synthesis
Bidirectional delay, bore resonance, woodwind acoustics
Formant Synthesis
Vocal tract resonances, vowel space, F1-F2
Vowel Space
F1-F2 plane, IPA vowels, diphthong trajectories
Thermodynamic Cycles
PV diagrams, Carnot efficiency, Otto/Diesel/Stirling
Critical Phenomena
Universality classes, power laws, scaling near T_c
Landau Theory
Order parameter, free energy, symmetry breaking
Ginzburg-Landau Theory
Superconductor vortices, coherence length, Type I/II
BCS Superconductivity
Cooper pairs, energy gap, phonon-mediated pairing
Fermi Liquid Theory
Quasiparticles, effective mass, Fermi surface
Band Structure
Energy bands, gaps, metals vs insulators
Brillouin Zone
Reciprocal lattice, zone folding, k-space topology
Tight-Binding Model
Hopping integrals, bandwidth, van Hove singularities
Hubbard Model
Mott transition, U vs t competition, correlations
Quantum Hall Effect
Landau levels, σ_xy = ne²/h, TKNN invariant
Frustrated Magnets
Geometric frustration, spin ice, kagome lattice
Magnetic Skyrmions
Topological spin textures, DM interaction, winding number
Dirac Cone
Linear dispersion, pseudospin, Klein tunneling
Graphene Band Structure
Honeycomb lattice, K/K' valleys, Dirac points
Crystallographic Symmetry
Point groups, space groups, Bravais lattices
Space Groups
Screw axes, glide planes, 230 space groups
Miller Indices
Crystal planes, hkl notation, d-spacing
X-Ray Diffraction
Bragg peaks, structure factor, powder pattern
Bragg's Law
Constructive interference, d-spacing, rocking curve
Nucleation & Growth
Critical nucleus, Gibbs barrier, supersaturation
Colloidal Crystals
Self-assembly, opal structure, photonic bandgap
Photonic Crystal
Photonic bandgap, mode confinement, slow light
Metamaterial Optics
Negative index, split-ring resonators, sub-wavelength
Negative Refraction
Veselago lens, reversed Snell's law, perfect focus
Transformation Optics
Coordinate folding, invisibility, Maxwell in curved space
Cloaking Device Physics
Scattering suppression, plasmonic, carpet cloak
Plasmonics
Surface plasmon polariton, confinement, SPR sensor
Surface Plasmon Resonance
Kretschmann config, evanescent coupling, biosensing
Mie Scattering
Sphere scattering, resonances, rainbow, vs Rayleigh
Optical Fiber
Total internal reflection, modes, numerical aperture
Total Internal Reflection
Critical angle, evanescent wave, frustrated TIR
Evanescent Wave
Exponential decay, tunneling, TIRF microscopy
Fiber Bragg Grating
Periodic refractive index, reflection spectrum, sensing
Second Harmonic Generation
χ⁽²⁾, phase matching, frequency doubling crystals
Optical Parametric Amplifier
Signal/idler, energy conservation, entangled photons
Four-Wave Mixing
χ⁽³⁾, FWM in fiber, cross-phase modulation
Soliton Propagation
SPM vs GVD, NLSE, shape-preserving pulses
Kerr Effect
n = n₀ + n₂I, self-phase modulation, B-integral
EIT
Dark state, slow light, quantum interference
Slow Light via EIT
Group velocity, pulse delay, light storage
Laser Cooling
Doppler cooling, radiation pressure, sub-Doppler
Magneto-Optical Trap
3D trapping, Zeeman shift, optical molasses
Bose-Einstein Condensate
Macroscopic wavefunction, evaporative cooling
Atom Interferometry
Matter waves, Sagnac effect, gravitational sensing
Optical Lattice Clock
Magic wavelength, 10⁻¹⁸ accuracy, redshift
Delayed Choice Experiment
Wave-particle duality, Wheeler, retroactive choice
Wheeler's Cosmic Experiment
Gravitational lensing, delayed choice, quasar
Which-Path Information
Complementarity, D² + V² ≤ 1, quantum eraser
Complementarity Principle
Bohr, position-momentum, time-energy uncertainty
Quantum Contextuality
Kochen-Specker, measurement context, KCBS
Bell Inequality Violation
CHSH, local realism, 2022 Nobel Prize
GHZ Entanglement
3-qubit entanglement, all-or-nothing nonlocality
Quantum Teleportation
Bell measurement, classical channel, no-cloning
E91 Protocol
Ekert QKD, entanglement-based, Bell test security
Quantum Threshold Theorem
Fault tolerance, error correction, p < p_th
Fault-Tolerant QC
Magic states, transversal gates, overhead
Topological Quantum Computing
Anyons, braiding, decoherence immunity
Anyons & Braiding
Fractional statistics, worldlines, topological phase
Majorana Fermions
Self-conjugate, Kitaev chain, topological qubits
Polymer Chains
Random walk conformations of flexible polymers — end-to-end distance, radius of gyration
Metamaterial Lens
Negative refractive index bends light backwards — Pendry's perfect lens beyond diffraction
2D Electron Gas
Quantum Hall effect — Landau levels, quantized conductance σ = νe²/h, edge states
Glacier Dynamics
Ice sheet flow, accumulation zones, basal sliding, response to temperature change
Ocean Circulation
Thermohaline conveyor belt — density-driven AMOC, heat transport, freshwater collapse
Spectral Decomposition
SVD reveals hidden structure — rank reduction, image compression, singular value spectrum
Virus Spreading
SIR epidemic on a contact network — R₀ effects, herd immunity threshold, outbreak waves
Quantum Spin Chain
Heisenberg model — ferromagnet vs antiferromagnet, magnon dispersion, Monte Carlo dynamics
Adiabatic Computing
Quantum annealing — tunneling through energy barriers to find the ground state
DNA Replication
DNA polymerase in action — leading strand, Okazaki fragments, helicase unwinding
Protein Folding
Energy funnel landscape, Levinthal paradox, HP lattice model, simulated annealing
Mitosis Phases
Cell division animated — prophase through cytokinesis, spindle assembly checkpoint
Homeostasis
Negative feedback regulation — set point, response delay, body temperature and blood glucose
Lymphocyte Activation
T-cell meets antigen — costimulation, clonal expansion, B-cell help, antibody production
Synapse Formation
Synaptogenesis — axon contact, vesicle docking, receptor clustering, CaMKII localization
Myelin Sheath
Saltatory conduction — myelinated axons propagate action potentials 100× faster
Cortical Column
6-layer neocortex — thalamic input to Layer IV, corticospinal output, feedback loops
Hippocampal Circuit
Memory's highway — perforant path → DG → CA3 → CA1, place cells, theta rhythm
Dopamine Pathway
Reward prediction error — mesolimbic circuit, VTA firing, unexpected reward vs omission
Memory Consolidation
Sleep replay transfers memories from hippocampus to neocortex — systems consolidation
Sleep Oscillations
EEG across sleep stages — slow waves, sleep spindles, K-complexes, REM desynchrony
Pain Signaling
Gate control theory — C fibers, Aδ fibers, dorsal horn, descending PAG inhibition
Wound Healing
Four-phase repair — hemostasis, inflammation, proliferation, and collagen remodeling
Stem Cell Niche
Wnt and Notch signals control self-renewal vs differentiation — asymmetric division
Antibody Binding
Affinity maturation — somatic mutation sharpens paratope fit, Kd improves 1000-fold
Viral Entry
Spike protein binds receptor → membrane fusion → endosomal escape → genome release
Apoptosis Cascade
Programmed death — intrinsic mitochondrial path meets extrinsic FasL, caspase cascade
Chromosome Packing
50,000× compaction — DNA wraps nucleosomes, folds into TADs, condenses to chromosome
Motor Protein Walk
Kinesin hand-over-hand stepping on microtubule — ATP hydrolysis, stall force, processivity
Mitochondrial Dynamics
Fission and fusion balance — membrane potential, ROS stress, cristae remodeling, mitophagy
Electroweak Symmetry
Higgs field potential and spontaneous symmetry breaking — W/Z boson mass generation
Gravitational Wave Detector
LIGO-style interferometer — strain, arm-length changes, and frequency spectrum
Tidal Disruption Event
Star torn apart at the Roche limit — debris stream and accretion onto a black hole
Dark Energy Equation of State
w parameter shapes cosmic expansion — phantom energy, quintessence, Big Rip
Quasar Spectrum
Broad emission lines redshifted by cosmic expansion — Lyman-alpha forest revealed
Supernova Explosion
Core collapse, bounce shock, neutrino burst, nucleosynthesis, and light curve
Planetary Migration
Type I and II disk migration — resonance capture and the Grand Tack scenario
Asteroid Belt & Kirkwood Gaps
Jupiter mean-motion resonances carve gaps in the asteroid belt
Protoplanetary Disk
Dust grain growth, streaming instability, and pebble accretion forming planets
Aurora Dynamics
Solar wind electrons spiral along field lines — color by altitude and collision species
Ionosphere Structure
D, E, F layers — plasma frequency determines radio wave reflection vs penetration
Troposphere Dynamics
Convection cells, dry adiabatic lapse rate, cloud formation at the LCL
Mesosphere
Meteor ablation, noctilucent clouds at the mesopause, and gravity wave propagation
Thermosphere
UV heating to 2000°C, satellite drag, and density vs solar activity
Radial Velocity Method
Stellar wobble from an orbiting planet — Doppler shift reveals mass and period
Direct Imaging of Exoplanets
Coronagraph suppresses starlight — contrast ratio, inner working angle, speckle noise
Circumstellar Disk
Gap opening by embedded planet, spiral density waves, and dust ring accumulation
Magnetar
Extreme neutron star with B~10¹⁵ G — starquakes, X-ray flares, and field decay
Coronal Mass Ejection
CME propagation through solar wind — shock front and geomagnetic storm severity
Heliosphere
Termination shock, heliopause, heliosheath — Voyager's journey to interstellar space
Pulsar Glitch
Superfluid vortex avalanche causes sudden spin-up — timing residual visualization
X-ray Binary
Compact object accretes from companion — hot disk, jets, and X-ray light curve
Cataclysmic Variable
White dwarf nova cycle — hydrogen accumulation, thermonuclear runaway, outburst
White Dwarf Cooling
Mestel cooling law, crystallization delay, and the oldest stars in the galaxy
Stellar Winds
CAK line-driven winds — mass loss rates from main sequence to Wolf-Rayet stars
QCD Confinement
Gluon flux tube stretching between quarks — string breaking and hadronization
Feynman Diagram Builder
Interactive diagram builder — draw fermions, photons, gluons, and Higgs propagators
Path Integral
Sum over all paths weighted by exp(iS/ℏ) — stationary phase gives classical limit
Quantum Field Vacuum
Virtual pair creation, Casimir effect — zero-point fluctuations between conducting plates
Frame Dragging
Lense-Thirring effect — gyroscope precession in rotating Kerr spacetime
Lattice QCD
Quark confinement on a discrete spacetime grid
Wilson Loop
Area law vs perimeter law — string tension in gauge theory
Confinement–Deconfinement
Polyakov loop order parameter and QCD phase transition
Quark-Gluon Plasma
RHIC fireball, jet quenching, and hydrodynamic flow
Chiral Symmetry Breaking
Mexican hat potential, pion Goldstone bosons, chiral condensate
Axion Field
Peccei-Quinn symmetry, strong CP problem, dark matter relic
Monopole Dynamics
Dirac magnetic monopole, field lines, and dyon quantization
Soliton Kink
φ⁴ kink solution, kink-antikink collision, topological charge
Theta Vacuum
Instanton gas, θ angle, CP violation, strong CP problem
Anomaly Cancellation
Triangle diagrams, chiral anomaly, Standard Model consistency
Calabi-Yau Compactification
Complex 3-fold, Hodge numbers, and the string landscape
String Amplitude
Veneziano amplitude, Regge trajectories, and string tension
Brane World
Randall-Sundrum bulk, brane tension, and the hierarchy problem
Extra Dimensions
Kaluza-Klein tower, compactification radius, graviton excitations
AdS Boundary
AdS₃ Poincaré coordinates, geodesics, and conformal structure
Ryu-Takayanagi
Minimal surface in AdS and holographic entanglement entropy
Entanglement Wedge
Wedge reconstruction, modular Hamiltonian, quantum error correction in AdS
MERA & CFT
Multi-scale tensor network, scale invariance, discrete holography
Bulk Reconstruction
HKLL smearing kernel, local bulk operator from boundary CFT
Stabilizer Code
Pauli stabilizers, syndrome measurement, 5-qubit perfect code
Logical Qubit
Surface code, code distance, and the threshold theorem
Topological Quantum Computing
Non-Abelian anyons, topological degeneracy, and braiding gates
Anyonic Braiding
Abelian anyon worldlines, phase accumulation, fusion rules
Fibonacci Anyons
τ⊗τ=1⊕τ fusion rules, pentagon equation, and the golden ratio
Quantum Teleportation
Bell measurement, classical correction, animated state transfer
Bell Inequality (CHSH)
Quantum vs classical bounds — nonlocality and Bloch sphere settings
Floquet Theory
Periodic driving, quasi-energies, and topological Floquet phases
Decoherence Channels
Bit flip, phase flip, depolarizing — Bloch sphere decay
Many-Body Localization
Disorder-induced localization, ETH breakdown, area-law entanglement
Open Quantum Systems
System-bath coupling, Redfield equation, non-Markovian memory
Satisfiability
DPLL SAT solver with live clause propagation
Constraint Propagation
AC-3 arc consistency on constraint graphs
Backtracking Search
N-Queens with live pruning visualization
Branch and Bound
Knapsack search with bounding and pruning
Memoization
Fibonacci call tree with cache hit highlights
Greedy Algorithm
Interval scheduling via earliest-finish heuristic
Divide and Conquer
Merge sort and quicksort with animated splits
Amortized Analysis
Dynamic array resizing and amortized O(1) cost
B-Tree
Insertion with node splitting and order control
Count-Min Sketch
Approximate frequency counting with hash tables
HyperLogLog
Cardinality estimation via leading-zero hashing
LSH Similarity
Locality-sensitive hashing for Jaccard similarity
Consistent Hashing
Ring with virtual nodes and key distribution
Distributed Consensus
Paxos propose/accept phases with quorum voting
Raft Protocol
Leader election, log replication, network partition
Gossip Protocol
Epidemic information spread and convergence
Vector Clocks
Causality arrows and concurrent event detection
Merkle Tree
Hash tree proof verification and tamper detection
Patricia Trie
Compressed prefix tree with insert and lookup
Rope Data Structure
Binary tree for efficient string split and concat
Finger Tree
2-3 tree deque with O(1) push/pop at both ends
Persistent Data Structure
Path-copying BST with full version history
Functional Zipper
Navigate and edit trees with context stack
Monad Composition
Maybe, List, and State monad bind chains
Parser Combinator
Recursive descent arithmetic parse tree
Register Allocation
Graph coloring for variable-to-register mapping
Lambda Calculus
Beta-reduction steps with Church numerals
Continuation-Passing Style
CPS transform vs direct style call stacks
Garbage Collection
Mark-sweep, ref counting, generational GC
Memory Allocator
Free list, buddy system, fragmentation metrics
Neural Architecture Search
Automated design of neural network architectures
Meta-Learning (MAML)
Learning to learn — inner/outer loop gradient visualization
Few-Shot Learning
N-way K-shot prototypical networks in embedding space
Self-Supervised Learning
Contrastive learning — pulling positive pairs, pushing negatives
Knowledge Distillation
Teacher-student training with soft targets and temperature
Variational Autoencoder
Latent space visualization, interpolation, and reparameterization
β-VAE
Beta parameter controls disentanglement via KL pressure
Disentangled Representations
Factor traversal — one latent dim, one generative factor
World Model
Latent dynamics model — imagination rollouts and uncertainty
Dreamer Agent
Actor-critic trained entirely in latent imagination space
Model Predictive Control
Horizon optimization with replanning at every step
Offline Reinforcement Learning
Conservative Q-learning prevents OOD action overestimation
Reward Shaping
Potential-based F(s,s') = γΦ(s') − Φ(s) — policy invariant
Multi-Agent RL
CTDE framework — centralized training, decentralized execution
Cooperative Navigation
Agents cover landmarks with collision avoidance and communication
Emergent Communication
Symbols arise from referential games — compositionality emerges
Causal World Model
SCM with do-calculus — interventions cut incoming edges
Neuro-Symbolic AI
Neural perception feeds symbolic reasoning for hybrid inference
Program Synthesis
Version space narrows with each input-output example
Neural Program Induction
Neural Turing Machine — differentiable external memory tape
Transformer Attention
Multi-head attention maps — each head learns different patterns
Positional Encoding
Sinusoidal, relative, and learned PE heatmaps compared
Rotary Embeddings (RoPE)
Complex-plane rotation — dot product depends only on relative position
FlashAttention
IO-aware tiling — SRAM blocks avoid N×N HBM materialization
Mixture of Experts
Top-k routing with load balancing and capacity factor
Sparse Attention
Local, strided, global, BigBird, and Longformer patterns
Linear Attention
Kernel trick turns O(N²) into O(N) via associativity
State Space Model
SSM dynamics with HiPPO initialization for long-range memory
Mamba Selective SSM
Input-dependent Δ,B,C — content-aware memory updates
In-Context Learning
Implicit gradient descent — attention implements optimization
Pascal's Fluid Law
Hydraulic press: small piston force multiplied by area ratio
Roche Lobe Transfer
Binary stars exchange mass when one fills its gravitational lobe
Precession & Wobble
Spinning top precesses: angular momentum resists gravity's torque
Hamiltonian Chaos
Double pendulum: Poincaré sections reveal KAM tori and chaos
Foucault Pendulum
Earth's rotation slowly turns the pendulum's plane of swing
Optical Aberrations
Chromatic and spherical aberration traced through imperfect lenses
Plasma Z-Pinch
Current through plasma creates magnetic field that pinches it inward
Detonation Wave
Chapman-Jouguet detonation: supersonic combustion vs deflagration
Shear & Couette Flow
Viscous fluid between plates: laminar profile transitions to turbulence
Ferrofluid Spikes
Rosensweig instability forms hexagonal spike arrays above critical field
Schlieren Imaging
Visualize invisible air density gradients from heat and shock waves
Piezoelectric Resonance
Crystal vibrates at resonant frequency under AC voltage with sharp Q peak
Photoacoustic Effect
Laser pulses absorbed by matter generate heat, pressure waves, and sound
Magnetostriction
Ferromagnetic materials change shape as magnetic domains align with applied field
Electrowetting
Voltage reshapes a droplet via Young-Lippmann equation; contact angle decreases
Dielectrophoresis
Neutral particles drift in non-uniform fields toward maxima or minima
Triboelectric Series
Rubbing materials transfers charge; polarity depends on series position
Ionic Wind
Electric field accelerates ions that drag neutral air, creating silent thrust
Corona Discharge
Sharp conductors exceed breakdown, forming Townsend avalanche and glow discharge
Hall Effect Sensor
Magnetic field deflects charge carriers, creating transverse Hall voltage V_H = IB/nqt
Peltier Cooler
Current pumps heat across a junction — thermoelectric cooling with COP
Thermoelectric Generator
Seebeck effect converts temperature gradient to voltage — ZT figure of merit
Superconductor Levitation
Flux pinning in type-II YBCO floats a magnet above the Meissner expulsion layer
NMR Pulse Sequence
90° RF pulse tips spins into xy-plane; watch FID decay and Fourier spectrum emerge
Synchrotron Radiation
Relativistic electrons bending in B-field emit a 1/γ cone of brilliant X-rays
Drift Tube Linac
RF-timed drift tubes accelerate protons — each tube grows longer as the beam speeds up
Wiggler / Undulator
Alternating magnets make electrons wiggle — coherent harmonic radiation from K parameter
Pair Production
Gamma ray near a nucleus births an electron-positron pair curving opposite ways in B-field
Compton Scattering
Photon scatters off a free electron — Δλ = λ_c(1−cosθ) wavelength shift, interactive angle
Bremsstrahlung Radiation
Electron decelerated by a nucleus emits continuous X-rays with characteristic line spectrum
Photon Momentum
Mirror recoil and solar sail thrust from p = h/λ photon kicks
Radiation Pressure
Gaussian beam traps a dielectric sphere via gradient and scattering forces
Doppler Cooling
Red-detuned lasers slow atoms below 1 mK using the Doppler effect
Atom Interferometer
Matter-wave Sagnac fringes measure rotation with atomic de Broglie waves
Stern-Gerlach Experiment
Spin quantization splits silver atoms into discrete beams in a magnetic gradient
Squeezed Light
Phase-space ellipse beats the standard quantum limit in one quadrature
Photon Number States
Fock state Wigner functions reveal quantum negativity in phase space
Vacuum Fluctuations
Virtual particle pairs pop from the quantum vacuum carrying ½ħω per mode
Casimir Effect
Two uncharged plates attract via mode-counting of vacuum fluctuations
Lamb Shift
QED virtual photons lift 2s above 2p in hydrogen by 1057.8 MHz
Photon Antibunching
HBT interferometer proves light is quantized via g²(0) dip
Hong-Ou-Mandel Effect
Two identical photons always exit a beamsplitter together
Bell State Generation
CNOT on |+⟩|0⟩ creates maximally entangled qubit pairs
GHZ State
Three-qubit all-or-nothing entanglement violates Mermin inequality
Stabilizer Formalism
Pauli stabilizers detect qubit errors without collapsing logical state
Quantum Annealing
Quantum tunneling finds ground states by slowly turning off transverse field
Variational Eigensolver
Classical optimizer tunes a quantum ansatz to find H₂ ground energy
Grover's Oracle
Amplitude amplification finds marked items in √N quantum steps
Shor's Algorithm
Quantum period finding factors integers exponentially faster than classical
Quantum Supremacy
Random circuit sampling generates distributions no classical computer can match
Band Structure of Solids
Nearly-free electron model with band gaps, Fermi level, and metal-insulator toggle.
Magnon Spectrum
Spin waves in ferromagnets: ω=Dk² dispersion, density of states, animated spin chain.
Quantum Spin Liquid
Frustrated triangular antiferromagnet with disordered ground state and spin correlations.
Charge Density Wave
Peierls instability: 2k_F charge modulation, Lindhard divergence, and CDW gap opening.
Spin Density Wave
Itinerant antiferromagnetism via Fermi surface nesting and SDW band structure gap.
Peierls Distortion
1D chain dimerization opens a gap and lowers total energy — spontaneous symmetry breaking.
Jahn-Teller Effect
Octahedral complex distorts to lift e_g degeneracy; double-well adiabatic potential surface.
Polaronic Transport
Self-trapped electrons in polar crystals with thermally activated hopping conductivity.
Magnetoresistance
Giant MR spin-valve structure and ordinary MR — R(B) curves and spin-dependent scattering.
Colossal Magnetoresistance
Manganite phase diagram, double-exchange mechanism, and dramatic ρ(T,B) resistance collapse.
Kuramoto Synchronization
N coupled oscillators spontaneously synchronize above critical coupling K_c.
Power-Law Distribution
Heavy tails revealed on log-log plots — Zipf, Pareto, earthquake magnitudes.
Rich-Club Network
High-degree hubs preferentially connect to each other, forming an elite club.
Cascading Failure
Interdependent networks collapse in avalanches — Buldyrev 2010 click simulation.
Multiplex Network
Same nodes, different edge types — friends, work, family layers stacked in 3D.
Adaptive Network
SIS epidemic on a rewiring network — topology and disease co-evolve in real time.
Network Epidemics
SIR on heterogeneous networks — hub immunization beats random immunization.
Graph Spectrum
Adjacency eigenvalues reveal structure — Wigner semicircle vs scale-free spike.
Laplacian Matrix
Heat diffusion, Fiedler vector, and algebraic connectivity on interactive graphs.
Network Entropy
Degree entropy and Von Neumann entropy compare ordered, random, scale-free graphs.
Smale's Horseshoe Map
Stretch, fold, repeat — the canonical geometric construction of chaos
Space-Filling Curves
Hilbert and Peano curves that pass through every point in a square
Chirikov Standard Map
KAM tori dissolve into chaos above the critical stochasticity threshold
Twist Map & KAM Theory
Invariant circles rotate by frequency ω(r) — rational ones break, irrational survive
Fiber Bundles
Möbius strip as a non-trivial bundle — one twist that changes everything
Saddle Connection
A heteroclinic connection where unstable and stable manifolds meet exactly
Heteroclinic Orbit
A trajectory connecting two saddle equilibria through the double-well
Andronov–Hopf Bifurcation
A stable spiral loses stability and a limit cycle is born from equilibrium
Transcritical Bifurcation
Two equilibria pass through each other and exchange stability at μ=0
Pitchfork Bifurcation
One stable fixed point splits into two — symmetry breaking, supercritical and subcritical
Neuron Cable Equation
Voltage spreading along a dendrite — length constant λ and passive decay
Bursting Neuron
Conductance-based burst-pause rhythm via slow K⁺ current adaptation
Cortical Columns
V1 orientation pinwheel — 18 columns rotate through 0–180° around a singularity
Thalamic Relay
Thalamocortical loop gating sensation — relay, spindle oscillations, burst modes
Basal Ganglia Circuit
Direct and indirect pathways — dopamine shifts balance to control movement
Cerebellar Learning
Purkinje cell LTD and eyeblink conditioning — climbing fiber teaches the cerebellum
Grid Cell Geometry
Hexagonal firing fields in entorhinal cortex — spacing, orientation, and phase
Head Direction Cell
Neural compass attractor ring — drag to rotate heading, watch the activity bump
Vestibular Integration
Semicircular canals, cupula deflection, and the VOR stabilizing gaze
Olfactory Bulb
Glomerular odor map — present rose, garlic, coffee, and see chemical feature coding
Three-Species Lotka-Volterra
Plant–herbivore–predator food chain with 3D phase portrait and ODE sliders.
Food Web Dynamics
12-species trophic network with cascading extinctions and energy flow.
Coevolution Arms Race
Red Queen dynamics: prey and predator speeds escalate in endless coevolution.
Metacommunity Dynamics
Patch occupancy, rescue effect, and dispersal in fragmented habitats.
Keystone Predator
Remove the sea star and watch mussels collapse a rocky intertidal community.
Biodiversity–Productivity
Intermediate disturbance hypothesis: hump-shaped diversity–disturbance curve.
Trait Evolution
Breeders' equation R=h²S: watch a trait distribution shift under selection.
Allopatric vs. Sympatric
Geographic isolation vs. disruptive selection driving genetic divergence.
Runaway Sexual Selection
Fisher's runaway: preference–trait covariance drives escalation to extremes.
Kin Selection & Hamilton's Rule
Altruism spreads when rb>c — watch helpers invade or collapse with relatedness.
Voting Paradoxes
Arrow's theorem and Condorcet cycles made interactive
Euler Fluid
Incompressible vortex dynamics with divergence-free projection
Soap Bubble Minimal Surface
Plateau's problem — mean curvature zero via Laplace relaxation
Predator-Prey Chaos
Spatial Lotka-Volterra spiral waves and spatiotemporal turbulence
Optical Interference
Double-slit, gratings, and thin films — wave-particle duality
Turing Patterns 3D
Gray-Scott reaction-diffusion on a rotating sphere or torus
Hopfield Network
Associative memory — store patterns, corrupt them, watch attractors recall
Brownian Motion 2D
Einstein relation, MSD, drift, and diffusion coefficient live
Double Pendulum Chaos
Lyapunov divergence — two pendulums, exponentially separating trajectories
Sandpile SOC
Bak-Tang-Wiesenfeld model — power-law avalanches and self-organized criticality
Hyperbolic Tilings
Navigate infinite {p,q} tilings in the Poincaré disk
Topological Defects 2D
XY model vortices and the BKT transition with temperature slider
Kuramoto Synchronization
50 coupled oscillators — watch phase order emerge as coupling grows
Anisotropic Heat Diffusion
Draw heat sources and watch directional spreading via diffusivity tensor
Fractal Flame
IFS with log-density rendering, swirl/sphere/polar variations
Reaction-Diffusion Manifold
Gray-Scott patterns on torus and Klein bottle topology
Sorting Networks
Batcher bitonic and odd-even mergesort — animate parallel comparators
Gravitational Waves
Inspiral chirp signal, spectrogram, and matched filter detection
Magnetohydrodynamics
2D MHD simulation — Alfvén waves, current density, magnetic pressure
3D Percolation
Bond percolation on a cubic lattice with spanning cluster detection
Polymer Folding
HP model Monte Carlo minimization on a 2D lattice
Spatial SIR Epidemic
Agent-based epidemic spread with infection radius and vaccination patches
Morse Theory
Height functions, critical points, level set sweeping, and Euler characteristic
Keplerian Orbits
Kepler's three laws — ellipses, equal areas, T²∝a³ with N-body simulation
Econophysics Wealth
Random exchange model, Pareto tail emergence, Gini coefficient, taxation
Protein Surface
Hydrophobic core formation and electrostatic potential mapping
Quantum Decoherence
Bloch sphere, environment coupling, coherence decay, and pointer states
Maxwell FDTD 2D
Finite-difference EM wave solver with dielectrics and PML boundaries
Nonlinear Waves
KdV solitons, NLS breathers, Burgers shocks — dispersion vs nonlinearity
Turing Machine
Tape visualizer with Busy Beaver machines, Rule 110, and space-time diagram
Arctic Circle Theorem
Random domino tilings reveal an emergent circular boundary
Riemann Zeta Visualizer
Domain coloring of ζ(s), zeros on the critical line, winding numbers
Hodgkin-Huxley Neuron
Full 4-variable ODE model of action potentials with phase plane
Minimal Spanning Tree
Animated Kruskal and Prim algorithms on random geometric graphs
Phase Space Flow
Vector fields, nullclines, and trajectories for classic 2D ODE systems
Stochastic PDE
Additive noise, multiplicative noise, and KPZ interface roughening
Topological Insulators
SSH chain band structure, winding number, and protected edge states
Hamming Codes
Encode messages, flip bits, syndrome decode, and explore perfect codes
Fourier Transform Visualizer
Draw signals, see their DFT, filter interactively, uncertainty principle
Mandelbrot / Julia Sync
Hover Mandelbrot to watch the corresponding Julia set update live
Collatz Conjecture
Follow the 3n+1 sequence through its mysterious stopping times
Boolean Circuit Simulator
Drag-and-drop logic gates, wire them up, watch signals propagate
Banach-Tarski Paradox
How a ball can be decomposed and reassembled into two identical balls
Gödel Incompleteness
Gödel numbering, self-reference, and the unprovable true statement
Halting Problem
Turing's diagonal argument and the undecidability of halting
Rule 30 Entropy
Wolfram's chaotic CA — Shannon entropy, autocorrelation, pseudorandomness
Langton's Ant 3D
3D turmite with multi-color states — watch emergent highways form
Polymer Radius of Gyration
Self-avoiding walks and the Flory exponent governing polymer scaling
LWR Traffic Flow
Shock waves and rarefaction fans in the Lighthill-Whitham-Richards PDE
Bridge Resonance
Driven oscillator, Tacoma Narrows dynamics, Bode plot frequency sweep
Penrose Quasicrystal
Aperiodic tilings with 5-fold symmetry via de Bruijn pentagrid
Solar System 3D
All 8 planets + Pluto with correct orbital periods and 3D tilt controls
Black Hole Lensing
Schwarzschild ray tracing — photon sphere, Einstein ring, accretion disk
Neutron Star Pulsar
Rotating pulsar lighthouse beam, P-Ṗ diagram, spin-down energy and glitches
Big Bang Nucleosynthesis
Primordial abundances of H, He, D, Li vs baryon density — the first 20 minutes
Riemann Hypothesis
|ζ(½+it)| on the critical line, non-trivial zeros, and π(x) vs Li(x)
Goldbach's Conjecture
Goldbach comet and pair decompositions for every even number up to 2000
Modular Arithmetic
Cayley tables, primitive roots, quadratic residues, and element orders mod n
Integer Sequences
OEIS explorer — Fibonacci, primes, Catalan, Bell, partition numbers and more
Figurate Numbers
Triangular, pentagonal, star and 3D figurate numbers as animated dot arrangements
Magic Squares
Generate and explore the ancient mathematics of magic squares
Catenary Chain
Hanging chains and catenary curves — drag endpoints, compare with parabola, Gateway Arch
Soap Film — Plateau's Problem
Minimal surfaces spanning wire frames — catenoid, helicoid, Enneper surface, Scherk
Isoperimetric Problem
Among all curves with equal perimeter, the circle maximizes area — Dido's problem
Cycloid & Tautochrone
Rolling circle traces the brachistochrone — balls from any height reach bottom simultaneously
Spirograph Mathematics
Hypotrochoids and epitrochoids — gear ratios, petal counts, Lissajous connections
Epicycloid & Hypocycloid
Cardioid, nephroid, deltoid, astroid — circles rolling on circles with tangent lines
Rhodonea Curves
Rose curves r = cos(kθ) — petal count rules, rational k, animated families
Butterfly & Algebraic Curves
Fay's butterfly, lemniscate, folium of Descartes, strophoid, witch of Agnesi
Parametric Surfaces
Torus, Klein bottle, Boy surface, Roman surface — interactive 3D rotation and rendering
Klein Bottle 3D
Explore the one-sided non-orientable surface immersed in 3D
Lorenz Weather System
Butterfly attractor with sensitive dependence on initial conditions
Torus Knots
Explore (p,q) torus knots with tube rendering and knot invariants
Möbius Transformations
Conformal maps of the complex plane — circles to circles
Projective Plane
Real projective plane RP², vanishing points, and homogeneous coordinates
Convex Hull Algorithms
Graham scan, Jarvis march, QuickHull animated step by step
Computational Geometry
Intersections, triangulation, visibility polygon, Minkowski sum
A* Pathfinding
Interactive grid maze with heuristic comparison and wall painting
Dijkstra's Algorithm
Shortest path tree animation with Bellman-Ford comparison
B-Tree Visualizer
Insert, delete, and search with animated node splits and merges
p-adic Integers
Ultrametric tree of ℤₚ — explore p-adic distance and residue class balls
Modular Forms
Visualize Δ(τ), j-invariant, and Eisenstein series on the upper half-plane
Lattice Gas Automaton
FHP model — emergent fluid dynamics from simple hexagonal collision rules
Persistent Homology
Vietoris-Rips filtration with live barcode diagram — topology across scales
Navier-Stokes Vorticity
2D vorticity-stream function solver — click to inject vortex pairs
Spin Glass
Edwards-Anderson model with frustrated bonds — Metropolis Monte Carlo
Hyperbolic Tessellation
{p,q} tilings of the Poincaré disk — infinite hyperbolic plane in finite space
Wave Packet Spreading
Free-particle quantum dispersion — split-operator method with phase portrait
Quantum Tunneling
Wave packet through a barrier — transmission and reflection probabilities live
Lorenz Attractor
Strange attractor, Lorenz map, and sensitive dependence on initial conditions
de Rham Cohomology
Differential 1-forms on surfaces — closed vs exact, cohomology classes, and the topology of holes
Abelian Sandpile
Chip-firing on a grid — self-organized criticality and fractal patterns from local toppling rules
Spectral Graph Theory
Graph Laplacian eigenvectors — Fiedler vector, algebraic connectivity, and community detection
Quantum Harmonic Crystal
Phonon modes in a 1D chain — dispersion relation, zero-point motion, and Bose-Einstein populations
Bravais Lattices
All 5 planar crystal symmetries with their reciprocal lattices — direct and Fourier dual side by side
Wigner Semicircle Law
Random matrix eigenvalue statistics — the universal semicircle emerges from GOE, GUE, and sparse ensembles
Floquet Topological Phases
Periodically driven quantum bands — anomalous edge modes, winding numbers, and the full phase diagram
Penrose Tiling
Aperiodic quasicrystal tiling via deflation rules — 5-fold symmetry and the golden ratio in every tile count
Mandelbulb
3D fractal ray marching — sphere-traced Mandelbulb with adjustable power and distance-estimate shading
Artin Braid Group
Strand diagrams for Bₙ — type braid words, see crossings, and explore knot closures interactively
Symplectic Billiard
Billiard dynamics in convex domains — area-preserving reflection maps, KAM tori, and Poincaré sections reveal chaotic and integrable regions
3D Game of Life
Conway's rules on a cubic lattice — choose survival/birth counts and watch 3D cellular automata evolve explosive, stable, or oscillating patterns
Frustrated Ising Model
Antiferromagnetic Ising spins on a triangular lattice — geometrical frustration prevents long-range order even at low temperature
Riemannian Curvature
Parallel transport on curved surfaces — a vector carried around a closed loop returns rotated by the holonomy angle, equal to enclosed Gaussian curvature
Galaxy Merger N-Body
Two galaxies colliding under gravity — tidal arms, stellar bridges, and eventual coalescence in a live N-body simulation with massive black holes
Crystal Phonon Dispersion
Lattice vibrations in mono- and diatomic 1D crystals — acoustic and optical branches, animated atom displacements, and dispersion curves in real time
Delaunay & Voronoi 3D
3D point sites with Voronoi cells and dual Delaunay triangulation — circumsphere conditions and spatial duality visualized in rotatable 3D
3D Fourier Epicycles
Any closed 3D space curve — trefoil knot, torus knot, Lissajous — decomposed into rotating circles; add terms to watch the curve assemble
Cohomology Spectral Sequences
Serre spectral sequences for classic fibrations — step through E₂, E₃, … E∞ pages and watch differentials cancel cohomology groups
Jacobi Elliptic Functions
sn, cn, dn computed via AGM — deform from trigonometric to hyperbolic as modulus k→1, with exact pendulum solution and phase portraits
Bloch Sphere Qubit
Single-qubit state on the Bloch sphere — apply Hadamard, Pauli, and phase gates, watch rotations animate, and collapse via Born-rule measurement
Riemann Zeta Zeros
Trace ζ(1/2+it) as a curve in the complex plane — zeros appear as crossings through the origin, with known non-trivial zeros highlighted
Erdős–Rényi Phase Transition
Watch a giant connected component suddenly emerge in G(n,p) as mean degree c=np crosses 1 — the sharp percolation threshold in random graphs
Escher Circle Limit
Hyperbolic tessellations in the Poincaré disk — BFS-reflected {p,q} tilings where every tile is the same hyperbolic size, shrinking toward the infinite boundary
Rule 110 — Universal CA
The Turing-complete 1D cellular automaton — complex gliders and structures emerge from a 3-cell neighborhood rule, with all 256 Wolfram rules selectable
Kuramoto Model 3D
N oscillators on a sphere synchronize as coupling K increases past the critical threshold — order parameter r measures the phase transition in real time
Legendre Transform
Visualize convex duality: f*(p) = sup[px−f(x)] — animated tangent lines sweep the primal, tracing the dual curve, with thermodynamic analogies
Barnsley Fern 3D
Four affine IFS maps iterated via the chaos game generate a perfect fern fractal — extended to 3D with depth twist, rotation, and multiple fern variants
Einstein Ring
Gravitational lensing simulation: ray-shoot through a point mass lens, see Einstein rings form at perfect alignment and arc into images as source offset increases
Poisson Disk Sampling
Bridson's O(n) blue noise algorithm — no two points closer than r, no empty region larger than r, visualized with Voronoi, Delaunay, and exclusion disks
Catalan Numbers
Ballot sequences, Dyck paths, mountain ranges, and polygon triangulations — all counted by C(n) = (2n)!/(n+1)!n!
Schwarzschild Geodesics
Light and massive particle paths around a black hole — photon sphere at 3M, ISCO at 6M, light bending and orbital precession
Toda Lattice Solitons
Integrable nonlinear lattice with exact soliton solutions — compression waves that pass through each other without dispersion
Lorenz Poincaré Section
Lorenz attractor return map — z_max vs z_{n+1} reveals the strange attractor's 1D fractal structure beneath the chaos
Dirichlet Eta Function
η(s) = (1−2¹⁻ˢ)ζ(s) visualized via domain coloring, partial sum spirals, and the critical line Re(s)=½
Spinor Double Cover
SU(2) → SO(3): the belt trick shows 360° is non-trivial but 720° unwinds — spinors acquire a phase flip after one full rotation
Hadamard Matrices
Walsh functions, fast Hadamard transform, and Reed-Muller error-correcting codes — built from H ⊗ H Sylvester construction
Topological Defect Vortices
Vortices and antivortices in the 2D XY model — Monte Carlo simulation of the Kosterlitz-Thouless transition near T≈0.89J
Liouville's Theorem
Phase space volume conservation — watch a blob of trajectories deform but never shrink or grow under Hamiltonian flow
Schwarz-Christoffel Mapping
Conformal maps from the upper half-plane to polygons — orthogonal grids mapped angle-preservingly into squares, triangles, and L-shapes
Poincaré Disk Geodesics
Draw geodesics and triangles in the hyperbolic plane — where the angle sum is less than π and infinitely many parallels pass through every point
Fourier Transform Pairs
Visualize time-frequency duality live — switch between rect, Gaussian, sinc, chirp, and triangle signals and watch the spectrum update instantly
Quantum Double-Slit
Two slits, one particle, both paths at once — adjust slit geometry and wavelength to reshape the interference fringes
Continued Fractions
Convergents, best rational approximations, and the Stern-Brocot tree — explore π, φ, e, √2, or any real number you choose
Kirchhoff's Laws
Interactive resistor networks with live current flow — nodal analysis solves Wheatstone bridges, ladders, and parallel circuits in real time
Cayley Graphs of Groups
Visualize S₃, D₄, ℤ/8ℤ, Q₈, A₄, and the Klein four-group — colored edges are generators, and the graph IS the group
Gauss Map Iteration
T(x) = {1/x} generates continued fraction coefficients — cobweb diagrams, orbit histograms, and the Gauss-Kuzmin invariant density
Surface of Revolution
Draw any profile curve and rotate it around an axis — generates vases, spheres, tori, and cones with Phong-shaded 3D rendering
Stirling's Approximation
n! ≈ √(2πn)(n/e)ⁿ — compare exact log-factorials against Stirling and the refined series, with relative error and ratio plots
Fokker-Planck Evolution
Probability density evolving in a potential — watch diffusion spread and drift focus the distribution toward the Boltzmann steady state
Ricci Flow on Surfaces
Hamilton's ∂g/∂t = −2Ric — watch a 2D metric evolve toward constant curvature via scalar curvature diffusion, the engine behind Perelman's proof
Tensor Network Visualizer
MPS, MERA, PEPS, and tree tensor networks — drag nodes, animate index contractions, and see how bond dimension χ controls entanglement
Bernoulli Numbers
Bₙ via generating function t/(eᵗ−1) — explore connections to ζ(2n), power sums, Pascal's triangle, and superexponential growth of |B₂ₙ|
2D Diffusion Equation
∂u/∂t = D∇²u on a 2D domain — click to add heat sources, switch boundary conditions, and watch Gaussian spread in vivid colormaps
Newton Fractal — Extended
Newton's method for degree-5+ polynomials — pixels colored by convergence root and speed, with relaxation parameter α to trigger bifurcations
Spectral Graph Clustering
Graph Laplacian eigenvectors embed nodes into Rᵏ — watch the Fiedler vector reveal optimal cuts and k-means find clusters in eigenspace
Minimal Surfaces — Soap Film
Mean curvature flow H=0 — simulate soap films on wire frames, render catenoids, helicoids, and Enneper surfaces with live curvature maps
Bott Periodicity
πₖ(U) has period 2; πₖ(O) has period 8 — interactive tables of homotopy groups, K-theory of spheres, and the topological insulator classification
Floer Homology
Gradient flow trajectories on a potential — watch Morse critical points, gradient descents, and the chain complex ∂² = 0 come alive
Integer Partitions
Young diagrams, Hardy-Ramanujan asymptotics, Euler's generating function, hook length formula, and Ramanujan rank distribution for p(n)
3D Brownian Motion
Particles undergoing random thermal diffusion in 3D — drag to rotate, watch mean square displacement grow as ⟨r²⟩ = 6Dt
Soap Bubble Cluster
Plateau's laws in action — bubbles meet at 120°, minimize total surface area via Young-Laplace pressure balance
Quantum Entanglement
Bell states on the Bloch sphere — measure qubit A and watch B collapse, violating Bell's inequality up to 2√2
Hodgepodge Machine
Excitable cellular automaton that spontaneously forms spiral waves — analogous to the Belousov-Zhabotinsky reaction
Apollonius Gasket
Apollonian gasket via Descartes' circle theorem — a fractal of tangent circles with Hausdorff dimension ≈ 1.3057
Euler Product Formula
ζ(s) = Σn⁻ˢ = Π(1−p⁻ˢ)⁻¹ — watch partial sums and prime products converge together, animated Sieve of Eratosthenes
Crystal Growth — DLA
Diffusion-Limited Aggregation grows fractal crystals with dimension ≈ 1.71 — snowflake symmetry, multi-seed, and stickiness control
Fluid Vortex Ring
Toroidal vortices self-propel via Biot-Savart; two rings leapfrog and interact — tracers reveal the induced velocity field
Laplace Equation Solver
Draw conductors at fixed potentials, watch equipotential lines and electric field emerge via SOR iteration of ∇²φ = 0
Ising — Wolff Cluster
Ising model with Wolff cluster flips eliminates critical slowing down near T_c ≈ 2.269 — compare with Metropolis
Optical Caustics
Light rays reflect off a curved mirror to form brilliant caustic curves — the same optics that create rainbows and swimming-pool shimmer
Fourier Series & Gibbs
Add harmonics to square, sawtooth, and triangle waves — watch the ~8.9% Gibbs overshoot persist at every discontinuity no matter how many terms
Quantum Hall Edge States
Chiral edge currents flow without backscattering in Landau levels — Hall conductance σ_xy = νe²/h is exact to one part in 10⁹
Barnsley Chaos Game
Four affine contractions applied randomly generate exact fractal attractors — fern, tree, dragon, and Sierpinski from nothing but probability
2D Wave Equation Membrane
Strike a drum or excite pure (m,n) modes — the 2D wave equation solved in real time with a colorfield or 3D surface view
Padé Approximants
Rational [m/n] approximants vs Taylor polynomials — Padé converges far beyond the Taylor radius, especially near poles
Julia Set Parameter Sweep
Click the Mandelbrot set to pick c and watch the Julia set transition between connected and Cantor-dust regimes in real time
Abelian Sandpile (BTW)
Drop grains until cells topple — avalanche sizes follow a power law P(s) ~ s^{-1.2}, the canonical example of self-organized criticality
Diffusion-Limited Aggregation
Random-walking particles stick on contact, growing fractal dendrites with Hausdorff dimension d_H ≈ 1.71 via tip-screening instability
Random Matrix Level Spacing
Eigenvalue spacings from GUE and GOE obey Wigner surmise — level repulsion is universal across quantum chaos, nuclear spectra, and prime gaps
Riemann Zeta Zeros
Visualize |ζ(1/2+it)| on the critical line, watch the function graze zero at t ≈ 14.13, 21.02 … and explore the Riemann Hypothesis in action.
Quantum Eraser Experiment
Toggle the which-path detector and watch double-slit interference fringes collapse and restore — photon by photon.
Fibonacci Spiral Growth
Place seeds at the golden angle 137.5° and watch interlocking Fibonacci spirals emerge from a single arithmetic rule.
Atmospheric Circulation Cells
Animated cross-section of Earth's Hadley, Ferrel, and Polar cells — the global wind engine driven by solar heating and the Coriolis effect.
Gravitational Lensing Arcs
Click to place background galaxies and watch them arc into Einstein rings around a massive foreground cluster via the Schwarzschild lens equation.
Bioluminescent Waves
Move through dark ocean water to excite dinoflagellates — each particle flashes cyan-blue and fades as its luciferin depletes.
Navier-Stokes Streamlines
Potential flow around a cylinder or airfoil via complex analysis — drag the obstacle and watch streamlines and pressure fields respond.
Eigenvalue Perturbation
Perturb a 2×2 matrix with sliders and watch eigenvalues trace paths in the complex plane, avoiding each other near degeneracy.
Solar Wind & Magnetosphere
Animated proton stream from the Sun compresses Earth's magnetic field, forms a bow shock, and funnels auroral particles to the poles.
Phononic Crystal Bandgap
Drive a diatomic spring chain at different frequencies to see propagating waves vs. exponential evanescent decay inside the phononic bandgap.
Geodesic Dome Geometry
Explore icosahedron subdivision and the elegant geometry of geodesic spheres — drag to rotate, adjust frequency to subdivide faces.
Plasma Wave Dispersion
Dispersion relations ω(k) for Langmuir, whistler, and lower hybrid plasma waves — click the curve to see any wave oscillate in real time.
Monte Carlo π Estimation
Random points rain into a square — the ratio inside the quarter-circle converges to π/4. Watch live convergence toward the true value.
Muon Decay Simulation
Relativistic muons created at 15 km altitude survive to Earth's surface via time dilation. Compare lab frame vs muon rest frame side by side.
Sierpinski Carpet Generator
Generate the Sierpinski carpet iteratively up to depth 6. Each level is colored distinctly; fractal dimension log(8)/log(3) ≈ 1.893.
Ocean Tidal Forcing
Animate tidal bulges from the Moon and Sun. See spring tides when they align and neap tides when perpendicular — drag the Moon around Earth.
Polymer Random Coil
Freely-jointed chain polymer as a random walk — end-to-end distance scales as √N. Generate ensembles and compare to the Gaussian theory.
Fourier Epicycles Drawing
Draw any closed shape and watch rotating epicycles reconstruct it via DFT. Adjust the number of frequency terms to control accuracy.
Lévy Flight Random Walk
Power-law step lengths produce rare giant jumps and clustered self-similar patterns. Compare Lévy and Gaussian walks side by side.
Standing Wave Resonance
Toggle harmonics on a fixed string and tune amplitudes — watch superposition build complex waveforms from pure sinusoidal modes.
Ising Model 2D
Watch magnetic phase transitions emerge from local spin interactions near the critical temperature T_c ≈ 2.269. Metropolis Monte Carlo on an 80×80 lattice.
Lorenz Butterfly Attractor
The strange attractor of the Lorenz system traced by RK4. Drag to rotate in 3D — two nearby trajectories diverge exponentially, revealing deterministic chaos.
Galton Board
Balls cascade through 12 rows of pegs, randomly bouncing left or right. The bins accumulate into a binomial distribution — the central limit theorem made visible.
Magnetic Field Visualizer
Magnetic dipole field lines traced by numerical integration. Click to place a second magnet and watch the fields interact. Toggle field vectors and dipole mode.
Predator-Prey Phase Space
Lotka-Volterra dynamics shown as a time series and phase portrait simultaneously. Adjust α, β, γ, δ to deform the closed orbits of perpetual oscillation.
Voronoi Lloyd Relaxation
Random seed points converge to a perfectly uniform distribution via Lloyd's algorithm — each seed migrates to the centroid of its Voronoi cell, iteration by iteration.
Brownian Motion & Einstein Relation
Einstein's 1905 result ⟨r²⟩ = 4Dt confirmed live: particle paths on the left, mean-squared displacement vs time on the right, with theoretical line overlaid.
Airy Function & Caustics
Ai(x) and Bi(x) plotted via RK4 integration of y''=xy. The intensity |Ai(-x)|² governs light near a fold caustic — the classical turning point lives at x=0.
4D Hypercube Projection
A tesseract with 16 vertices and 32 edges, projected from 4D to 2D through two simultaneous rotation planes. Drag to rotate in 3D while the 4D spin continues.
Wave Packet Dispersion
A Gaussian wave packet under linear vs quadratic dispersion — the free quantum particle (ω=k²/2) spreads as it propagates while the non-dispersive case holds shape.
Maxwell-Boltzmann Distribution
Explore how temperature shapes the speed distribution of gas molecules.
Epicycloid Spirograph
Two rotating circles trace mesmerizing parametric curves in real time.
Quantum Particle in a Box
Wavefunctions, energy levels, and quantum beating in the infinite square well.
Coriolis Effect
Watch straight-line motion curve dramatically in a rotating reference frame.
Minimal Surface — Catenoid
The catenoid soap film and its snap to two flat discs at the Goldschmidt discontinuity.
Feigenbaum Bifurcation
Period-doubling cascade into chaos in the logistic map, with delta approx 4.669.
Electric Field Multipoles
Field lines and equipotentials for monopole, dipole, quadrupole, and octopole charges.
Nonlinear Pendulum Phase Portrait
Click the phase plane to launch trajectories through libration, separatrix, and rotation.
Diffraction Grating Spectrum
Constructive interference peaks sharpen as slit count N increases.
Higgs Mexican Hat Potential
Spontaneous symmetry breaking, the vacuum manifold, Goldstone mode, and Higgs mass.
Sandpile & Self-Organized Criticality
Watch avalanches obey a power law as the sandpile hovers at the edge of chaos.
Wigner Semicircle Law
Random matrix eigenvalues converge to the semicircle distribution as matrix size grows.
Elastic Collision Billiard
Hard disk billiards demonstrating ergodicity and the emergence of Maxwell-Boltzmann statistics.
Kármán Vortex Street
Vortices shed alternately behind a cylinder, forming the iconic alternating street pattern.
Poincaré Section — Hénon-Heiles
KAM tori dissolve into chaos as energy increases past the critical threshold.
Quantum Tunneling & WKB
A wavepacket splits at a barrier — part reflects, part tunnels through with WKB transmission T.
Fractal Tree Branching
Recursive branching geometry generates diverse tree morphologies with tunable Hausdorff dimension.
Chaos Game — IFS Fractals
Random jumps toward attractor points trace Sierpinski triangles, ferns, and custom fractals.
Network Percolation Threshold
Watch a giant connected component snap into existence at the percolation phase transition.
KdV Solitons
Two solitons collide and pass through each other unchanged — the miraculous elastic KdV property.
Geodesic Flow on a Torus
Watch straight-line geodesics wind ergodically across a curved surface.
Game of Life — 3D Spacetime
Conway's Game of Life stacked into a 3D history you can rotate and explore.
Spin Echo NMR
Animate the Hahn echo: spins dephase, a 180° pulse refocuses them into a perfect echo.
Atmospheric Gravity Waves
Internal gravity waves in a stratified atmosphere — tilted phase lines, perpendicular group velocity.
Erdős–Rényi Random Graph
Watch the giant component explode into existence right at p_c = 1/n.
Quantum Decoherence & Pointer States
Coherences vanish as the Bloch vector collapses inward — environment selects pointer states.
Torus Knot Visualization
Trefoils, cinquefoils, and links — parametric (p,q) knots winding around a transparent torus.
Maxwell's Demon
A demon sorts hot and cold particles — until Landauer's erasure principle saves the 2nd Law.
Hopf Bifurcation
Stuart-Landau dynamics: fixed point to limit cycle as μ crosses zero — supercritical and subcritical.
Ikeda Attractor
A laser-cavity map spiraling into a strange attractor — beautiful chaos from a simple rule.
Langton's Ant
A simple cellular automaton that spontaneously builds a periodic highway after 10,000 chaotic steps. Multi-ant mode shows emergent collective behavior.
Schwarzschild Orbit Precession
GR's 1/r³ correction turns closed Newtonian ellipses into precessing rosettes — explaining Mercury's 43 arcsec/century anomaly.
Acoustic & Optical Phonons
Diatomic lattice dispersion: acoustic and optical branches, bandgap at the zone boundary, and animated atomic displacements for each mode.
Vortex Ring — Smoke Ring
A smoke ring self-propels via Biot-Savart induction. Two rings leapfrog forever — each passing through the other in alternation.
Continued Fraction Approximation
Convergents pₙ/qₙ bracket the target with exponentially shrinking error. φ needs all 1s; π's [3;7,15,1,292] gives 355/113 to 7 decimal places.
Belousov-Zhabotinsky Spiral
Oregonator reaction-diffusion: an excitable chemical medium spontaneously forms rotating spiral waves — order from chemistry.
Schwarzschild Penrose Diagram
Conformal compactification of the Kruskal extension: four regions, event horizons, singularities, and infalling worldlines — all of spacetime on one page.
Neural Net Loss Landscape
A 2D training surface with saddle points and local minima. Watch GD, SGD+Momentum, and Adam navigate simultaneously — rotate the 3D view by dragging.
Quantum Random Walk
Hadamard coin walk spreads ballistically (σ∝N) with interference fringes — radically different from the classical Gaussian (σ∝√N).
Hyperbolic Space — Poincaré Disk
The {7,3} tiling packs heptagons where only hexagons fit in flat space. Geodesics are circular arcs; triangle angles sum to less than 180°.
Spinor & The Belt Trick
Why electrons need 720° to return to their original state — the topology of spinors made visual.
✦ Lab #4000 — Emergence ✦
The 4000th exploration: four simultaneous emergence phenomena — flocking, memory, crystallization, prime spirals.
Pendulum Phase Portrait
Animate a pendulum and trace its path through phase space — separatrix and chaos.
Logistic Map Bifurcation
Zoom into the bifurcation diagram to reveal the Feigenbaum constant and fractal chaos.
Cellular Automata Rule 110
Wolfram's Rule 110 — proven Turing complete — generates rich patterns from three-cell logic.
Sierpiński Triangle 3D
Chaos game on a 3D tetrahedron — rotating fractal point cloud with Hausdorff dimension 2.
Julia Set Explorer
Move your mouse over the Mandelbrot minimap to morph the Julia set in real time.
Particle Life Simulation
Emergent cells and ecosystems from simple species attraction-repulsion matrices.
Brusselator Oscillator
Reaction-diffusion Turing patterns and chemical waves on a 2D grid you can poke.
Fractal Tree Generator
L-system branching trees with adjustable angle, depth, and randomness — ferns to snowflakes.
Langevin Brownian Motion
Thermal particles with live MSD plots confirming the Einstein relation D = k_BT/γ.
Topological Defects XY Model
Watch vortex-antivortex pairs proliferate above the Berezinskii-Kosterlitz-Thouless transition.
Eigenvalue Visualization
Watch eigenvectors glow and rotate as you reshape a 2×2 matrix live.
Schrödinger 1D Well
Quantum bound states: wavefunctions, energy levels, and time evolution.
Fourier Transform Drawing
Draw any curve; watch epicycles reconstruct it harmonic by harmonic.
Spring-Mass Lattice
Pluck a 2D crystal lattice and watch phonon waves propagate and decay.
Gravitational N-Body
Galaxy collision simulation with orbital trails and tidal disruption.
Metabolic Network Flow
Animate glycolysis and Krebs cycle flux; tune enzymes to control ATP yield.
SDE Stochastic Paths
Brownian motion, geometric BM, and Ornstein-Uhlenbeck with confidence bands.
Cellular Potts Model
Cell sorting via differential adhesion — Steinberg's hypothesis animated.
Knot: Trefoil 3D
Rotate glowing 3D knots; switch types and view topological invariants.
Maxwell's Demon
Gas sorting by speed: entropy violation, Landauer's principle, information cost.
Soap Bubble Minimal Surface
Mean curvature flow finds minimal area surfaces spanning wire frames.
Voronoi Growth Dynamics
Animated cellular territories compete and conquer with moving seeds.
Epidemic SIR Network
Disease spreads through a small-world network; watch SIR curves evolve.
Magnetic Pendulum Fractal
Three magnets create fractal basin boundaries from chaotic trajectories.
Chord Harmony Visualizer
Circle of fifths, harmonic waves, and chord progressions with audio.
Hopfield Network Memory
Associative memory: noisy patterns recover via energy minimization.
Fluid Vortex Street
Kármán vortex street: alternating vortices shed behind a cylinder in flow.
Strange Nonchaotic Attractor
Fractal geometry with negative Lyapunov exponents — not chaos, not order.
Polymer Chain Folding
HP lattice model: hydrophobic collapse folds proteins by Monte Carlo search.
Nonlinear Schrodinger Soliton
Solitons propagate without dispersion; two collide and pass through intact.
Topology: Euler Characteristic
Explore V−E+F=χ across polyhedra; visualize genus and homeomorphism classes in 3D.
Klein Bottle 4D
Rotate a Klein bottle through 4D space; self-intersection slides as artifact of projection.
Random Walk & Lévy Flight
Compare Brownian motion with Lévy flight; fat tails cause superdiffusion and infinite variance.
Zeta Function Zeros
Visualize ζ(s) in the complex plane; all known non-trivial zeros sit on Re(s)=½.
Amoeba Division Simulation
Cells grow, deform, and divide; click to add nutrients and watch a colony bloom.
Collatz Conjecture Tree
Trace 3n+1 sequences converging to 1; visualize the unsolved conjecture as a branching tree.
Lorenz-96 Weather Model
Atmospheric chaos on a ring of K variables; tuning forcing F drives order into full chaos.
Escape Velocity & Orbits
Place bodies and set velocities; watch circular, elliptical, and hyperbolic trajectories unfold.
Dirichlet Series Explorer
Visualize L-functions and Euler products; spiral partial sums converge to zeros and poles.
Phase Transition: Potts Model
Q-state generalization of Ising; watch first-order vs continuous transitions with live order parameter.
Topological Sort DAG
Animate Kahn's and DFS algorithms on an interactive directed acyclic graph.
Crowd Flocking: Vicsek Model
Spontaneous symmetry breaking — disorder to ordered flocking via noise and alignment.
Spin Glass: Sherrington-Kirkpatrick
Frustrated random couplings, rugged energy landscape, and replica overlap order parameter.
Fourier Heat Diffusion 2D
Draw heat sources and watch ∂T/∂t = α∇²T diffuse with spectral decomposition.
Moiré Interference Pattern
Two overlapping grids beat to form emergent fringes; tune spacing, angle, and type.
Neural Oscillator Synchrony
Kuramoto model: watch oscillators spontaneously synchronize above the critical coupling K_c.
Hyperbolic Geometry: Poincaré Disk
Draw geodesics and tessellations in the hyperbolic plane; infinite parallel lines abound.
Crystallization & Nucleation
Dendritic crystal growth, grain boundaries, and symmetry controlled by anisotropy.
Adaptive Optics Wavefront
Atmospheric turbulence distorts starlight; watch Zernike correction sharpen the PSF.
Prime Gaps Visualization
Scatter plots of prime gaps, Cramér's conjecture, record gaps, and animated Sieve.
Continued Fractions Visualizer
Euclidean rectangle subdivision reveals the deep structure of every real number.
Boolean Satisfiability: 3-SAT
Flip variables to satisfy clauses; watch the phase transition at α≈4.27 emerge.
Tidal Locking Simulation
Tidal torque gradually synchronizes a moon's spin with its orbital period.
Maxwell's Equations: Waves
FDTD simulation of EM waves — click to place sources, draw conductors, watch diffraction.
Euler Fluid 2D
Incompressible Euler equations with Kelvin-Helmholtz instability and vortex dynamics.
Galton Board & Central Limit Theorem
Binomial trials through pins converge to a perfect Gaussian — the CLT made visible.
Lindenmayer System 3D
Parallel string rewriting generates fractal trees, dragon curves, and Hilbert curves in 3D.
Variational Autoencoder Latent Space
Click in 2D latent space to decode shapes; interpolate through smooth learned representations.
Spin Waves & Magnons
Heisenberg ferromagnet: flip one spin and watch magnons propagate with dispersion ω(k)=4J sin²(k/2).
Conway's Surreal Numbers
Day-by-day construction of the surreal tree: reals, infinitesimals, and ω from two empty sets.
Atmospheric Convection Cells
Rayleigh-Bénard instability drives convection rolls above critical Ra≈1708
Network Centrality Measures
Degree, betweenness, PageRank reveal which nodes matter most in a network
Acoustic Chladni Patterns
Sand settles at nodal lines of vibrating plates, tracing hidden standing waves
Conformal Mapping: Complex Analysis
Analytic functions deform grids while preserving angles — conformality visualized
Entropy & Maxwell-Boltzmann
Colliding gas particles relax to the Maxwell-Boltzmann distribution as entropy rises
Optical Tweezers Trap
A focused laser gradient force traps a particle against Brownian thermal kicks
Markov Chain Mixing
Probability flows through states toward stationary distribution; watch it converge
Soliton: KdV Equation
KdV solitons collide elastically and emerge unchanged — nonlinear wave magic
Wright-Fisher Genetics
Genetic drift randomly fixes or loses alleles; smaller populations drift faster
Hypercube: Tesseract
4D hypercube projected to 2D, rotating through all six planes of 4D space
Penrose Tiling: Aperiodic
Quasicrystal tilings cover the plane without ever repeating, with 5-fold symmetry
Black Hole Photon Orbit
General relativistic photon paths bend, orbit, and spiral into a Schwarzschild black hole
Tournament Sorting Network
Fixed comparator networks sort any input in parallel, revealing depth vs comparators
Fire Spread Percolation
Forest fire model self-organizes to criticality; fire sizes follow a power law
Electrochemical Diffusion
Nernst-Planck ion transport: diffusion battles electromigration forming Debye layers
Barnsley Fern IFS
Four random affine transformations generate a perfect fractal fern attractor
Cayley Graph of a Group
Visualize group structure geometrically: nodes are elements, edges are generators
Solar Wind & Magnetosphere
Solar wind compresses Earth's magnetic field into a bow shock and magnetotail
Lie Algebra Commutator
SO(3) generators: two rotations fail to commute by exactly a third-axis rotation
Verlet Molecular Dynamics
Lennard-Jones particles crystallize, melt, and boil as temperature changes
Chua's Circuit Chaos
Simplest electronic circuit with chaos — double-scroll strange attractor in 3D
Ice Nucleation & Growth
Dendritic crystal growth with six-fold symmetry — snowflake shapes emerge from physics
Social Contagion Threshold
Watts cascade model: when do ideas spread globally vs. fizzle out locally?
Quantum Zeno Effect
Frequent measurement freezes quantum evolution — watching a qubit collapses it
Vortex Filament Reconnection
Two vortex rings stretch, approach, and reconnect — topology change in turbulence
Eigenface Decomposition
PCA reveals faces live on a low-dimensional manifold — reconstruct from eigenfaces
Rössler Attractor
Single-scroll strange attractor with period-doubling route to chaos visualized live
Lattice Boltzmann Fluid
D2Q9 fluid simulation — draw walls, watch Kármán vortex street form behind cylinders
Cosmic Inflation: Slow Roll
Inflaton field rolls down potential driving exponential cosmic expansion — 60 e-folds
Autocatalytic Origin of Life
Kauffman's model: catalytic closure emerges spontaneously above a critical threshold
Wave Function Collapse
Constraint propagation procedural generation — entropy-minimizing tile placement
Electron Orbital Shapes
Hydrogen wavefunctions |ψₙₗₘ|² — Laguerre polynomials and spherical harmonics
Geodesic Dome Generator
Buckminster Fuller's icosahedron subdivision — Euler's V−E+F=2 in 3D
Acoustic Resonance Modes
Standing wave eigenmodes of a 2D membrane — Chladni figures and drum harmonics
Protein Folding Lattice
HP model Monte Carlo — hydrophobic core formation on a self-avoiding lattice walk
Liquid Crystal Simulation
Nematic director field, Frank elastic energy, and topological ±½ disclinations
Spin Glass / Ising Model
Metropolis Monte Carlo near the Curie temperature — spontaneous symmetry breaking
Fourier Sound Synthesizer
Additive synthesis from harmonics — build sawtooth, square, and triangle waves
Neural Wave Propagation
Hodgkin-Huxley excitable media — action potentials, spiral waves, reentrant rhythms
Euler Fluid Simulation
Incompressible 2D flow — semi-Lagrangian advection and Helmholtz pressure projection
Galton Board Statistics
Binomial distribution emerging from random binary choices — Central Limit Theorem in action
Complex Domain Coloring
Visualizing complex functions — hue encodes phase, brightness encodes modulus
Neural Turing Machine Tape
Differentiable memory access — soft attention reads and writes an external tape
Cell Division — Mitosis
Chromosome condensation, spindle assembly, and cytokinesis animated through all five phases
Geodesic Shortest Path
Dijkstra's algorithm on weighted terrain — distance wavefronts propagating across the grid
Hilbert Curve Fractal
Space-filling curve visiting every point — locality-preserving fractal of dimension 2
Lorenz System Bifurcation
Strange attractor with tunable σ, ρ, β — sensitive dependence on initial conditions
Compton Scattering
Photon momentum transfer — wavelength shift Δλ = λ_c(1−cosθ) proves wave-particle duality
Earthquake Fault Slip
Stick-slip elastic rebound — tectonic stress builds until frictional threshold triggers rupture
Elastic Wave Reflection
P-wave reflection and refraction at interfaces — Snell's law, critical angle, and total internal reflection
Pendulum Wave Machine
15 pendulums of tuned lengths creating hypnotic interference patterns and wave superposition
Hénon Map Fractal Attractor
Strange attractor with Hausdorff dimension 1.261 from a deceptively simple 2D map
Virus Spread — SIR Model
Particle-based epidemic simulation with live S/I/R curves and R₀ calculation
Sierpiński Triangle Chaos Game
Random iterations that deterministically produce a fractal — IFS attractors visualized
Traffic Flow — Nagel-Schreckenberg
Cellular automaton showing phantom traffic jams emerging from random braking
Solar System N-Body Gravity
All 8 planets integrated with velocity-Verlet symplectic gravity using real masses and orbits
Spectral Gap & Graph Laplacian
Fiedler vectors, algebraic connectivity, and Cheeger inequalities on interactive graphs
Ising Model Percolation Clusters
2D Ising phase transition at T_c=2.269 with Metropolis and Wolff cluster algorithms
Foucault Pendulum & Coriolis
Earth's rotation makes pendulum planes precess — period 24h/sin(φ) depends on latitude
Maxwell's Demon & Entropy
Landauer's principle: sorting fast/slow molecules costs k_B T ln2 per bit of demon memory
Lenia Cellular Automaton
Continuous smooth life-like automaton with exotic self-organizing creatures emerging from convolution kernels
Schwarzschild Geodesics
Orbit a black hole — GR perihelion precession, ISCO instability, and photon sphere revealed
Navier-Stokes Vortex Street
Kármán vortex street behind a cylinder — Lattice-Boltzmann simulation of viscous flow at tunable Reynolds number
Forest Fire Percolation
Critical phase transition at p_c≈0.593 — power-law cluster sizes and fractal fire fronts near the percolation threshold
Neural Network Decision Boundary
Watch backpropagation carve complex boundaries through spiral, XOR, and moon datasets in real time
Hamiltonian Chaos & Poincaré Section
Hénon-Heiles system — KAM tori dissolve into chaos as energy crosses the 1/6 threshold
Runge-Kutta Integrator Comparison
Euler vs RK4 vs symplectic leapfrog — see energy drift accumulate and why symplectic methods preserve Hamiltonians
Landau Phase Transition
Free energy landscape bifurcates at Tc — tune between 1st and 2nd order transitions with b and c coefficients
Galaxy Collision Simulation
N-body gravitational encounter — tidal tails, bridges, and merger remnants from two colliding disk galaxies
Carnot Engine Cycle
Live P-V and T-S diagrams of the ideal Carnot cycle — efficiency η=1−Tc/Th is the thermodynamic maximum
Quantum Interference
Double-slit wave interference, single-photon buildup, and which-path measurement collapse
Torus Knot Visualizer
T(p,q) torus knots and links in 3D — trefoil, cinquefoil, and the topology of winding numbers
Mandelbrot–Julia Morph
Hover the Mandelbrot set to watch its Julia sets transform — connected sets vs. Cantor dust
Bernoulli Equation Flow
Venturi effect in a pipe: pressure drops where velocity rises — manometers, particles, and continuity
Strange Attractors
Lorenz, Rossler, Thomas, Halvorsen, Chen, and Dadras attractors — drag to explore chaotic phase space
Polyhedra Dual Explorer
Platonic solids and their duals — morph between cube and octahedron, dodecahedron and icosahedron
Atmospheric Lapse Rate
Dry and moist adiabatic lapse rates, ISA standard atmosphere, LCL cloud base, and atmospheric stability
Random Walk Dimensions
1D, 2D, and 3D random walks — Polya's recurrence theorem and mean-square displacement scaling
Geodesic Flow on Surfaces
Geodesics on sphere, torus, saddle, and cylinder — Christoffel symbols and curvature made visual
Logistic Map Orbit Diagram
Full bifurcation diagram of x to rx(1-x) — period-doubling, Feigenbaum constant, chaos onset
Turing & the Halting Problem
Step through Turing machines including the Busy Beaver — visualizing undecidability and diagonal arguments
Crystal Lattice Phonons
Acoustic and optical phonon modes in a 1D crystal — dispersion relation ω(k) and quantized vibrations
Schrödinger's Cat Superposition
Bloch sphere, density matrix, and decoherence — quantum measurement collapse visualized in real time
Sierpinski Carpet 3D
Fractal extrusion with Hausdorff dimension log(8)/log(3) ≈ 1.893 — drag to rotate the 3D structure
2D Heat Diffusion
Paint heat sources and cold sinks, watch Fourier's equation ∂T/∂t = α∇²T evolve in real time
Orbital Resonance
Two-body mean-motion resonance via Kepler's third law — Laplace resonance, Kirkwood gaps, Lissajous patterns
Nernst Equation
Electrochemical cell potential vs concentration — E = E° − (RT/nF)ln(Q) with animated half-cell visualization
Stress & Strain Tensors
Mohr's circle, principal stresses, Hooke's law, and von Mises yield criterion for 2D plane stress
Complex Network Percolation
Bond percolation phase transition on square, triangular, Erdős-Rényi, and scale-free networks
Standing Wave Harmonics
Superpose harmonics on a fixed string — Fourier decomposition, nodes, antinodes, and spectrum visualization
Topological Phase Transitions
BKT vortex-antivortex unbinding, winding numbers, and the transition from quasi-long-range order to disorder
Quantum Dots Energy Levels
Particle-in-a-spherical-box energy levels and wavefunctions — Brus equation, size-tunable quantum confinement
Entropy Maximization
MaxEnt principle: Gaussian, exponential, Laplace distributions as least-biased solutions to constrained entropy problems
Self-Organized Criticality Sandpile
Bak-Tang-Wiesenfeld abelian sandpile with power-law avalanche distribution and emergent criticality
Blackbody Radiation – Planck
Planck's law vs Rayleigh-Jeans catastrophe — Wien's displacement, Stefan-Boltzmann, and visible color of hot objects
Spectral Graph Clustering
Graph Laplacian eigenvectors reveal community structure — Fiedler vector, spectral gap, and Cheeger inequality
Arctic Circle Theorem
Random domino tilings of the Aztec diamond — frozen corners, temperate chaos zone, and the inscribed arctic circle
Contact Angle & Wetting
Young-Dupré equation, spreading coefficient, Wenzel roughness — hydrophilic to superhydrophobic droplet physics
Symplectic Integrators
Euler vs RK4 vs Verlet on Kepler orbits — energy drift in non-symplectic vs bounded oscillation in symplectic methods
Mode Locking & Arnold Tongues
Circle map dynamics, winding number locking to rationals, and the Arnold tongue structure in parameter space
Genetic Drift
Wright-Fisher model showing allele fixation and loss — smaller populations drift faster, visualizing Kimura's neutral theory
Phononic Band Gap
Dispersion relation for a 1D diatomic lattice — acoustic vs optical branches, band gap opening, and animated atomic motion
Diffusion Green's Function
Fundamental solution of the heat equation — Gaussian spreading with σ~√(Dt), superposed point sources, and a heatmap
Taylor Series Convergence
Polynomial approximations to sin, exp, ln and arctan — radius of convergence, Gibbs-like overshoots, interactive term count
Topological Defects in Nematics
Half-integer and integer winding number defects in liquid crystals — place ±½ and ±1 charges, watch director field respond
Drude Model of Conductivity
Classical electron transport — electrons scatter off ion cores with mean free time τ, visualizing drift and AC conductivity σ(ω)
Morse Potential
Anharmonic diatomic molecular vibrations — exact quantum energy levels, turning points, dissociation, and bond animation
Brusselator Oscillator
Autocatalytic chemical oscillations — Hopf bifurcation at B=1+A², limit cycles in phase space, Belousov-Zhabotinsky analog
Optical Fiber Modes
LP modes in step-index fiber — Bessel function field distributions, V-number, single-mode condition, and mode cutoff
Quantum Walk 1D
Discrete-time quantum walk with Hadamard coin — ballistic σ~t/√2 spreading vs classical √t, interference patterns and spacetime diagram
Josephson Junction
AC/DC Josephson effects and phase dynamics in superconducting junctions via RCSJ model
Ecological Succession
Probabilistic cellular automaton of pioneer to climax forest succession with disturbance
Mersenne Prime Sieve
Lucas-Lehmer primality test visualized across exponents — hunting extraordinarily rare primes
Liquid Drop Nuclear Model
Bethe-Weizsäcker mass formula: volume, surface, Coulomb, asymmetry, and pairing terms
Soap Film & Plateau's Rules
Minimal surfaces and Steiner trees — all junctions meet at 120° to minimize total length
Cyclotron Motion in Plasma
Larmor gyration, E×B drift, and charged particle trajectories in magnetic fields
Birth-Death & Coalescent
Kingman coalescent trees and stochastic birth-death phylogenies with phase transitions
Lattice Gauge Theory U(1)
Compact QED on a 2D lattice — Wilson action, plaquettes, and Metropolis Monte Carlo
Gravitational Redshift
Schwarzschild metric and photon redshift near neutron stars, black holes, and the Sun
Opinion Dynamics: Deffuant
Bounded confidence model — how society fragments into clusters when ε < 0.5
Credit Risk Network
Cascade failure animation through correlated financial exposures — watch systemic risk propagate firm-to-firm.
Mandelbrot & Julia Connection
Click the Mandelbrot set to pick c; the corresponding Julia set renders live — connected vs. Cantor dust.
Entropy & the Arrow of Time
Gas particles spread from order to disorder. Flip all velocities and watch entropy briefly reverse.
Boolean Network Dynamics
Kauffman NK random Boolean networks settle into periodic attractors — the mathematics of biological order.
Lévy Flight Explorer
Heavy-tailed step distributions produce dramatic long jumps. Tune α from Brownian motion to Cauchy chaos.
Simulated Annealing — TSP
Watch the travelling salesman route evolve as temperature cools and the algorithm escapes local minima.
Voter Model Dynamics
Opinion dynamics on a 2D grid: agents copy neighbors until consensus emerges. Compare update rules.
Random Walk Absorption
1D and 2D random walks with absorbing boundaries — build the hitting-time distribution over many trials.
Watts-Strogatz Network
Rewire a ring lattice to find the small-world sweet spot: high clustering and short path lengths.
Zipf's Law Explorer
Paste any text to see rank-frequency on a log-log plot — the universal power law of natural language.
Hopf Bifurcation Explorer
Watch a stable fixed point transform into a limit cycle as μ crosses zero — phase portrait with live trajectory tracing.
Runge-Kutta Visualizer
Compare Euler, RK2, and RK4 integration on classic ODEs — step size slider reveals how error accumulates.
Automata Rule Space
Wolfram's 256 elementary cellular automata rules — toggle bits, explore Classes I–IV from uniformity to chaos.
Turing Pattern Generator
Gray-Scott reaction-diffusion — tune feed and kill rates to grow spots, stripes, spirals, and coral from random noise.
Predator-Prey Phase Portrait
Lotka-Volterra dynamics with nullclines and period measurement — click to launch orbits from any initial condition.
Knot Crossing Number
3D perspective projection of classical knots — drag to rotate trefoil, figure-eight, torus knots with Jones polynomial info.
Airy Function Explorer
Plot Ai(x) and Bi(x) with WKB approximation overlay and zeros marked — connection to quantum tunneling at linear potentials.
Convex Hull Animation
Add points by clicking, then animate Graham scan, Jarvis march, or QuickHull step by step with hull area displayed.
Fibonacci Spiral & Phyllotaxis
Golden angle sunflower with n-seed slider — perturb the divergence angle to see why 137.51° is the optimal packing angle.
Hodge Decomposition
Decompose any 2D vector field into curl-free, divergence-free, and harmonic components — draw fields or choose presets.
Geodesic Dome Explorer
Subdivide an icosahedron into geodesic spheres at frequencies 1–5. Rotate in 3D, toggle faces/edges/vertices, verify Euler's formula V−E+F=2.
Spectral Clustering Visualizer
Build a graph Laplacian from 2D point clouds, compute Fiedler eigenvectors via power iteration, and watch clusters emerge in spectral embedding space.
Crystallization Simulation
Lennard-Jones molecular dynamics: cool particles from liquid to crystal, watch grain boundaries form, and track the local order parameter in real time.
Boids Obstacle Course
Classic Reynolds flocking with click-to-place obstacles, tunable separation/alignment/cohesion, and a predator that scatters the flock on demand.
Language Tree Branching
Explore the Indo-European phylogenetic tree: click branches to expand, read about Grimm's Law, laryngeals, and Proto-Indo-European reconstructions.
Bell Inequality Visualizer
Adjust detector angles and watch the CHSH parameter cross the classical bound of 2 into quantum territory — up to the Tsirelson bound of 2√2.
Josephson Junction I-V Curve
RCSJ model simulation: tune bias current and McCumber parameter to see supercurrent, hysteresis, and voltage-state phase dynamics.
Lissajous 3D Spirograph
Parametric curves x=sin(at+d), y=sin(bt), z=sin(ct) traced in 3D with fading trails. Tune a:b:c ratios to reveal knot topologies and closed orbits.
Polyhedron Dual Explorer
Animate morphing between dual pairs — cube-octahedron, tetra-tetra, dodecahedron-icosahedron — while verifying Euler's formula for each.
Chebyshev Polynomial Approx
Approximate |x|, step functions, Runge's function, and more with N Chebyshev terms. Compare to Taylor series and see the minimax error advantage.
Spring-Mass Network
Drag nodes to excite normal modes in a spring-coupled mass network. Watch energy propagate and dissipate.
Asteroid Orbital Resonance
Jupiter's gravity clears Kirkwood gaps at mean-motion resonances. Watch the asteroid belt histogram develop.
Dendrite Growth Simulation
Diffusion-limited aggregation with anisotropy. Tune crystal axes to grow snowflake-like fractal dendrites.
River Network Erosion
Stream power law erodes random terrain into branching river networks. View flow accumulation and watersheds.
Synapse Plasticity Model
STDP learning rule: weight change depends on spike timing. Explore BCM and Hebbian rules with live traces.
Opinion Cascade Threshold
Watts threshold model: seed adopters and watch global cascades spread through social network topology.
Cantor Set Explorer
Animated Cantor set construction with fat variants and 2D Cantor dust. Measure decays; dimension ≈ 0.631.
Penrose Tiling Generator
P2 Penrose tiling via deflation substitution. Zoom into perfect 5-fold symmetry with no translational period.
Rayleigh-Bénard Convection
Fluid heated from below forms convection rolls above Ra≈1708. Track Nusselt number as rolls develop.
Epidemic Threshold Network
SIR model on ER, BA, and ring networks. R₀ = β⟨k⟩/γ governs epidemic takeoff. Visualize the phase transition.
Streamlines
Interactive 2D vector field visualizer. Choose dipole, vortex, saddle, or source fields. Drop tracers and watch them flow.
Galaxy Collision Simulator
Two-galaxy N-body collision with tidal tail formation. Stars colored by origin galaxy. Adjust mass ratio and impact parameter.
Möbius Strip Geodesic
3D Möbius strip with an ant walking a geodesic path. Watch the normal vector reverse — proving non-orientability.
Laplacian Growth / DLA
Dielectric breakdown model. Growth probability ∝ |∇φ|^η. Tune η from random DLA to sharp dendritic crystals.
Persistence Diagram
Topological Data Analysis: Vietoris-Rips filtration on point clouds. Birth-death pairs reveal loops and components.
Absorbing Markov Chains
Gambler's ruin and absorbing states visualized. Compute fundamental matrix N=(I−Q)⁻¹ and expected absorption times.
Quantum Tunneling
Gaussian wavepacket hits a potential barrier. Split-operator Schrödinger solver shows transmission and reflection probabilities.
Hopfield Network Attractors
Store patterns as neural attractors. Corrupt a memory and watch the network retrieve it. Energy landscape descends monotonically.
Origami Folding
Miura-ori fold animation with rigid-foldability. Drag to rotate the 3D surface. Kawasaki's theorem governs every vertex.
Population Bottleneck
Wright-Fisher genetic drift model. Trigger a bottleneck to watch diversity collapse. Compare drift vs. selection coefficient.
SPH 2D Fluid Simulation
Smoothed Particle Hydrodynamics fluid sim with gravity, viscosity, surface tension, and pressure color coding.
Knot Invariants Explorer
Explore trefoil, figure-8, and torus knots with crossing number, Alexander polynomial, Jones polynomial, and Reidemeister moves.
Dragon Curve Fractal
Animated dragon curve construction up to iteration 16 via paper-folding sequences, with zoom, pan, and color modes.
Voting Power: Shapley & Banzhaf
Shapley-Shubik and Banzhaf power indices for weighted voting games. Power diverges dramatically from weight.
Minimax Game Tree
Interactive minimax tree with alpha-beta pruning animation. Watch branches get cut and the optimal path highlighted.
Zero-Sum Game Matrix
Edit payoff matrices, find Nash equilibria (pure or mixed), visualize saddle points and best-response correspondences.
Bayesian Update Visualizer
Watch Beta posterior distributions update in real time as you add observations. Credible intervals and conjugate families.
Neural ODE Phase Portrait
Train a neural network to learn vector field dynamics. Compare learned vs target phase portraits for spirals, limit cycles, and more.
Classical Scattering Cross Section
Particle scattering off Coulomb, hard-sphere, Yukawa, and Lennard-Jones potentials. Deflection angle θ(b) and dσ/dΩ polar plot.
Thermodynamic Cycle P-V Diagram
Interactive P-V diagram with draggable state points. Carnot, Otto, Diesel, Stirling, Brayton, Rankine presets. Compute work and efficiency.
Crystal Lattice Phonons
2D spring-coupled crystal with acoustic and optical phonon modes. Excite wave modes, adjust temperature, and visualize the dispersion relation.
Protein Folding — HP Model
Hydrophobic-Polar lattice model of protein folding. Monte Carlo search finds native state by maximizing hydrophobic contacts.
Sandpile Self-Organized Criticality
Abelian sandpile model. Add grains one at a time and watch power-law avalanches emerge spontaneously at the critical state.
Cellular Potts Model
Glazier-Graner-Hogeweg model of cell sorting. Differential adhesion drives A and B cells to segregate via Metropolis dynamics.
Random Sequential Adsorption
Irreversible adsorption of disks, dimers, needles, and trimers. Coverage saturates at shape-dependent jamming limits.
Frustrated Magnet — Triangular Lattice
Antiferromagnetic Ising on a triangular lattice. Geometric frustration prevents long-range order — compare with the unfrustrated square lattice.
EM Algorithm — Gaussian Mixtures
Click to place data points, then watch Expectation-Maximization iteratively fit Gaussian mixture models with animated E and M steps.
Game of Life Variants
Conway's Life plus HighLife, Seeds, Brian's Brain, Day & Night, and Morley. Draw patterns, load gliders and oscillators, compare rules.
2D Ising Model — Monte Carlo
Ferromagnetic Ising model with Metropolis and Wolff cluster algorithms. Temperature slider crosses Tc ≈ 2.269. Watch the phase transition.
Continued Fraction Clock
Live clock displaying current time as continued fractions. Convergents shown as best rational approximations. Compare φ, π, e, and √2.
Voronoi Crystal Growth
Crystal nucleation and Voronoi cell growth simulation
Schrödinger Wave Packet
Quantum wave packet propagation in a potential well
Kuramoto Oscillators
Coupled neural oscillators synchronizing via Kuramoto model
Archimedean Spiral Waves
Rotating spiral waves on a 2D excitable medium
Runge-Kutta Orbits
Orbital mechanics solved with RK4 integration
Mandelbrot ↔ Julia
Click the Mandelbrot set to reveal its Julia set
Harmonic Oscillator Phase
Phase portraits and Lissajous figures of oscillators
Hypocycloid Spirograph
Spirograph patterns from adjustable gear ratios
Coriolis Projectile
Projectile deflection due to Earth's rotation
Blackbody Radiation
Wien's law and Planck curves across temperatures
Fluid Dam Break
SPH simulation of dam break and fluid spreading
Harmonic Series & Music
Additive synthesis building waveforms from overtones
Wireworld CA
Electrons flowing through editable wire logic circuits
Hamiltonian Billiards
Billiard trajectories showing chaos vs. integrability
Cantor Set Dust
Fractal Cantor set construction and dimension
Maxwell Wave (FDTD)
2D electromagnetic wave simulation with FDTD method
Crystal Dislocations
Edge and screw dislocations with Burgers vector
Airy Diffraction
Airy disk and Rayleigh criterion for telescope resolution
Lorenz Climate Chaos
Butterfly effect and sensitive dependence in climate model
Elastic Wave Medium
P-waves and S-waves propagating through elastic medium
Topological Band Gap
SSH model topology, edge states, and band inversion
Hexagonal Phonons
Phonon dispersion on honeycomb lattice with normal modes
Polymer Random Walk
Self-avoiding walks and polymer conformation statistics
Fractal Coastline
Richardson effect — how coastline length depends on ruler size
Lévy Flight Foraging
Optimal foraging via Lévy flights vs. Brownian motion
Rayleigh-Bénard Onset
Convection cell formation at critical Rayleigh number
Penrose Inflation
Penrose tiling via substitution inflation rules
Duffing Chaos
Duffing oscillator strange attractor and Poincaré section
Langton's Ant Variants
Multi-color ant rules and emergent highway patterns
Bifurcation Normal Forms
Saddle-node, pitchfork, Hopf — interactive bifurcation explorer
Phase-Field Solidification
Dendrite growth and solidification via Allen-Cahn model
Active Matter Motility
Self-propelled particles and motility-induced phase separation
Random Graph Percolation
Giant component emergence in Erdős-Rényi random graphs
Chemical Potential Gradient
Diffusion driven by chemical potential and Fick's law
Magnetic Skyrmion
Topological spin textures with DM interaction
Chaos Game Fractals
IFS chaos game generating Sierpiński triangle and variants
Topological Defect Annihilation
XY model vortex-antivortex pair creation and annihilation
Rossby Waves
Atmospheric Rossby waves on beta-plane, westward propagation
Geodesic Flow on Torus
Geodesics on torus — dense windings vs. closed orbits
Nonequilibrium Fluctuations
Jarzynski equality and work distributions far from equilibrium
Tidal Locking Resonance
Spin-orbit resonance and tidal locking capture dynamics
Gradient Descent Optimizers
SGD, Adam, momentum racing across loss landscapes
FitzHugh-Nagumo Excitability
Excitable cell model, action potentials, and spiral waves
Kolmogorov Cascade
Turbulent energy cascade and k⁻⁵/³ spectrum
Hyperbolic Space Navigation
Navigate the Poincaré disk and trace hyperbolic geodesics
Diffusion Map Manifold
Diffusion maps unfolding Swiss roll and torus manifolds
Floquet Driven Quantum
Periodically driven qubit and Floquet quasienergies
Euler Column Buckling
Critical load, buckling modes, and post-buckling shapes
Network Motif Dynamics
Feed-forward loops and network motif signal processing
Newton Fractal Basins
Newton's method fractal basins of attraction
Viscous Fingering
Hele-Shaw cell Saffman-Taylor fingering instability
Muscle Cross-Bridge Cycle
Huxley model of actin-myosin force generation
Thermoacoustic Engine
Heat-driven acoustic oscillation and Rijke tube waves
Rigid Origami Folding
Miura-ori and Yoshimura rigid fold kinematics
Network SEIR Epidemic
Epidemic spreading on heterogeneous contact networks
Two-Stream Instability
Plasma electron beam instability and Langmuir waves
Fibonacci Quasicrystal
Quasicrystal diffraction and 10-fold symmetry
Ant Colony Pheromones
Pheromone reinforcement and shortest path emergence
Takens Attractor Reconstruction
Delay embedding and attractor reconstruction from time series
Minimal Surface Catenoid
Catenoid and minimal surfaces with zero mean curvature
Topological Surface States
Dirac cone and bulk-boundary correspondence in topological insulators
Stochastic Block Model
Community structure detection in random block graphs
Magnetic Hysteresis Loop
B-H curve, coercivity, and Preisach hysteresis model
Rich-Club Networks
Rich-club phenomenon and degree correlations in networks
Wavelet Transform
Continuous wavelet transform and signal scalogram
Bounded Confidence Opinions
Opinion cluster formation with bounded confidence dynamics
Kelvin-Helmholtz Instability
Shear instability and vortex roll-up at fluid interfaces
Dielectric Breakdown
Fractal lightning patterns from dielectric breakdown model
Random Matrix Theory
Wigner semicircle law and GOE/GUE level spacing statistics
Flocking & Murmuration
Reynolds rules: alignment, cohesion, separation in flocks
Negative Index Metamaterial
Negative refraction, Snell's law reversal, and perfect lens
Porous Media Flow
Darcy vs. Forchheimer flow regimes in porous media
Allometric Scaling
Kleiber's ¾ law and fractal vascular scaling in biology
Boolean Network Attractors
Kauffman NK networks and attractor landscape dynamics
Dissipative Solitons
Localized structures sustained by gain-loss balance in CGLE
Network Controllability
Minimum driver nodes for structural network controllability
Crystal Field Splitting
d-orbital splitting and transition metal colors
Chaos Synchronization
Pecora-Carroll synchronization of coupled chaotic systems
Excitable Media Spirals
Spiral wave tip meander in FHN excitable medium
Weisfeiler-Leman Test
Graph isomorphism via color refinement algorithm
Kondo Effect
Magnetic impurity screening and resistance minimum
Turing Morphogenesis
Activator-inhibitor stripe and spot pattern formation
OGY Chaos Control
Stabilizing unstable periodic orbits in chaotic systems
Fiber Bundle Topology
Möbius band, Hopf fibration, and fiber bundle structure
Stochastic Petri Nets
Token firing and stochastic reaction network simulation
Phononic Crystal Bandgap
Acoustic wave propagation through phononic crystal
Life: Glider Guns
Conway's Life with Gosper glider gun and spaceship library
Quantum Discord
Quantum discord vs. entanglement in two-qubit states
Kerr Nonlinear Optics
Self-phase modulation and optical soliton formation
Elementary CA Spacetime
Wolfram's 256 elementary CA rules and spacetime diagrams
Reaction Telegraph Wave
Wave-diffusion transition via the telegraph equation
Fluid Vortex Sheet
Kelvin-Helmholtz instability rolling up at a shear layer
Stochastic Resonance
Noise enhances weak signal detection in neurons
Fractal Mountain Erosion
Diamond-square terrain carved by simulated water flow
Magnetic Pendulum Basins
Fractal attraction basins above three magnets
Rayleigh-Bénard Convection
Thermal convection rolls from bottom heating
Activator-Inhibitor Patterns
Turing's reaction-diffusion generates spots and stripes
Quantum Billiards
Chaotic vs integrable trajectories in 2D billiard tables
Equatorial Kelvin Wave
Ocean wave trapped at the equator, propagating eastward
Saffman-Taylor Fingering
Viscous fingering instability in a Hele-Shaw cell
Lorenz Water Wheel
Malkus waterwheel — physical realization of chaos
Quantum Harmonic Oscillator
Coherent state wave packets in a parabolic potential
Crystal Growth (DLA)
Fractal crystals formed by diffusion-limited aggregation
Burgers Equation Shock
Shock wave formation and viscous smoothing
Cardiac Spiral Wave
FitzHugh-Nagumo spiral waves — cardiac arrhythmia model
Wave Interference
2D wave equation with multiple point sources
Collatz Sequences
The 3n+1 conjecture — all roads lead to 1
Prime Sieve Animated
Sieve of Eratosthenes finding primes in real time
Binary Star & Lagrange Points
Restricted 3-body problem with L1–L5 equilibria
Plasma Drift Wave
Drift wave instability in magnetized plasma
Geodesics on Curved Surfaces
Shortest paths on sphere, torus, and saddle — bent by curvature
Pendulum Wave Machine
15 pendulums creating mesmerizing collective wave patterns
Nematic Liquid Crystal Defects
Topological disclinations annihilating in a liquid crystal
Ising Renormalization Group
Block spin decimation and RG flow near the critical point
Voronoi Lloyd Relaxation
Centroidal Voronoi tessellation via iterative Lloyd's algorithm
Lenia
Continuous cellular automaton with self-organizing creatures
Airy Function Waves
Ai(x) and Bi(x) — oscillation meets exponential decay at a turning point
Morse Potential
Anharmonic molecular bond with quantized vibrational levels
Logistic Map Cobweb
Period doubling cascade to chaos — live cobweb and bifurcation diagram
Hopf Bifurcation
Fixed point → limit cycle transition with phase portrait
Geodesic Dome & Polyhedra
Platonic solids and geodesic sphere subdivisions — Euler's V-E+F=2
Plateau's Minimal Surfaces
Soap films relaxing to zero mean curvature
Wigner Level Spacing
Random matrix eigenvalue repulsion — the Wigner surmise
Joukowski Airfoil
Conformal mapping transforms circles into airfoil shapes
Elastic Wave Dispersion
P-waves, S-waves, and dispersive water wave packets
Polymer Conformations
Self-avoiding random chains and Flory scaling R ~ N^ν
Epidemic on a Network
SIR disease spreading through random and scale-free networks
Geodesic Flow on a Torus
Rational vs irrational winding numbers — periodic vs dense orbits
Drum Eigenmodes
Laplacian eigenfunctions — can you hear the shape of a drum?
Johnson-Lindenstrauss Lemma
Random projections preserve pairwise distances
Lattice Boltzmann Flow
D2Q9 fluid simulation with vortex shedding behind a cylinder
Frobenius Coin Problem
Which amounts can be made with given coin denominations?
Symplectic Integrators
Leapfrog vs Euler — energy conservation over long simulations
Boolean Network Attractors
Kauffman RBN attractors and basins of attraction
Spectral Clustering
Graph Laplacian eigenvectors reveal non-convex clusters
Continued Fractions & Farey
Stern-Brocot tree and rational approximations
Wavelet Transform
Scalogram reveals time-frequency structure invisible to Fourier
Lid-Driven Cavity Flow
Navier-Stokes benchmark — recirculation zones at increasing Re
Metabolic Flux Balance
FBA on a toy metabolic network — visualizing reaction fluxes
Graph Isomorphism
Animated bijection mapping between isomorphic graphs
SSH Topological Band Structure
Edge states emerge when SSH chain enters topological phase
Epidemic Percolation
SIR disease spreading as bond percolation near p_c
Spring-Mass Normal Modes
Standing wave eigenmodes of a coupled oscillator chain
Algebraic Curves
Implicit polynomial curves — elliptics, lemniscates, and more
Maxwell Construction
Equal-area rule resolves the van der Waals S-curve
Elementary Cellular Automata
All 256 Wolfram rules — from chaos to Turing completeness
Mean-Field Phase Transition
Self-consistency equation and bifurcation at T_c
Hénon Map Attractor
Strange attractor with fractal Cantor-like cross-sections
Fokker-Planck Diffusion
Probability density evolution in a double-well potential
Fisher Information Geometry
Statistical manifold of Gaussians and geodesics in (μ,σ) space
Euler Column Buckling
Post-buckling shapes and critical load bifurcation
Schnakenberg Turing Patterns
Spots and stripes self-organizing from reaction-diffusion
Newton Fractal Basins
Fractal boundaries between Newton's method convergence zones
Quantum Eraser
Which-path information destroys — and erasure restores — interference
Critical Opalescence
Diverging correlations near T_c scatter light at all wavelengths
Ising Wolff Clusters
Cluster flip algorithm reveals critical fluctuations at T_c
Feynman Path Integral
Quantum amplitudes sum over all paths — classical emerges from stationary phase
Vortex Reconnection
Vortex tubes reconnect, changing topology and helicity
Green-Kubo Diffusion
Velocity autocorrelation integrates to give the diffusion coefficient
Mathieu Parametric Resonance
Ince-Strutt stability diagram — resonance tongues in parameter space
Catenary & Calculus of Variations
Hanging chain minimizes potential energy — Euler-Lagrange gives the catenary
Kac-Rice Zero Crossings
Expected zeros of a random function from spectral moments
Wigner Semicircle Law
Eigenvalue density of large random matrices converges to a semicircle
Modular Arithmetic Patterns
Multiplication tables mod n create stunning chord diagram patterns
KdV Soliton Collision
Two solitons pass through each other elastically — exact 2-soliton solution
Heat Kernel Diffusion
Heat spreading from a point on different geometries
Kuramoto Synchronization
Phase oscillators spontaneously synchronize above a coupling threshold
Lévy Flights
Power-law jump lengths create clustering with rare giant leaps
Liouville's Theorem
Phase-space volume is conserved — ensemble density stays constant
Quantum Chaos Stadium
Scarred eigenstates and Wigner-Dyson statistics in the stadium billiard
Sinai Billiard (Dispersing)
Circular obstacles create ergodic chaos — trajectory fills the table
Ornstein-Uhlenbeck Process
Mean-reverting noise — the stationary Gaussian stochastic process
Geometric Brownian Motion
Log-normal stock price model — foundation of Black-Scholes
Euler Characteristic χ
V−E+F stays constant under deformation — topology in action
Continued Fraction Convergents
Best rational approximations to π, φ, √2 — staircase of convergents
Two-Stream Instability
Counter-propagating electron beams form unstable plasma modes
Duffing Oscillator Chaos
Strange attractor and Poincaré sections in a driven anharmonic oscillator
Eigenvalue Power Iteration
Animated convergence of power method and QR algorithm
Monodromy & Floquet Theory
Floquet multipliers determine stability of periodic orbits
Percolation Cluster Hull
Fractal frontier of the spanning cluster — dimension d_f = 7/4
Strange Attractor Circuit
Lorenz, Rössler, and Thomas attractors as electronic circuits in phase space
Symplectic Pendulum Phase Space
Pendulum phase portrait with Liouville area preservation and Hamiltonian orbits
Arnold Tongues
Frequency locking and the devil's staircase in the driven circle map
Ising Critical Exponents
2D Ising model near T_c with Metropolis algorithm and real-time susceptibility
Hénon Map Bifurcation
Strange attractor and bifurcation diagram of the area-contracting Hénon map
Wavelet Multiresolution
Haar and Daubechies DWT decomposition with scalogram and denoising threshold
Topological Defects & Vortices
Vortex-antivortex pairs in the 2D XY model and the BKT phase transition
Stochastic Resonance
Noise enhances weak signal detection — SNR peaks at optimal noise amplitude
Rayleigh-Bénard Convection
Thermal convection cells emerge above Ra_c = 1708 with Nusselt number tracking
Poincaré Section & KAM Tori
Standard map chaos onset — KAM tori dissolve as the Chirikov parameter grows
Box-Counting Fractal Dimension
Measure fractal dimension via log N(ε) / log(1/ε) — Koch, Sierpinski, coastlines visualized
Logistic Map & Feigenbaum Universality
Period-doubling cascade xₙ₊₁=rxₙ(1−xₙ), Feigenbaum constant δ≈4.6692 in the bifurcation diagram
Hopf Bifurcation Normal Form
Limit cycles born at μ=0 — stable spirals become oscillators in the supercritical Hopf bifurcation
Viscous Fingering — Saffman-Taylor
DLA-style diffusion-limited growth models fingering instability when low-viscosity fluid invades high-viscosity medium
Sandpile Avalanches — SOC
Bak-Tang-Wiesenfeld model self-organizes to criticality — avalanche sizes follow a power law with no tuning
Voronoi — Fortune's Sweep Line
Animated O(n log n) Voronoi construction via parabolic beach line, with Delaunay triangulation dual
Chua's Double-Scroll Attractor
Chua's circuit — the simplest chaotic electronic circuit — traces a double-scroll strange attractor in 3D
Perlin Noise Terrain Generator
Fractal Brownian Motion via layered Perlin octaves generates infinite procedural mountain landscapes
Quantum Harmonic Ladder Operators
Eigenstates ψₙ(x) with Hermite polynomials, a± ladder operators, and coherent state oscillation
Sierpinski Gasket — Chaos Game
IFS chaos game builds Sierpinski's fractal from random vertex jumps — colored by orbit history
Kelvin–Helmholtz Vortex Sheet
Birkhoff–Rott vortex sheet roll-up — tangential shear drives the Kelvin-Helmholtz instability into spiraling vortices
Geodesic Sphere Subdivision
Icosahedron subdivided into geodesic spheres at frequency k — 3D rotation, wireframe and solid, 20·4^k faces
Nonequilibrium Entropy Production
Driven Brownian particles via Langevin dynamics — Jarzynski equality, Crooks theorem, and work distribution
SIR Epidemic Spreading Threshold
SIR model on Erdős–Rényi, scale-free, and lattice networks — R₀ threshold, herd immunity, and scale-free divergence
Fourier Series & Gibbs Phenomenon
Square, sawtooth, and triangle wave approximations — the ~8.9% Gibbs overshoot that never disappears
Polymer Chain — Self-Avoiding Walk
SAW polymer model measuring Flory exponent ν≈0.588 in 3D and ν=3/4 in 2D from end-to-end distance scaling
Riemann Zeta — Critical Line
ζ(½+it) as a complex spiral — non-trivial zeros, partial sum approximation, and the Riemann Hypothesis visualized
Complex Newton Fractal
Newton's method basin boundaries for degree-n polynomials — fractal Julia set frontiers, damping, and custom roots
3-Color Totalistic Cellular Automaton
Wolfram's 3-color outer totalistic CA — 2187 rules, spacetime diagrams, rule 357 and emergent complexity
Spectral Clustering Demo
Graph Laplacian eigenvectors partition point clouds — Fiedler vector, algebraic connectivity, and k-NN similarity graphs
Orbital Mechanics — Kepler's Laws
N-body Keplerian orbits with conic sections, perihelion/aphelion markers, vis-viva equation, and Kepler's three laws visualized live
Fluid Vorticity & Streamlines
2D potential flow around a cylinder — streamlines, vorticity field, Magnus effect, and Kutta-Joukowski lift theorem
Monte Carlo π & Lattice Field
Monte Carlo π estimation with convergence tracking alongside a 2D Ising-like lattice updated by the Metropolis algorithm
4D Hypercube Projection
Tesseract rotating in all six 4D planes — projected to 2D with perspective, color-coded by fourth coordinate
Brownian Motion & Diffusion
Free, confined, anomalous, and superdiffusion regimes — MSD ∝ tᵅ fit, diffusion coefficient, and log-log convergence plot
Magnetic Field Visualization
Dipole, current loop, solenoid, and wire field lines with color-mapped |B| intensity and interactive pole placement
Waveguide Modes & Resonance
TE and TM modes in rectangular waveguides — 2D field patterns, cutoff frequency, and dispersion relation ω(k)
Lotka-Volterra Phase Plane
Predator-prey dynamics with full phase portrait, nullclines, limit cycles, RK4 integration, and stochastic noise toggle
Gradient Descent Loss Landscape
SGD, Momentum, and Adam racing across Rosenbrock, Himmelblau, Ackley, and Rastrigin landscapes simultaneously
Shannon Entropy & Information Channel
H(X), mutual information I(X;Y), channel capacity for BSC, BEC, and AWGN channels — live capacity-noise curves
Spacetime Diagram
Minkowski spacetime, worldlines, light cones, Lorentz boosts with time dilation and length contraction readouts
Crystallographic Symmetry
2D Bravais lattices, point groups, and wallpaper group animations — rotation axes, mirrors, and glide reflections
Heisenberg Spin Chain
1D quantum spin model with nearest-neighbor coupling, magnon dispersion, and spin correlation functions
Atmospheric Rossby Waves
β-plane planetary waves, dispersion relation, phase vs group velocity, and simulated weather system patterns
Cosmic Web & Voids
Zel'dovich approximation and adhesion model — dark matter sheet collapse, caustics, and large-scale void hierarchy
Ising & Percolation Crossing
2D Ising model at T_c with FK random cluster visualization, Cardy crossing formula, and critical cluster statistics
Game of Life Glider Gun
Conway's Life with Gosper glider gun, selectable patterns, draw mode, and live population chart
Neural Field Bump Attractor
Wilson-Cowan neural field model with bump attractor dynamics — working memory and input-driven spatial tuning
Quantum Chaos: Stadium Billiard
Bunimovich stadium ray tracing, Poincaré sections, and Wigner-Dyson vs Poisson level spacing statistics
Lyapunov Stability & Basins
Nonlinear 2D systems, basins of attraction, Lyapunov exponent fields, nullclines, and interactive trajectory launching
Mode Locking & Circle Map
Arnold tongue diagram, devil's staircase winding number, and quasiperiodic-to-locked transition via the circle map
Discrete Quantum Walk
1D quantum walk with Hadamard/Grover coins, ballistic spreading vs classical diffusion, and interference patterns
Kauffman NK Boolean Network
Random Boolean networks, attractor landscapes, and criticality at K=2 — the edge of chaos
Plateau–Rayleigh Instability
Fluid cylinder breakup driven by surface tension, fastest-growing wavelength λ_max=2π√2·r, and droplet formation
Heat Equation & Heat Kernel
Gaussian heat kernel diffusion, multiple initial conditions, boundary types, and 1D/2D visualization of ∂ₜu=α∇²u
Julia Set Family
Julia sets J_c as c varies across the Mandelbrot set — smooth coloring, orbit traps, and interactive parameter picking
KPZ Interface Growth
Kardar-Parisi-Zhang roughening with β=1/3 scaling, Family-Vicsek collapse, and comparison with Edwards-Wilkinson
Spin Glass & Replica Symmetry
Edwards-Anderson spin glass, frustration, P(q) order parameter, and simulated annealing on random graphs
Hamiltonian Monte Carlo
HMC leapfrog trajectories on banana, funnel, and mixture targets — acceptance rate and comparison with random-walk MH
Isomap & Manifold Learning
Geodesic distances on Swiss roll and S-curve, k-NN graph, and Isomap vs PCA unfolding comparison
Aperiodic Penrose Tiling
Penrose P3 rhombus tiling via substitution rules — quasiperiodic 5-fold symmetry, τ-scaling, never repeating yet almost periodic
Double-Slit Interference
Young's experiment with variable wavelength and slit separation — fringe pattern, diffraction envelope, single vs double slit
Chemical Reaction Networks
Brusselator, Lotka-Volterra, and Oregonator oscillators — phase portraits, limit cycles, and Hopf bifurcation in CRNT
Gravitational Lensing
SIS and point-mass lens profiles — Einstein ring formation, arc magnification, and critical curve geometry
Polymer Reptation
de Gennes reptation model — entangled chain confined to tube, Rouse-to-reptation MSD crossover regimes
Cyclic Cellular Automaton
Greenberg-Hastings excitable medium — cyclic states, spontaneous spiral waves, and adjustable threshold rules
Critical Phenomena & Scaling
Universal critical exponents β, γ, ν — order parameter, susceptibility, and correlation length diverge at Tc with Ising simulation
Duffing Oscillator
Driven double-well oscillator — resonance, bistability, period-doubling cascade to chaos, and Poincaré section evolution
Network Motifs & Feedback
Feedforward loops, negative feedback, bistable switches — Boolean signal dynamics and motif computation in biological networks
Wavefunction Collapse
Quantum measurement: superposition, Born rule probabilities, decoherence, density matrix, and many-worlds branching
Fluid Turbulence Cascade
Kolmogorov energy cascade E(k)~k^-5/3, 2D turbulence with point vortices, inverse cascade and inertial subrange live in your browser
Knot Invariants
Trefoil, figure-eight, and torus knots rendered in 3D — Jones polynomial, writhe, crossing number, and Reidemeister invariants
First-Order Phase Transition
Double-well free energy landscape, nucleation and growth, Maxwell construction, hysteresis loop, and metastability interactively explored
Cellular Potts Morphogenesis
Glazier-Graner-Hogeweg model: cell sorting by differential adhesion, Metropolis dynamics, and tissue-level pattern formation
Nonlinear Wave & Shock
Burgers equation shock formation via method of characteristics, Rankine-Hugoniot conditions, viscous vs inviscid comparison
Quantum Decoherence (Lindblad)
Lindblad master equation for an open qubit: density matrix evolution, Bloch sphere, dephasing and relaxation channels in real time
Random Graph Giant Component
Erdős-Rényi G(n,p) percolation transition at p=1/n — giant component emergence, component size distribution, live phase diagram
Faraday Instability
Parametric Faraday waves on a vibrated fluid: Mathieu stability tongues, hexagonal and stripe pattern selection, resonance visualization
Vicsek Flocking Model
N self-propelled particles with velocity alignment — order parameter phase transition, critical noise, density-noise phase diagram
Frustrated Triangular Lattice
Antiferromagnetic Ising model on a triangular lattice — geometric frustration, residual entropy, Pauling estimate, and spin correlations
Graph Coloring & Chromatic Polynomial
Greedy and backtrack coloring animated, Brooks' theorem, four-color theorem, chromatic number.
Itô vs Stratonovich SDEs
Geometric Brownian motion paths, Itô correction, Fokker-Planck density, noise-induced transitions.
Laser Mode Competition
Two-mode rate equations, gain saturation, winner-take-all bistability, and phase-space portrait.
Bicoid Morphogen Gradient
Drosophila positional information: diffusion-degradation profile, hunchback threshold, dosage robustness.
Gutenberg-Richter & Seismic SOC
log N = a − bM law, Omori aftershock decay, spring-block self-organized criticality model.
Scalar φ⁴ Field Theory
Spontaneous symmetry breaking, kink solitons, effective potential, and Wilson-Fisher RG flow in 1+1D.
Information Geometry & Fisher Metric
Statistical manifold of Gaussians, Poincaré geodesics, natural gradient vs SGD, KL divergence.
Kuramoto Synchronization
N phase oscillators with coupling K, order parameter r(t), synchronization transition at K_c = 2/πg(0).
Run-and-Tumble Bacteria
Active particle dynamics, effective diffusion, boundary accumulation, and chemotaxis gradient climbing.
Anyonic Braiding & Topological Gates
Non-Abelian anyon worldlines, Fibonacci/Ising braid group representations, fault-tolerant quantum gates.
Abelian Sandpile
Self-organized criticality via toppling grains
Sierpinski Gasket IFS
Fractal from the chaos game — iterated function system
Cantor Dust
Middle-thirds removal — fractal with measure zero
Fermi Paradox
Drake equation sliders — how many civilizations exist?
Penrose Tiling
Aperiodic quasicrystal with 5-fold symmetry
Coupled Pendula
Normal modes and energy exchange via resonance
Stochastic SIR Epidemic
Agent-based infection spreading on a grid
Voronoi Fortune Sweep
O(n log n) nearest-neighbor diagram with sweep animation
Collatz Conjecture
The 3n+1 mystery — every orbit reaches 1 (unproven)
Lattice Boltzmann Fluid
Mesoscopic fluid simulation with streaming and collision
Random Walk & Brownian Motion
Walkers stepping randomly; displacement grows as √t
Langton's Ant
Simple local rules produce emergent highway after ~10,000 steps
Bell Inequality & CHSH Test
Quantum correlations exceed the classical bound S ≤ 2
Hyperbolic Geometry — Poincaré Disk
Geodesics, triangles, and tessellations in curved space
Prime Gap Distribution
Gaps between consecutive primes; Cramér's conjecture: max ~ (ln p)²
Schelling Segregation Model
Mild preference for similar neighbors produces strong macro-segregation
Traveling Salesman Problem
Nearest-neighbor heuristic then 2-opt improvement, animated
Catastrophe Theory — Cusp
V(x)=x⁴+ax²+bx: fold bifurcations and sudden equilibrium jumps
PCA / Eigenface Demo
Mix principal components to reconstruct synthetic faces
Zipf's Law
Rank-frequency power law in language, cities, and income
Solar System Orrery
All 8 planets with correct relative orbital periods — click for details
Fourier Series — Square Wave
Rotating epicycles approximating a square wave, Gibbs phenomenon
Escape Velocity & Gravity
Launch projectiles into elliptic, parabolic, or hyperbolic orbits
Markov Chain Text Generator
n-gram transition matrices generating probabilistic text sequences
Genetic Algorithm — Knapsack
Evolutionary search solving the NP-hard 0/1 knapsack problem
Neural Network XOR Visualizer
2-2-1 MLP learning XOR via backprop with live decision boundary
Boids — Emergent Flocking
Reynolds' 3-rule algorithm: separation, alignment, cohesion
Bernoulli's Principle
Fluid velocity and pressure through a constricted pipe
Topological Knot Theory
Trefoil, figure-eight, and torus knots with Jones polynomial info
Mandelbrot & Julia Duality
Hover the Mandelbrot set to see the linked Julia set update live
2D Wave Interference
Circular waves from two draggable sources showing constructive and destructive interference with nodal lines
Lissajous Curves & Phase
Animated Lissajous figures with continuously sweeping phase — x=A·sin(aτ+δ), y=B·sin(bτ)
Barycentric Coordinates
Click any point near a triangle to see its barycentric coords — centroid, incenter, circumcenter, orthocenter
Hénon Map & Bifurcation
Strange attractor and bifurcation diagram for the Hénon map — chaos from a simple quadratic recurrence
Dirac Delta & Convolution
Animated convolution integral with sliding kernel — shows delta as identity element and step-by-step integration
Apollonian Gasket
Infinite circle packing via Descartes' Circle Theorem — recursive tangent circles with integer curvatures
Morse Code Encoder
Type text to hear and see Morse code with audio, visual flash, and full decode mode — complete alphabet chart
Brownian Bridge
Random walks pinned at both endpoints with covariance Cov(Wₛ,Wₜ)=s(1−t) — compared to free Brownian motion
Steiner Tree Problem
Minimum spanning tree vs Steiner tree — adding Steiner points reduces total cable length by up to 13.4%
Wallpaper Groups
All 17 plane symmetry groups as animated tilings — rotations, reflections, and glide reflections
Dadras Strange Attractor
3D chaotic attractor projected and rotating in real time with speed coloring
Condorcet Paradox
See how plurality, IRV, Borda, and Condorcet methods pick different winners
Geodesic Sphere & Dome
Buckminster Fuller icosahedron subdivisions — Euler's V−E+F=2 in 3D
Seeds Cellular Automaton
Explosive Life-like growth: Seeds, HighLife, Day & Night, Replicator rules
Joukowski Airfoil Map
Conformal mapping w=z+1/z turns a circle into a wing — drag to reshape
Chaos Game Variations
Random vertex jumps converge to Sierpinski, pentaflake, and carpet fractals
Quantum Harmonic Oscillator
Wavefunctions ψₙ, energy levels, and oscillating superposition wave packets
Network Epidemic Spread
SIR model on random, small-world, and scale-free networks with live curves
Signal Convolution & Filter
Kernel sliding over a signal — box, Gaussian, edge, sharpen in time and frequency
Fractal Coastline Measure
Richardson effect: ruler length changes measured coastline — log-log reveals dimension
Torus Knot
Parametric curves winding around a torus — adjust p,q winding numbers in 3D
Maxwell's Equations
Electric field lines, magnetic fields, and EM wave propagation — the four laws
Small-World Epidemic
Watts-Strogatz rewiring generates small-world networks — run SIR epidemic on them
Slime Mold Network
Physarum polycephalum agents find shortest paths via emergent pheromone networks
GCD & Number Theory
Euclidean algorithm, rectangle packing, Stern-Brocot tree, and coprime lattice
Boolean SAT
DPLL algorithm on 3-CNF formulas — unit propagation, backtracking, phase transition
Cosmic Microwave Background
CMB angular power spectrum with acoustic peaks — tune cosmological parameters
Bézier de Casteljau
Drag control points and watch repeated linear interpolation trace the curve
Phase Plane Dynamics
Vector fields, nullclines, and trajectories for pendulum, Van der Pol, Lotka-Volterra
Escher Tessellation
17 wallpaper groups — deform square, hex, and triangular tilings into interlocking shapes
Hilbert Curve
Space-filling curves — Hilbert, Peano, Z-order — animated construction through all orders
Heat Equation
1D diffusion PDE solved by finite differences — click to add heat, watch it spread
Double Pendulum Chaos
Sensitive dependence on initial conditions — tiny angle differences diverge exponentially
N-Body Gravity
Verlet integration of gravitational orbits — click to add planets, watch energy conserve
Color Metamers
CIE 1931 chromaticity — two different spectra perceived as identical color
Turing Machine
Visualize tape, head, and state transitions — busy beaver, binary increment, and more
Rayleigh-Bénard Convection
Buoyancy-driven convection cells form above critical Rayleigh number Ra ≈ 1708
Crystal Point Groups
32 crystallographic point groups — animated stereographic projections of symmetry
Tax Incidence
Supply and demand elasticities split the tax burden — deadweight loss triangle visualized
Optimal Foraging
Marginal Value Theorem — Charnov's tangent construction for when to leave a patch
Rössler Attractor
Strange attractor with chaotic dynamics
Magnetic Pendulum
Fractal basins of attraction
Kármán Vortex Street
Fluid vortices behind a cylinder
Hyperbolic Tiling
Poincaré disk model geometry
Abelian Sandpile
Self-organized criticality
Ising-Kawasaki
Conserved magnetization dynamics
Fractal Canopy
Branching tree with wind animation
Euler Spiral
Clothoid with growing curvature
Newton's Cradle
Elastic collision physics
Strange Repeller
Escape-time Hamiltonian chaos
Harmonograph
Lissajous curves with decaying oscillators
Snowflake Dendrite
Diffusion-limited solidification
Hele-Shaw Growth
Viscous fingering patterns
Hopf Fibration
S³→S² fiber bundle visualization
Chladni Figures
Plate vibration mode shapes
Bifurcation Zoom
Feigenbaum structure deep dive
Hénon Map
Poincaré section orbits
Kuramoto Sync
Phase synchronization emergence
Vicsek Flocking
Alignment-driven collective motion
Lévy Flight
Heavy-tail vs Brownian motion
XY Model Defects
Vortex-antivortex topological defects
Airy Function
Quantum tunneling in linear potential
Langevin Dynamics
Thermal noise in a potential well
Born Rule
Wavefunction collapse statistics
Sharkovskii's Theorem
Period-3 implies all periods
Bell States
Quantum entanglement and CHSH
Cantor Set
Devil's staircase and zero measure
Mandelbrot Deep Zoom
Smooth coloring to 1e-12 depth
Continued Fractions
Rational convergents on number line
Tesseract
4D hypercube projected to 2D
Lorenz Section
3D attractor with Poincaré slice
Poisson Process
Arrival times and exponential gaps
Entropy Mixing
Gas spreading and Shannon entropy
Boids Flocking
Separation, alignment, cohesion rules
Newton Fractals
Basins of attraction for polynomials
IFS Fractals
Affine transform fractal builder
Morse Potential
Diatomic molecule quantum vibrations
Hopf Bifurcation
Spiral to limit cycle transition
Modulation Instability
NLS breather formation
Spin Precession
Bloch sphere Rabi oscillations
Logistic Harvesting
Sustainable yield and extinction bifurcation
Navier-Stokes Flow
2D vorticity-streamfunction solver
Leslie Matrix
Age-structured population dynamics
Infinite Square Well
Quantum eigenstates and superposition
Walk Recurrence
Pólya theorem in 1D/2D/3D
Geodesic Sphere
Icosahedron subdivision and great circles
Torus Knot Flow
Parametric (p,q) knot animation
Klein Bottle
Non-orientable surface in 3D
Wallpaper Groups
All 17 2D symmetry patterns
Floquet Theory
Quasi-energy bands, periodic driving
Stadium Billiards
Chaotic Hamiltonian billiards
Allometric Scaling
Metabolic rate power laws
Mean-Field Ising
Spontaneous magnetization bifurcation
Superellipse
Lamé curves morphing diamond to square
Adiabatic Invariant
Action angle in slowly varying pendulum
Quantum Circuit
Drag-and-drop gate builder
Ptolemy's Epicycles
Fourier series drawing machine
1D Cellular Automata
Wolfram's 256 rules explorer
SSH Edge States
Topological zero mode visualization
Hamming Code
Error correction visualized
Replicator Dynamics
Evolutionary game theory in action
Parrondo's Paradox
Two losing games that win together
Duffing Oscillator
Nonlinear resonance and chaos
Anderson Localization
Random potential eigenstate trapping
Gradient Descent
SGD vs momentum vs adaptive methods
Phase Portraits
Nullclines and vector fields
Quantum Annealing
Transverse field Ising optimization
Four-Color Theorem
Interactive planar graph coloring
Sznajd Model
Opinion consensus dynamics
Maxwell's Demon
Information and thermodynamic entropy
Liquid Crystals
Director field and topological defects
Scale-Free Network
Barabási-Albert preferential attachment
Elliptic Pendulum
Exact solution via elliptic integrals
Padé Approximants
Rational convergence beyond Taylor
Heat Equation
Green's function thermal diffusion
Random Matrix GOE
Wigner semicircle and level repulsion
Winding Number
Homotopy classes of closed curves
Brachistochrone
Least-action cycloid solution
Information Bottleneck
Compression vs relevance tradeoff
Poincaré Recurrence
Gas particles almost-periodic return
Voronoi Relaxation
Lloyd's algorithm iteratively relaxes Voronoi cells toward their centroids.
Topological Band Theory
SSH model: topological vs trivial phases and protected edge states.
Diffusion-Limited Aggregation
Particles random-walk until sticking, forming beautiful fractal clusters.
Entropy Production
Non-equilibrium entropy production in a driven system, Clausius inequality.
Spatial Prisoner's Dilemma
Strategies compete on a grid: TFT, defectors, cooperators, Pavlov.
Molecular Orbital Theory
LCAO: hydrogen bonding and antibonding molecular orbitals visualized.
Tidal Locking
Moon's spin slows under tidal torque until synchronous rotation locks in.
Power-Law Distributions
Interactive Pareto distributions: exponent, log-log plots, heavy tails.
Polymer Chain Statistics
Freely jointed chain: end-to-end distance, persistence length, Rouse modes.
Lorenz '96 Weather Model
Atmospheric predictability, Lyapunov exponent, ensemble divergence.
Stochastic SIR
Gillespie algorithm epidemics: R₀ threshold, extinction probability, outbreak variability.
3D Ising Model
Metropolis Monte Carlo phase transition: magnetization, susceptibility near Tc.
Fourier Heat Diffusion
Eigenmode decomposition of heat equation: high-k modes decay fastest.
Neural Bursting
Hindmarsh-Rose neuron: fast-slow dynamics, tonic spiking vs bursting bifurcation.
Quantum Harmonic Lattice
1D phonon chain: dispersion relation, zero-point fluctuations, quantum occupation.
Lagrangian Chaos
Particles advected in 2D flow: chaotic mixing, Poincaré sections, stretching.
2D Drum Modes
Circular drum: Bessel eigenmodes, Chladni patterns, mode superposition.
Optical Interference
Multi-slit interference: wavelength, slit spacing, coherence length effects.
Stochastic Resonance
Noise enhances weak signal detection — SNR peaks at optimal noise amplitude.
Abelian Sandpile
Self-organized criticality: toppling rules, power-law avalanche size distribution.
FitzHugh-Nagumo Waves
Reaction-diffusion: traveling waves, spiral waves, and Turing patterns.
Rule 30 Automaton
Wolfram's Rule 30: randomness and complexity from a dead-simple rule.
Double Pendulum Chaos
Lagrangian dynamics: sensitive dependence on initial conditions visualized.
Spin Wave Magnons
Ferromagnetic spin waves: dispersion relation, group vs phase velocity.
Galton Board / CLT
Bean machine: inelastic collisions converge to Gaussian by the central limit theorem.
Hénon-Heiles Chaos
Poincaré sections reveal KAM tori and chaotic seas as energy increases.
Granular Gas Cooling
Inelastic collisions, Haff's law T∝t⁻², clustering instability in granular gases.
XY Model Defects
Vortex-antivortex topological defects and the BKT transition in 2D.
Debye-Hückel Screening
Electrostatic screening in electrolytes: Debye length vs ion concentration.
Algebraic Surfaces
Ray-marched Cayley cubic, Kummer surface, Barth sextic — implicit surface explorer.
Navier-Stokes 2D
Stream function/vorticity flow past cylinder: von Kármán vortex street.
Stadium Billiard
Bunimovich stadium: ergodic dynamics, Lyapunov exponent, Poincaré map.
Kuramoto 2D Sync
Kuramoto oscillators on a 2D grid: synchronization order parameter dynamics.
Spectral Clustering
Graph Laplacian Fiedler vector reveals community structure in networks.
Morse Theory
Gradient flow on surfaces: critical points, saddles, handle decomposition.
Van der Waals Gas
P-V isotherms, Maxwell construction, liquid-vapor coexistence curve.
Acoustic Impedance
Wave transmission through layered media: impedance matching and resonances.
Quantum S-Matrix
1D scattering: transmission/reflection vs energy, resonances through barriers.
Phase Portrait Explorer
Nonlinear ODE vector fields: nullclines, fixed points, limit cycles.
Bond Percolation
Square lattice bond percolation: spanning cluster near critical p_c ≈ 0.5.
Network Epidemics
SIR on Erdős-Rényi vs scale-free networks: R₀ threshold and cascade dynamics.
Kuramoto Mean Field
Synchronization transition at K_c: order parameter r and Lorentzian vs Gaussian distributions.
Quantum Walk 1D
Quantum vs classical random walk: ballistic spreading, coin operators, interference.
Diffusion Maps
Heat kernel on point clouds reveals intrinsic geometry via Laplace-Beltrami eigenfunctions.
Persistence Barcodes
Vietoris-Rips filtration: track β₀ and β₁ as the scale parameter grows.
Rayleigh-Taylor
Heavy fluid over light: fastest-growing wavelength, spikes and bubbles.
BZ Reaction Spirals
Oregonator model: spiral waves, target patterns, oscillating chemistry in 2D.
Entropy & Mixing
Shannon entropy, KL divergence, and information flow as distributions mix.
Logistic Map Chaos
Period-doubling bifurcations, Feigenbaum constant δ≈4.669, Lyapunov exponent.
Brusselator Patterns
Turing instability in the Brusselator: stripe vs spot selection, parameter space.
Airy & WKB Tunneling
Quantum tunneling visualized through Airy functions and WKB approximation
BTW Sandpile
Bak-Tang-Wiesenfeld abelian sandpile — self-organized criticality
Chimera States
Coexistence of synchronized and desynchronized oscillator populations
Hyperbolic Tiling
Poincaré disk model: infinite tilings in bounded space
Lissajous Curves
Parametric Lissajous figures from coupled harmonic oscillators
Spectral Clustering
Graph Laplacian eigenvectors reveal hidden cluster structure
Euler's Identity
e^(iπ)+1=0 animated on the complex plane
Winding Numbers
Topological winding number and charge in 2D fields
Apollonian Gasket
Infinitely nested circle packing via Descartes' theorem
Weierstrass Function
Continuous everywhere, differentiable nowhere — a fractal function
Hilbert Curve
Space-filling curve construction animated step by step
Möbius Strip
One-sided, one-edged surface: topology made visual
Four-Wing Attractor
Chaotic four-wing strange attractor projected from 3D
Fourier Epicycles
Rotating circles trace any path via Fourier decomposition
Runge-Kutta Portraits
ODE integration and phase portrait visualization with RK4
Hopf Bifurcation
Equilibrium destabilizes into limit cycle as parameter crosses threshold
Attractor Finance
Strange attractor dynamics applied to financial time series
Newton-Raphson Fractal
Complex root-finding iteration produces fractal basin boundaries
Gyrator & Monopole
Gyrator circuit analog of the magnetic monopole
Power Laws
Power-law distributions and scale-free network generation
Geodesic Sphere
Icosahedron subdivision into geodesic dome — 3D wireframe
Vector Field Flows
Streamlines through 2D vector fields with saddle points and spirals
Drum Eigenmodes
Laplacian eigenfunctions: the modes a drum can vibrate in
Ising Metropolis
2D Ising model with Metropolis sampling through the phase transition
2D Fluid Sim
Navier-Stokes grid-based 2D fluid simulation with vorticity
Shor's Algorithm
Quantum period finding and integer factorization visualized
Bell Inequality
Quantum entanglement violates local hidden variable bounds
Partition Function
Statistical mechanics: Z, free energy, and thermodynamic observables
Klein Bottle
Non-orientable surface with no inside — immersed in 3D
4D Tesseract
Rotating hypercube projected from 4D to 2D
Carnot Engine
Carnot cycle PV diagram with efficiency η = 1 − T_c/T_h
Maxwell's Demon
Szilard engine: information erasure and thermodynamic cost
Liquid Crystal Defects
Topological defects in nematic liquid crystals: ±½ disclinations
Voter Model
Opinion dynamics: consensus formation on random networks
Tipping Points
Bifurcations and catastrophic tipping points in dynamical systems
3D Cellular Automata
Conway's Game of Life extended to three dimensions
Fiber Bundle & Gauge
U(1) gauge field as a fiber bundle: phase holonomy and curvature
Feynman Path Integral
Sum over all paths: quantum amplitudes from classical trajectories
Spin Glass
Rugged energy landscape of a frustrated spin glass system
Julia Set Family
Holomorphic dynamics: explore the Julia set family z²+c
Rabinovich-Fabrikant
Rabinovich-Fabrikant strange attractor: nonlinear wave interactions
Majorana Zero Modes
Topological qubits from Majorana fermion zero modes at wire ends
Rule 90 Automaton
Rule 90 CA generates Sierpiński triangle from a single cell
Symbolic Dynamics
Coding chaotic orbits as symbol sequences under shift maps
Correlation Decay
Exponential vs power-law correlation decay in critical systems
Quantum Decoherence
Wavefunction collapse through environmental entanglement
Entropic Gravity
Verlinde: gravity as an emergent entropic force from information
Langton's Ant
Simple rules, complex emergent highway behavior
FDTD Wave Equation
Wave equation solved with finite difference time domain on a 2D grid
Cosmic String
Topological defect as conical spacetime: deficit angle and lensing
Lattice QCD
Wilson loops, plaquettes, and quark confinement on a lattice
Dehn Surgery
Building 3-manifolds by cutting and regluing knot complements
Modular j-Function
SL(2,Z) symmetry and the j-function on the upper half-plane
Riemannian Curvature
Parallel transport around loops reveals intrinsic curvature
Pólya Urn
Reinforcement dynamics: rich-get-richer and power-law emergence
Voronoi & Delaunay
Voronoi diagram and Delaunay triangulation as dual planar graphs
Persistent Homology
Barcode diagrams reveal topological features across scales
Tropical Geometry
Piecewise-linear algebraic geometry: max-plus algebra curves
Noncommutative Torus
Weyl quantization and the rotation algebra Aθ
Hyperbolic Geodesics
Poincaré disk: geodesics, parallel lines, and angle defect
Cayley Tables
Finite group structure visualized as Cayley multiplication tables
Lie Algebra su(2)
Commutation relations [Jᵢ,Jⱼ]=iεᵢⱼₖJₖ and spin representations
Spectral Gap
Markov chain mixing time controlled by the spectral gap λ₂
Continued Fractions
Farey sequences, mediants, and the geometry of continued fractions
Algebraic K-Theory
K₀ and K₁ groups: classifying vector bundles algebraically
Étale Cohomology
Galois action on algebraic varieties over finite fields
Hodge Decomposition
Pure Hodge theory: H^n = ⊕ H^{p,q} on Kähler manifolds
GUE Random Matrices
Wigner semicircle law and eigenvalue repulsion in the GUE
Number Field Sieve
Visualizing the algebraic structure behind NFS integer factorization
Elliptic Curve Group
Point addition law on elliptic curves over ℝ and finite fields
Boolean Algebra
Boolean laws demonstrated interactively with Venn diagrams
Category Theory
Functors, natural transformations, and commuting diagrams
Topos Theory
Subobject classifiers and the internal logic of a topos
Serre Spectral Sequence
Filtering cohomology of fibrations page by page
Cobordism
Manifolds with boundary: handle decompositions and cobordism classes
η-Invariant
APS η-invariant: spectral asymmetry of self-adjoint operators
Cohomological FT
TQFT partition functions and cobordism categories on 2D surfaces
Arithmetic Dynamics
Iteration of rational maps over finite fields: periodic orbits and portraits
Formal Group Laws
Power series generalizations of group structure and their classification
Operator Spectrum
Eigenvalue distribution of self-adjoint operators and spectral theory
Fourier Transform
Building signals from sinusoids: Fourier synthesis and analysis animated
Nash Equilibrium
Find Nash equilibria in 2-player matrix games: pure and mixed strategies
Residue Theorem
Contour integration and residues in the complex plane
Brownian Motion
Stochastic paths, diffusion, and the Wiener process
Pushdown Automaton
Stack-based computation for context-free language recognition
Diophantine Geometry
Integer solutions on algebraic curves: lattice points and geometry
Generating Functions
Probability generating functions and branching process extinction
Fundamental Group
Loop spaces and π₁: the fundamental group of topological spaces
Manifold Atlas
Charts, transition maps, and smooth structure on manifolds
Lebesgue Measure
Outer measure, measurable sets, and Lebesgue integration
Ultrafilters
Ultrafilters and Tychonoff compactness: topology at the limit
Nonstandard Analysis
Infinitesimals made rigorous via the transfer principle
Constructive Math
Type theory and Curry-Howard: proofs as programs
Catastrophe Theory
Fold and cusp catastrophes: discontinuous jumps in smooth systems
Morse Theory
Gradient flows, critical points, and handle attachments in Morse theory
TDA
Vietoris-Rips filtration and birth-death pairs from point cloud data
Symplectic Topology
Lagrangian submanifolds and Gromov's non-squeezing theorem
Quantum Groups
Hopf algebras, q-deformation, and quantum symmetry
Quiver Representations
Gabriel's theorem: finite-type quivers and indecomposable modules
Sheaf Cohomology
Local-to-global: Čech cohomology and sheaf theory
Derived Categories
Triangulated structure, distinguished triangles, and derived functors
String Diagrams
Visual calculus for monoidal categories: wires, boxes, and tensor products
Operads
Tree composition and the algebraic structure of E∞ operads
Homotopy Type Theory
Path types, identity, and the univalence axiom in HoTT
Univalence Axiom
Equivalence of types as equality: (A≃B) ≃ (A=B)
Cubical Type Theory
The interval [0,1] gives computational content to univalence
Modal Logic
Kripke frames and possible worlds semantics for □ and ◇
Linear Logic
Proof nets and the geometry of interaction in linear logic
Lambda Calculus
β-reduction, normal forms, and Church numerals step by step
SKI Calculus
Combinatory logic: S, K, I eliminate all variables from terms
CA Rule Explorer
Beyond Conway's Life: explore B/S rule space of 2D cellular automata
Turmites
2D Turing machines on colored grids: complex highways from simple rules
Logic Circuit Sim
Boolean gates wired together: build and simulate digital circuits
Turing Machine
Step through a Turing machine: tape, head, states, and transitions
Busy Beaver
How many steps can a halting Turing machine take? The non-computable answer.
Church-Turing Thesis
Equivalent models of computation: λ-calculus, TM, recursive functions
Coq Proof Assistant
Interactive theorem proving: tactics, goals, and proof trees in Coq
Lean 4
Dependent types and formal verification in the Lean 4 proof language
Idris Types
First-class dependent types: Vect n a, Fin n, and type-safe programs
Agda
Total functional programming: termination, coverage, and type universes
4-Color Theorem
Greedy 4-coloring of planar graphs — every map needs at most 4 colors
Kruskal's MST
Minimum spanning tree built edge by edge with union-find
Dijkstra's Algorithm
Shortest paths on weighted graphs — priority queue expansion animated
Max-Flow Min-Cut
Ford-Fulkerson augmenting paths and the max-flow min-cut theorem
Traveling Salesman
NP-hard TSP: nearest neighbor heuristic and 2-opt improvement
Sorting Algorithms
Bubble, merge, quicksort, and heapsort side by side — watch them race
Dynamic Programming
Memoization and tabulation: overlapping subproblems visualized
Hash Tables
Collision resolution: chaining vs open addressing under load
IEEE 754 Floats
Sign, exponent, mantissa: floating point precision and rounding
Regex to Automata
Thompson's construction: regular expressions become finite automata
Decision Trees
Information gain and recursive splitting — how trees grow
Random Forest
Ensemble of decision trees with bagging and OOB error estimation
Attention Head
Scaled dot-product attention: Q, K, V matrices and softmax weights
CNN Convolution
Sliding kernel produces feature maps: visualize filters and activations
LSTM Gates
Forget, input, output gates and cell state flow in an LSTM
Autoencoder
Encoder-decoder bottleneck and latent space interpolation
GAN Training
Generator vs discriminator: minimax game and loss dynamics
Bayesian Network
Belief propagation through a directed graphical model
EM Algorithm
Gaussian mixture model fitting: E-step and M-step animated
Diffusion Models
Forward noising process and reverse denoising score matching
Quantum Circuit
Compose quantum circuits: gates, qubits, and state vector evolution
Quantum Gates
H, X, Y, Z, CNOT — unitary matrices and Bloch sphere rotations
Quantum Teleportation
Bell pair, measurement, and classical correction — step by step
VQE
Variational quantum eigensolver: minimize energy with parameterized circuits
QAOA
Quantum approximate optimization: MaxCut via alternating unitaries
BB84 Protocol
Quantum key distribution: polarized photons and eavesdropper detection
Surface Code
Logical qubit from a grid of physical qubits: X/Z stabilizer measurements
Topological Codes
Anyons, braiding, and topologically protected logical operators
Quantum Walk
Quantum walk on a graph: interference produces speedup over classical walk
Hubbard Model
Quantum simulation of the Hubbard model: Mott insulator and superconductivity
L-Systems
Interactive Lindenmayer system generator — turtle graphics rewriting rules produce fractal plants, dragon curves, and Koch snowflakes.
Random Walk Polymer
Polymer chains as 2D random walks. Watch end-to-end distance statistics obey the R~√N Rouse scaling law in real time.
Liquid Crystal Defects
Director field visualization with ±½ topological defects. Place charges, watch them attract, annihilate, or frustrate.
Sandpile Identity
The abelian sandpile group's identity element — a fractal configuration that emerges from purely local toppling rules.
Frustrated Kuramoto
Antiferromagnetically coupled oscillators on a triangle resist synchronization. Watch the order parameter R fluctuate as geometric frustration wins.
Hyperboloid Geodesics
Geodesic paths on a hyperboloid of one sheet — a doubly-ruled surface. Drag to rotate, explore the interplay of curvature and straight-line rulings.
Wavelet Transform
Haar, Mexican hat, and Morlet wavelets applied to chirps and impulses. The scalogram reveals when and at what scale events occur.
Sierpinski Chaos Game
Pick a random vertex, move halfway, plot. Repeat forever. The Sierpinski attractor emerges from pure randomness via iterated function systems.
Reaction-Telegraph Waves
Waves with finite propagation speed from the telegraph equation. Unlike the heat equation, disturbances travel at c=√(D/τ) with sharp fronts.
Stadium Billiards
Bunimovich's stadium billiard — ergodic and chaotic. Click to add trajectories; watch the Poincaré section fill as chaos ergodically covers phase space.
Devil's Staircase
The Cantor function: continuous, non-decreasing, yet with zero derivative almost everywhere. Zoom into the fractal staircase rising only on the Cantor set.
Penrose Rhombus Tiling
Aperiodic P3 tiling with thick and thin rhombuses built by deflation. The golden ratio τ governs splitting rules; global 5-fold symmetry emerges without periodicity.
Julia Set Explorer
Click the Mandelbrot set to pick c and see the corresponding filled Julia set. Inside the Mandelbrot set → connected Julia; outside → Cantor dust.
Fokker-Planck Drift
Probability density evolving under drift and diffusion in a double-well potential. Watch relaxation to the Boltzmann equilibrium as D and well depth change.
Ising Correlation Length
2D Ising model near the critical temperature T_c ≈ 2.269. Live spin-spin correlation length measurement shows divergence as T → T_c with exponent ν = 1.
Apollonian Gasket
Circle packing via Descartes' theorem: (k₁+k₂+k₃+k₄)² = 2(k₁²+…+k₄²). Integer curvatures throughout; fractal dimension ≈ 1.3057. Animate the recursive construction.
Network Motif Census
Count triangles, wedges, and paths in a random graph; compare to Erdős–Rényi expectations with Z-scores. Motif over-representation reveals non-random network structure.
Snowflake Crystal Growth
Reiter cellular automaton on a hexagonal lattice. Vapor diffuses and freezes at ice boundaries. Tune temperature β and supersaturation γ to grow dendrites or plates.
Spectral Graph Partitioning
The Fiedler vector (second Laplacian eigenvector) partitions a random geometric graph. Algebraic connectivity λ₂ measures how well-connected the graph is.
Airy Function and the Rainbow
Airy (1838) solved the rainbow caustic with y'' = xy. Intensity goes as Ai²(ξ) near the rainbow angle, replacing ray theory's infinity with finite supernumerary arcs.
N-Body Escape Velocity
Launch a probe into an N-body gravitational system. Tune planet masses and watch slingshot trajectories unfold — or the probe spiral in to its doom.
Tropical Polynomial Roots
In tropical algebra, + is min and × is +. Polynomials become piecewise-linear functions; their "roots" are breakpoints where two monomials tie — a polyhedral geometry.
Van der Pol Limit Cycle
The Van der Pol oscillator ẍ = μ(1−x²)ẋ − x has a globally attracting limit cycle. Watch relaxation oscillations emerge as μ grows: slow drift, fast spike, repeat.
p-adic Ultrametric Tree
The p-adic integers form an ultrametric space where numbers are close if they share many base-p digits. The Cantor-like tree makes this geometry visual and tangible.
Zero-Knowledge Proof Demo
Interactive graph 3-coloring ZKP: Peggy commits to a permuted coloring, Victor checks one edge per round. Watch soundness accumulate — or see what happens when Peggy cheats.
Schelling Segregation 3D
Agents on a 3D cubic lattice move if too few neighbors match them. Even mild thresholds produce stark segregation — Schelling's 1971 result made three-dimensional.
Prime Gaps Distribution
Histogram of gaps between consecutive primes up to N, compared to Cramér's probabilistic model. Twin primes, even-gap dominance, and the elusive structure of prime spacings.
Quantum Harmonic Oscillator Wigner Function
The Wigner quasi-probability distribution W(x,p) over phase space. Fock states show negative regions (blue) — a purely quantum signature invisible in classical probability.
Complex Polynomial Zeros
Drag roots in the complex plane — coefficients update via Vieta's formulas. Newton's method basins of attraction color the background, revealing fractal boundaries between roots.
Lissajous Resonance Modes
x=sin(aωt+δ), y=sin(bωt): when a:b is rational the curve closes into a knot-like figure. Explore all integer ratios in the grid, animate the phase, or spin a 3D Lissajous.
Hamiltonian Cycle Search
Backtracking algorithm hunts for a cycle visiting every vertex exactly once — watch it explore, deadend, and backtrack in real time across Petersen, cube, dodecahedron, and random graphs.
Diffusion Map Embedding
Nonlinear dimensionality reduction via a Markov diffusion kernel. Swiss rolls, tori, and helices are "unrolled" into flat 2D embeddings using the eigenvectors ψ₁, ψ₂ of the diffusion operator.
Mandelbrot External Rays
External rays at rational angles p/q land on parabolic points of the Mandelbrot set. Choose preset angle families (1/3, 2/3 — main bulb; 1/7 family — period-3 bulb) and zoom in to see them land.
Metabolic Scaling Law
Kleiber's law B ∝ M^(3/4) spans 27 orders of magnitude from mycoplasma to blue whales. Log-log plot with real animal data, adjustable exponent, and the West-Brown-Enquist fractal network derivation.
Spring Network Phonons
A 2D mass-spring lattice animated in real time — acoustic and optical normal modes, thermal superposition, and the full dispersion relation ω(k) along Γ→X→M→Γ in the Brillouin zone.
Boolean Perceptron & XOR
Single-layer perceptrons learn AND and OR live via gradient descent but fail on XOR — which needs 2 layers. Watch decision boundaries animate as the multilayer network trains, weight by weight.
Gravitational Wave Chirp
Binary black hole inspiral: frequency and amplitude sweep upward as the orbit decays. Adjust masses and distance, see the chirp signal h(t), spectrogram, and matched-filter SNR — like LIGO does it.
Maximum Entropy Inference
Jaynes' principle: given only moment constraints (mean, variance, skewness), choose the distribution with maximum Shannon entropy. Lagrange multipliers converge live; fixing mean+variance yields the Gaussian.
Origami Crease Patterns
Miura-ori, Yoshimura, and waterbomb base — rigid origami tessellations with mountain/valley folds. Animate the single-DOF fold from flat to 3D and back, as used in deployable satellite solar panels.
Saddle-Node Bifurcation
Normal form ẋ = r + x²: two fixed points collide and vanish at r = 0. Phase portraits, live particle trajectories, bifurcation diagram, and the ghost bottleneck that slows dynamics after the equilibria disappear.
Banach-Tarski Paradox
Decompose a ball into non-measurable pieces and reassemble into two identical balls — purely via rotations. Visualizes the free group F₂ ⊂ SO(3) that makes the paradox possible.
Menger Sponge
3D isometric fractal: subdivide a cube into 27, remove the face-centers and core. Drag to rotate, adjust iteration depth, slice with a cross-section. Dimension ≈ 2.727.
Two-Stream Instability
PIC-lite plasma simulation: two counter-streaming electron beams develop electrostatic instabilities. Watch exponential mode growth and phase-space vortex formation in real time.
Euler Characteristic χ = V−E+F
Triangulate surfaces and watch χ remain invariant under subdivision, refinement, and added faces. Switch between sphere (χ=2), torus (χ=0), and disk (χ=1).
Symbolic Regression
Genetic programming evolves expression trees to fit noisy data. Watch RMSE shrink as populations of math formulas breed, mutate, and compete across generations.
Fibonacci Lattice Sphere
Golden angle spiral packing achieves near-optimal uniform coverage of a sphere. Explore the sunflower disk, 3D sphere, and equirectangular density map. φ = (1+√5)/2 is why it works.
Wigner Semicircle Law
Generate random symmetric matrices, compute all eigenvalues, and watch the histogram converge to the semicircle ρ(x)=(2/π)√(1−x²) — regardless of the entry distribution.
Galton-Watson Branching
Each individual spawns random offspring. When mean μ ≤ 1, extinction is certain; when μ > 1, survival probability q satisfies q = G(q). Watch generation trees grow or die.
Chladni Figures
Vibrating plate modes cos(mπx)cos(nπy) reveal intricate nodal-line patterns where sand collects. Explore square and circular plates across 16 preset modes.
Topological Euler Number
Add handles (genus) and holes (boundary circles) to a surface and track χ = 2−2g−b live. Gauss-Bonnet: curvature integrates to exactly 2πχ for any metric.
Cantor Set: Measure Zero
Iterative middle-thirds removal — a fractal with zero length but uncountably many points.
Pendulum Chaos
Driven nonlinear pendulum — watch order dissolve into chaos as driving amplitude grows.
Substitution Tilings
Inflation rules generate quasiperiodic tilings with forbidden rotational symmetry.
SEIR Age Structure
Epidemic dynamics with age-stratified contact matrices — who infects whom matters.
Morse Code Fourier
Type text, hear it in Morse, and see the Fourier spectrum of the resulting waveform.
Cayley Graph
Visualise group structure as a coloured directed graph — generators become arrows.
Debye Shielding
Charged particles in a plasma rearrange to screen electric fields at the Debye length.
Bell Inequality
Simulate entangled qubit pairs and watch quantum correlations violate the CHSH bound.
Zipf's Law
Rank-frequency plot for classic texts — power-law fit and Mandelbrot extension slider.
3-State Potts Model
Metropolis simulation on a grid — sharper first-order phase transition than Ising.
Harmonic Analysis on the Circle
Build signals from Fourier harmonics, watch phasors rotate, and see amplitude/phase spectra update live.
Stirling Numbers
Triangles of Stirling numbers — counting set partitions (2nd kind) and permutation cycles (1st kind). Click cells to explore.
Boids Flocking
Three local rules — separation, alignment, cohesion — produce emergent flocking. Tune weights to break or strengthen the flock.
Heat Equation on a Sphere
Spherical harmonic initial conditions diffuse over time as e^{−αl(l+1)t}. Higher modes vanish first — the sphere smooths out.
Erdős–Rényi Phase Transition
Watch a giant connected component suddenly emerge as edge probability crosses the critical threshold 1/n.
Quantum Walk on a Line
Discrete-time quantum walker spreads ballistically (σ ~ t) via interference, versus classical diffusion (σ ~ √t).
Sphere Eversion
Smale's theorem: a sphere can be turned inside out without tearing. Animated halfway model shows the outside-in flip.
Parrondo's Paradox
Two individually losing games combine to win. The probability ratchet exploits capital modulo structure — watch capital rise.
Maxwell-Boltzmann Speed Distribution
Live particle simulation builds the speed histogram. Watch it converge to f(v) ∝ v²e^{−mv²/2kT} as temperature changes.
Stokes Drag and Terminal Velocity
A sphere falls through fluid from Stokes to Newton drag regimes. Reynolds number, drag coefficient, and force balance visualized.
Voter Model Opinion Dynamics
Each cell adopts a neighbor's opinion. Watch consensus emerge through coarsening — or add noise for a perpetually churning noisy voter model that never fully agrees.
Hamming(7,4) Error Correction
Encode 4 data bits into 7 using 3 parity bits. Flip bits to inject errors and watch the syndrome pinpoint and correct any single-bit mistake automatically.
Taylor-Green Vortex
An exact Navier-Stokes solution decaying under viscosity. Watch vorticity evolve and the energy spectrum develop a turbulent cascade toward the k⁻³ scaling law.
Random Fibonacci & Viswanath's Constant
xₙ = ±xₙ₋₁ + xₙ₋₂ with random signs. Despite randomness, |xₙ|^(1/n) → 1.1319882... (Viswanath 1999). Growth is deterministic even when sequence is not.
Three-Species Predator-Prey Chaos
Three-species Lotka-Volterra escapes the periodic orbits of two-species dynamics and becomes chaotic. Watch the strange attractor unfold in phase space.
Soap Film Minimal Surfaces
Plateau's problem: find a surface of minimal area spanning a wire boundary. Zero mean curvature everywhere. Explore catenoid, helicoid, Enneper, and monkey saddle.
Feigenbaum Constants
The period-doubling cascade to chaos is universal: δ≈4.669 and α≈2.502 apply to any smooth unimodal map. Click to zoom into the self-similar bifurcation structure.
Trefoil Knot — 3D
The simplest non-trivial knot: 3 crossings, cannot be untangled without cutting. Drag to rotate. Toggle the Seifert surface and explore knot invariants including the Jones polynomial.
Rate-Distortion Theory
Shannon's 1959 theorem: R(D) = ½log₂(σ²/D) bits for a Gaussian source. The curve separates achievable from impossible compression. Watch distortion degrade a real signal.
Icosahedral Symmetry Group
The icosahedron has 60 proper rotations (group I ≅ A₅). Visualize all symmetry axes: six 5-fold through vertices, ten 3-fold through faces, fifteen 2-fold through edges.
Continued Fraction Approximations
Convergents pₙ/qₙ are the best rational approximations to any real. Watch how √2, φ, and π are approximated, and trace the path through the Stern–Brocot tree.
Alexander Polynomial
Compute the Alexander knot invariant Δ(t) = det(S − tSᵀ) from the Seifert matrix. Explore trefoil, figure-eight, cinquefoil and more — see which knots the polynomial can distinguish.
Lindblad Master Equation
A qubit coupled to an environment: amplitude damping shrinks the Bloch sphere toward the ground state, dephasing collapses coherences to the z-axis. Watch the Bloch vector trajectory in real time.
Abelian Sandpile Toppling Waves
Add grains, watch avalanches ripple outward. The abelian property: toppling order doesn't affect the final state. Colors track wave-front generation, revealing fractal self-similarity.
Arithmetic Dynamics over 𝔽ₚ
Iterate rational maps over finite fields. Every orbit eventually cycles — the functional graph decomposes into ρ-shaped tails feeding into cycles. Explore fixed points, periods, and cycle structure.
Weierstrass Nowhere-Differentiable
W(x) = Σ aⁿcos(bⁿπx) is everywhere continuous but nowhere differentiable. Zoom in arbitrarily deep — it never straightens out. Adjust parameters to see Hölder exponent change.
Pólya Recurrence — 2D vs 3D Walk
A 2D random walk returns to the origin with probability 1; a 3D walk escapes with probability ≈ 0.66. "A drunk man finds home, a drunk bird may be lost forever." Watch the divergence live.
Brusselator Spiral Waves
The Brusselator reaction-diffusion PDE: past the Hopf bifurcation B > 1+A², spiral chemical waves self-organize and rotate. Click to seed new spirals, adjust parameters to control wave speed.
Lagrange Multipliers Visualized
Optimize f(x,y) on a constraint g=0. At the optimum, ∇f = λ∇g — gradients align. Watch gradient arrows become parallel as you tune the constraint, and see λ as the shadow price.
Convex Hull Algorithms
Graham scan, Jarvis march, and QuickHull running side-by-side, step by step. Compare O(n log n) vs O(nh) strategies on random, circular, clustered, and adversarial point sets.
Burning Ship Fractal
Escape-time rendering of the Burning Ship: z ← (|Re z|+i|Im z|)²+c breaks conjugate symmetry, producing an asymmetric ship-like form. Click to zoom, compare to Mandelbrot side-by-side.
Network Centrality Explorer
Eigenvector, betweenness, and PageRank centrality on Erdős–Rényi and Barabási–Albert graphs. Node size encodes importance — watch hubs emerge in scale-free networks.
Hopf Fibration S³→S²
Every point on S² lifts to a great circle on S³. Stereographic projection renders the fibers as linked torus knots in 3D. Drag to rotate, adjust fiber count and latitude.
Quasicrystal Penrose Diffraction
Penrose tiling (aperiodic, 5-fold symmetric) → Fourier transform → sharp 10-fold diffraction spots forbidden by classical crystallography. The Nobel-winning discovery of quasicrystals, live.
Stochastic Lotka-Volterra
Exact Gillespie SSA simulation of predator-prey dynamics. Discrete birth/death events introduce noise that can drive extinction even when the ODE predicts sustained oscillations.
Tropical Geometry Amoeba
The amoeba A(f) is the Log-image of a complex curve. Left: amoeba in ℝ². Right: tropical curve — the piecewise-linear skeleton where min-plus algebra replaces ordinary arithmetic.
Fourier Uncertainty Principle
Δt·Δf ≥ 1/(4π): compress a signal in time and it spreads in frequency. Gaussian wavepackets saturate the bound — the same math underlies Heisenberg's σ_x·σ_p ≥ ℏ/2.
Polymer Self-Avoiding Walks
The pivot algorithm generates self-avoiding walks on a 2D lattice. Measure the Flory exponent ν≈0.75 (exact: 3/4) from how radius of gyration scales with chain length N.
Barabási-Albert Scale-Free Growth
Watch a network grow by preferential attachment — new nodes link to well-connected hubs. The degree distribution converges to P(k)~k⁻³ in the live log-log plot.
Floquet Theory — Driven Qubit
A periodically driven two-level system: H(t+T)=H(t) → quasi-energies, Rabi oscillations, Mathieu stability tongues. Toggle between Bloch sphere, stroboscopic map, and stability diagram.
Kitaev Honeycomb Model
Interactive honeycomb lattice spin model with anyonic excitations. Tune Jx, Jy, Jz couplings and watch the phase diagram shift between gapped Abelian and gapless non-Abelian phases.
Elliptic Curve Cryptography
Visualize elliptic curves y²=x³+ax+b. Click to place points, watch point addition and scalar multiplication kP unfold geometrically — the foundation of modern public-key cryptography.
Brusselator Reaction-Diffusion
Two-species Brusselator system spontaneously forms Turing patterns via diffusion-driven instability. Tune A, B parameters and diffusion coefficients to explore spots, stripes, and chaos.
Abelian Sandpile Model
Click to drop grains — when a cell reaches 4 it topples, triggering avalanches with fractal structure. Self-organized criticality in action: power-law avalanche size distribution.
SIR Network Epidemics
Stochastic SIR epidemic spreading on a random network. Tune transmission rate β, recovery rate γ, and watch R₀ determine whether the infection spreads or fades out.
Langton's Ant
Simple local rules, emergent global order. After ~10,000 steps of apparent chaos, the ant spontaneously builds an infinite periodic highway. Multiple multi-color rule variants.
Wavelet Transform
Draw a signal and watch its Haar wavelet decomposition across 5 scales — approximation and detail coefficients visualized. Threshold sliders show compression and reconstruction.
Lyapunov Spectrum
Bifurcation diagram and Lyapunov exponent of the logistic map, computed side by side. Red = chaos (λ>0), blue = stable. Period-doubling route to chaos with Feigenbaum constant.
Dirac Magnetic Monopole
3D visualization of a Dirac monopole's radial magnetic field. Drag to rotate, tune the charge g — includes the Dirac string and the quantization condition eg = nℏc/2.
Quasiperiodic Fibonacci Tiling
Cut-and-project from a 2D square lattice at an irrational angle generates the Fibonacci quasicrystal. Animate the cut angle to see the transition between periodic and quasiperiodic.
Landauer's Principle
Bit erasure costs kT·ln2 energy — information is physical. Langevin particles in a double-well potential, erase to |0⟩ and watch entropy production and heat dissipation accumulate.
Berry Phase & Geometric Phase
A quantum spin traces a closed loop on the Bloch sphere, acquiring a geometric phase equal to half the solid angle enclosed — purely topological, independent of speed.
Kolmogorov Energy Cascade
Energy injected at large scales cascades down to the Kolmogorov microscale η, following E(k) ~ k⁻⁵/³. Animated vortex hierarchy and live log-log energy spectrum.
Diffusion-Limited Aggregation
Particles random-walk until they stick to the growing cluster, creating fractal crystal structures with dimension D_f ≈ 1.71. Rainbow-colored by arrival time.
Boids Flocking Simulation
Craig Reynolds' 1987 emergent flocking model — separation, alignment, and cohesion rules produce complex collective behavior from 120+ agents with no global coordination.
Maxwell's Demon
The demon sorts fast and slow particles across a partition, seemingly violating the 2nd law. Landauer's resolution: erasing the demon's memory costs kT·ln2 per bit.
Wigner Quasi-Probability
The Wigner function W(x,p) is the quantum phase-space distribution — it can go negative, signaling non-classical states. Coherent, Fock, cat, and squeezed states.
Tesseract: 4D Hypercube
A 4-dimensional hypercube rotating in all six independent planes simultaneously, projected to your screen. 16 vertices, 32 edges — control each rotation axis independently.
Heat Equation Diffusion
Draw any temperature profile and watch it diffuse via u_t = α u_xx. Live Fourier mode decomposition shows high-frequency modes decaying fastest — a spatial low-pass filter.
Fractal Flames (IFS)
Iterated Function Systems build fractals one random point at a time. Sierpinski triangle, Barnsley fern, dragon curve — density-mapped with logarithmic tone mapping.
Goldbach's Conjecture
Every even integer > 2 is the sum of two primes — verified to 4×10¹⁸, never proven. Arc diagram of all Goldbach pairs plus the Goldbach comet showing pair-count density.
Debye Model of Solids
Quantized lattice vibrations (phonons) explain solid heat capacity. Phonon density of states, C_V(T) from quantum T³ behavior to the classical Dulong-Petit limit.
Percolation & Giant Component
Bond percolation on a 2D grid — add bonds one-by-one, watch the giant cluster emerge at p_c ≈ 0.5, live cluster-size histogram and Kesten threshold crossing.
Restricted Boltzmann Machine
Visualize an RBM learning patterns via Contrastive Divergence — node activations, weight heatmap, Gibbs sampling, energy landscape. Sliders for learning rate and hidden units.
Geodesic Dome Explorer
Interactive 3D geodesic dome (frequency 1–5), icosahedral subdivision, Euler's V−E+F=2 verified live. Drag to rotate, color faces by apex distance.
SEIR Age-Structured Epidemic
Two-age-group SEIR epidemic model with a contact matrix. Stacked time-series for each compartment, sliders for β, σ, γ, mixing, and vaccination rate.
Möbius Strip & Non-Orientable Surfaces
Animated 3D Möbius strip, Klein bottle, and torus — drag to rotate, animate a normal vector flipping to prove non-orientability, wireframe mode.
String Resonance Modes
Vibrating string with standing wave modes n=1–8, click to pluck at any point, Fourier decomposition histogram, wave equation physics with damping and tension sliders.
2D Navier-Stokes Flow
Stam's stable fluids — drag to inject velocity, vorticity/velocity/density colormaps, vorticity confinement, circular obstacle toggle, viscosity and diffusion sliders.
Stochastic Differential Equations
Ensemble of SDE trajectories — Geometric Brownian Motion, Ornstein-Uhlenbeck, CIR. Mean ± std band, probability density evolution, drift and volatility sliders.
Random Matrix Theory
GOE, GUE, Wishart, and Poisson ensembles — eigenvalue distribution vs Wigner semicircle law, level-spacing histogram vs Wigner surmise, accumulate statistics across samples.
Forest Fire & Self-Organized Criticality
Drossel-Schwabl forest fire model — trees grow, lightning strikes, fires cascade. Log-log fire-size distribution reveals the power law of self-organized criticality.
Galaxy Collision N-Body
Two galaxies collide in a softened-gravity N-body simulation — tidal tails, merger dynamics, stars colored by galaxy of origin. Spin, separation, and velocity sliders.
Lissajous Figures
Parametric curves x=A·sin(aθ+δ), y=B·sin(bθ) — frequency ratio and phase sliders, fade trails in neon colors, rational vs irrational ratio behavior.
Bounded Confidence Opinions
Deffuant model: agents interact only when |opinions| < ε — watch polarization into clusters, opinion trajectory plot, live histogram, convergence threshold.
Chaos Synchronization
Two coupled Lorenz systems synchronize above a coupling threshold — time series, synchronization error, phase portrait, Pecora-Carroll transition.
Hadley Cell Circulation
Earth's atmospheric circulation cells animated on a globe cross-section — Hadley, Ferrel, and Polar cells, trade winds, ITCZ, hover for wind explanations.
Knot Theory Explorer
Trefoil, figure-eight, cinquefoil and more — Jones polynomial, Alexander polynomial, crossing number, chirality, animated Reidemeister moves I/II/III.
Turing Machine
Step-by-step Turing machine with animated tape and transition table — increment binary, palindrome checker, 3-state and 4-state Busy Beaver programs.
Wallpaper Symmetry Groups
All 17 crystallographic wallpaper groups as animated tilings — p1 through p6mm, unit cell overlay, scale and animation controls.
Autocorrelation & Time Series
White noise, AR(1/2), MA(1), ARMA, fractional Brownian motion, 1/f noise — signal, ACF, PACF, and power spectral density with parameter sliders.
Ising Model (Metropolis)
2D Ising model with Metropolis algorithm — watch the phase transition at T_c ≈ 2.269, live magnetization and energy plots, susceptibility divergence.
Gravitational Lensing
Einstein ring formation and multiple image lensing — drag the source, adjust lens mass, watch magnification and ring geometry change in real time.
Bragg Diffraction
2d·sin(θ)=nλ visualized — X-ray beams on crystal planes, constructive vs destructive interference, diffraction pattern intensity vs angle, d-spacing and wavelength sliders.
Mandelbulb 3D Fractal
Ray-marched cross-sections of the 3D Mandelbulb — the true 3D Mandelbrot set, colored by escape time
Wolfram 1D Cellular Automata
All 256 elementary rules — watch simple local rules generate complex global patterns including Rule 110 (Turing-complete)
Coupled Oscillators & Normal Modes
N masses on springs — compute and animate normal modes, frequency spectrum, and modal superposition
Quantum Dot Energy Levels
Particle in a 2D box — wavefunctions, quantized energy levels, and animated quantum superposition
Zipf's Law
Rank-frequency power laws in words, cities, earthquakes, and income — log-log plots with best-fit slope
Ice Crystal Growth
Hexagonal snowflake formation via the Reiter cellular automaton — hexagonal symmetry emerging from local diffusion rules
Network Motifs
Feed-forward loops, feedback, bi-fan, and more — animated signal propagation through recurring biological circuit patterns
Kepler's Laws & Solar System
Animated inner solar system with elliptical orbits, equal-area sweeps, and T²∝a³ verified in real time
Entropy of Mixing
Remove the wall between two gas chambers and watch Shannon entropy climb toward maximum — Boltzmann's H-theorem live
Rossler Attractor
Chaotic 3D trajectory with rotating projection and xy/xz/yz views — parameters a, b, c control the chaos
Higgs Mechanism & Symmetry Breaking
Ball rolling in the Mexican hat potential V = -mu^2|phi|^2 + lambda|phi|^4 — spontaneous symmetry breaking visualized
Hénon Map
Strange attractor from iterating xₙ₊₁ = 1 − axₙ² + yₙ — tune a and b to watch chaos emerge from order
Plasma Oscillations
1D jellium model — displace electrons from equilibrium and watch Langmuir waves at ωₚ = √(ne²/ε₀m)
Magnons & Spin Waves
Spin chain with precessing magnetic moments — dispersion ω(k), group vs phase velocity, FM and AFM coupling
Taylor Series Visualizer
Polynomial approximations of sin, cos, exp, ln and more — watch convergence radius and partial sums up to order 15
BCS Superconductivity
Cooper pairs, energy gap Δ(T), coherence peaks in the quasiparticle DOS — the gap vanishes at Tc
Morse Code
Type text, see the waveform, hear it play — animated dots and dashes with Web Audio API beeps
Benford's Law
Leading digit frequencies in powers of 2, Fibonacci, primes and more — the logarithmic law P(d) = log₁₀(1+1/d)
Ammann–Beenker Tiling
Aperiodic 8-fold symmetric tiling via substitution — squares and 45° rhombuses, silver ratio inflation
Ant Pheromone Trails
Stigmergic foraging — ants lay pheromone on food paths, trails self-reinforce, shortest routes emerge
Necklace Combinatorics
Count distinct necklaces via Burnside's lemma — visualize all rotational equivalence classes for n beads, k colors
Pitchfork Bifurcation
Symmetry-breaking bifurcation — watch the potential well split and fixed points appear as r sweeps through zero
Riemannian Manifold Curvature
Visualize geodesics on positively and negatively curved surfaces and how curvature bends straight lines.
Bernoulli Site Percolation
Watch the sharp phase transition at p_c ≈ 0.5927 as connected clusters suddenly span the lattice.
Chebyshev Polynomials & Runge's Phenomenon
See why equally-spaced interpolation nodes cause edge oscillations and how Chebyshev nodes fix it.
Vicsek Flocking Model
Tune noise to drive the collective motion phase transition in this landmark active matter model.
Crystal Symmetry & Space Groups
Explore all 17 wallpaper groups — the complete classification of 2D crystallographic symmetries.
Hamiltonian Cycle Visualization
Search for cycles visiting every vertex exactly once using backtracking on interactively built graphs.
Maxwell-Boltzmann Speed Distribution
Compare molecular speed distributions across gases and temperatures with the three characteristic speeds.
Penrose Tiling Generator
Generate aperiodic P2 and P3 Penrose tilings via inflation subdivision with 5-fold quasicrystal symmetry.
Brownian Bridge Process
Simulate conditioned Brownian motion pinned at both endpoints, the stochastic foundation of the KS test.
Galton Board / Quincunx
Drop balls through a peg board and watch the binomial distribution emerge toward a Gaussian in real time.
Conway's Game of Life 3D
Classic cellular automaton extended into three dimensions — drag to rotate and watch 3D life patterns evolve.
Fractional Calculus Visualizer
Explore the Grünwald-Letnikov fractional derivative D^α, interpolating smoothly between integration and differentiation.
Boids with Obstacle Avoidance
Reynolds flocking (separation, alignment, cohesion) extended with potential-field obstacle avoidance. Click to place obstacles.
Knot Invariants & Trefoil
3D tube rendering of classical knots with their Jones and Alexander polynomials, genus, and determinant.
Sphere Packing & Kepler
Visualize FCC, HCP, SC and BCC sphere packings — explore Kepler's conjecture, proved by Hales in 1998.
Spatial SIR Epidemic Model
Agent-based epidemic simulation with spatial spread — watch infection waves and herd immunity emerge in real time.
Airy Functions & Waveguide
Visualize Ai(z) and Bi(z), WKB turning points, quantum waveguide eigenstates, and the Tracy-Widom connection.
Riemann Sphere & Möbius
See how Möbius transforms map circles to circles on the Riemann sphere — live dual-panel complex plane visualization.
Ising Model Metropolis
2D ferromagnetic Ising model with Metropolis sampling — watch the phase transition at Onsager's critical temperature.
Diffusion-Limited Aggregation
Random-walk particles aggregate into fractal clusters with Hausdorff dimension ≈ 1.71 — models lightning, dendrites, coral.
Winding Number & Complex Analysis
Draw closed curves and watch them wind around poles — Cauchy integral formula and residue theorem made visual.
Hopf Fibration
S³ → S² — every fiber is a circle, every two fibers link exactly once. Stereographic projection into 3D.
Langton's Ant & Turmites
Emergent highways and chaos from local turning rules on a grid. Explore RL, figure-8s, and custom rule strings.
Kaprekar's Constant (6174)
Any 4-digit number reaches 6174 in at most 7 steps. Heatmap of step counts across all 9000 valid numbers.
Collatz Conjecture
3n+1 if odd, n/2 if even — does it always reach 1? Trace sequences, compare stopping times, explore the inverse tree.
Quantum Harmonic Oscillator
Hermite polynomial wavefunctions ψₙ(x), energy ladders, probability densities, and quantum superpositions.
Polar Rose Curves
Rhodonea r = cos(kθ) — k odd gives k petals, k even gives 2k. Fractional k reveals intricate layered symmetry.
2D Navier-Stokes Vorticity
Vorticity-streamfunction simulation — inject vortex pairs and watch them dance, merge, and dissipate.
Clifford Torus 4D Projection
A flat torus embedded in S³ ⊂ ℝ⁴ — stereographically projected into 3D and animated through 4D rotations.
Boolean Satisfiability (SAT)
Build 3-SAT formulas, assign variables manually, or watch DPLL search the solution tree. NP-complete, visually.
Laplace Equation Solver
Draw boundary conditions and watch the electrostatic potential field emerge via iterative relaxation. ∇²V = 0, solved live.
SPH Fluid Simulation
Smoothed Particle Hydrodynamics — pressure, viscosity, and gravity drive hundreds of particles. Click to splash.
Conic Sections Visualizer
Ellipse, parabola, hyperbola — explore foci, eccentricity, asymptotes, and directrices interactively.
Newton Fractal
Complex root-finding via Newton's method paints fractal basins of attraction. Zoom into infinite detail.
Metabolic Scaling Laws
Kleiber's Law: B ∝ M^(3/4) from mouse to whale. Adjust the exponent and see how life scales across 12 orders of magnitude.
Abelian Sandpile Toppling
Self-organized criticality in action — drop grains, trigger avalanches, and watch fractal patterns emerge at the edge of chaos.
DNA Codon Table
Explore the full genetic code: all 64 codons mapped to 20 amino acids. Click any codon for details, or paste a sequence to translate.
Lorenz Attractor
Deterministic chaos — drag to rotate the strange attractor in 3D and tune σ, ρ, β to reshape its wings.
Spherical Harmonics
Visualize Y_l^m(θ,φ) — the angular eigenfunctions of the Laplacian. Select l and m up to degree 5, rotate in 3D.
2D Phase Portrait & Nullclines
Vector fields, nullclines, and live trajectories for Van der Pol, Lotka-Volterra, Duffing, pendulum, and Hopf systems.
Tensor Field Visualizer
Stress and strain tensors rendered as ellipse glyphs — semi-axes show principal directions and eigenvalues. Animated fields with custom modes.
Riemann Zeta Animation
Sweep through the critical strip and watch ζ(s) spiral in the output plane. Non-trivial zeros appear as the spiral crosses the origin.
Graph Coloring
Compare greedy and backtracking chromatic coloring on random graphs. Step through the backtracking search or watch it animate live.
Autocorrelation & PSD
Wiener–Khinchin theorem in action — watch the autocorrelation and power spectral density update live for sine, chirp, noise, and mixed signals.
Interactive Fourier Transform
Draw any waveform and see its frequency spectrum instantly. Toggle individual harmonics on/off to reconstruct signals from partial spectra.
Geodesic Voronoi on a Sphere
Voronoi cells on a sphere using great-circle distances. Drag to rotate, randomize sites, or place Platonic solid vertices for symmetric partitions.
Parabolic Reflector & Ray Optics
Ray tracing through a parabolic mirror — parallel rays focus perfectly, point sources diverge. Adjust focal length and aperture live.
Mobius Strip
Parametric Mobius strip with adjustable twist count and width. Toggle wireframe, watch an ant traverse the single surface, drag to rotate in 3D.
Poisson Process Visualizer
Timeline arrivals, exponential inter-arrival histograms, Poisson count distributions, and 2D spatial Poisson — all updating live with adjustable rate lambda.
Betti Numbers & Simplicial Homology
Build simplicial complexes by clicking — compute beta-0 (components), beta-1 (loops), beta-2 (voids) via GF(2) boundary matrices. Vietoris-Rips filtration included.
LQR Optimal Control
Stabilize an inverted pendulum on a cart with a Linear Quadratic Regulator — tune Q and R weights and watch the cost-optimal feedback law in real time.
Manifold Learning & PCA
Rotate a 3D Swiss roll, sphere, torus, or helix and see how PCA projects it — revealing where linear dimensionality reduction succeeds and fails.
Quantum Circuit Simulator
Build single- and two-qubit circuits with H, X, Y, Z, S, T, Rx/Ry/Rz, and CNOT gates; watch Bloch spheres evolve and measure state probabilities.
Morse Theory: Critical Points
Sweep a height-slice across monkey saddles, tori, and volcanoes — minima, saddles, and maxima are detected live and the Euler characteristic χ is verified.
Fibonacci Spirals & Phyllotaxis
The golden angle 137.5° packs seeds most efficiently — drag the divergence slider away from φ² and watch the spiral arms collapse into spokes.
Bayes' Theorem Visualizer
Unit-square area diagrams make the base-rate fallacy visceral: a 99% accurate test for a 1% disease gives only ~17% posterior — adjust all three parameters live.
1D Heat Equation Solvers
Compare explicit FTCS and Crank-Nicolson solvers side by side — push the stability parameter r above 0.5 and watch the explicit scheme blow up while CN stays stable.
Markov Chain Visualizer
Edit the transition matrix, run the chain, and watch empirical frequencies converge to the stationary distribution π computed by power iteration.
PCA & Eigenvalue Decomposition
Click to place points or choose a dataset, then watch the covariance matrix, eigenspectrum, and principal component axes update live in 2D.
Schrödinger Particle in a Box
Mix quantum eigenstates, watch the wavefunction breathe as each mode rotates at its own energy frequency, and explore phase coloring and probability density.
Game Theory Payoff Matrix
Interactive payoff matrix — find pure and mixed Nash equilibria, dominant strategies, and best-response diagrams.
Chemical Reaction Network Dynamics
ODE simulation of CRNs with mass-action kinetics — Brusselator oscillator, Oregonator, toggle switch, Michaelis-Menten enzyme.
Nonlinear Regression & Curve Fitting
Levenberg-Marquardt fitting of sigmoid, Gaussian, Hill, Gompertz, and other models — click to add data, watch the fit update live.
Topological Phase: Toric Code
Kitaev's toric code on an interactive lattice — flip qubits, observe e/m anyonic syndromes, and apply MWPM error correction.
Euler Equations: Ideal Fluid Flow
2D incompressible Euler equations via spectral vorticity method — vortex dynamics, shear instability, streamlines and velocity fields.
Prime Number Theorem Visualizer
Explore π(x) ~ x/ln(x): prime counting, gap distribution, Ulam spiral, Li(x) approximation, and Riemann zeta zeros.
Spontaneous Symmetry Breaking (φ⁴)
Mexican hat potential, order parameter dynamics, Higgs mechanism, and phase transitions in φ⁴ field theory.
Random Forest Visualizer
Train a random forest on 2D datasets — visualize decision boundaries, individual trees, feature importance, and ensemble voting.
Numerical Integration Methods
Compare Riemann, Trapezoidal, Simpson's, and Gauss-Legendre quadrature — visualize error convergence rates on a log-log plot.
Complex Dynamics: Newton Basins
Newton's method in the complex plane — fractal basins of attraction for polynomial roots, with zoomable Julia-like boundaries.
Double Pendulum Bifurcation
Chaos and sensitivity in coupled pendula
Viral Infection Dynamics
SIR model spreading across a cellular grid
Predator-Prey Lattice
Spatial Lotka-Volterra population spirals
Escape Velocity Simulator
Orbital mechanics and planetary escape
Ising Magnetization
Phase transitions in a 2D spin lattice
Kuramoto Synchrony
Coupled oscillators self-organizing into phase lock
Elastic Collision Billiards
Energy-conserving billiard ball collisions
Reaction-Diffusion Glycolysis
Selkov model spiral waves and Turing patterns
Tidal Locking Evolution
A moon's rotation evolving toward tidal lock
Sandpile Model Avalanche
Self-organized criticality and power-law cascades
Mandelbrot–Julia Explorer
Live Julia sets as you explore the Mandelbrot boundary
Traffic Flow (LWR Model)
Backward-propagating shockwaves on a ring road
Bénard Convection Rolls
Thermal convection patterns from bottom heating
Solar System N-Body
Gravitational n-body simulation of our solar system
Reynolds Flocking (Boids)
Emergent flocking from separation, alignment, cohesion
Brusselator Reaction-Diffusion
Oscillating Turing patterns from the Brusselator model
Plasma Wave Simulation
Langmuir electron density waves in 1D plasma
Morse Potential Oscillator
Anharmonic oscillator showing dissociation threshold
Epidemic on Scale-Free Network
SIR spreading on a Barabási-Albert network
Diffusion & Entropy Growth
H-theorem: entropy increases as a packet diffuses
Josephson Junction Chaos
Poincaré sections of a driven Josephson junction
2D Sphere Packing
Random sequential adsorption approaching theoretical limits
Binary Star System
Two-body gravity with Lagrange points and test particles
Golden Ratio Spirals
Fibonacci, phyllotaxis, and golden angle patterns
Euler Fluid Vortices
Point vortex dynamics in 2D inviscid flow
Lightning Tree (DBM)
Dielectric breakdown model growing fractal lightning
Stadium Billiard
Ergodic mixing and Poincaré maps in a Bunimovich stadium
2D Drum Resonance Modes
Standing wave patterns and nodal lines of a vibrating membrane
Wilson-Cowan Neural Oscillator
Excitatory-inhibitory neural population dynamics
Archimedean & Log Spirals
Exploring Archimedean, logarithmic, Fermat's, and hyperbolic spirals
Hodgepodge Cellular Automata
Spiral patterns from Dewdney's excitable medium
Hall Effect Simulation
E×B drift and Hall voltage in crossed fields
Newton Fractal (4 Roots)
Convergence basins of Newton's method for z⁴=1
Sierpiński Carpet
Self-similar fractal with dimension log8/log3 ≈ 1.893
Firefly Synchronization
Kuramoto coupling driving fireflies to unison blinking
Quantum Well Bound States
Schrödinger equation eigenstates in a finite square well
Rayleigh-Taylor Instability
Mushroom fingers from heavy fluid atop light fluid
Möbius Strip Topology
Single-sided surfaces and topological curiosities
Penrose Tiling Inflation
Aperiodic tiling by kite-and-dart substitution rules
Galton Board
Binomial distribution emerging from random peg bounces
Hopf Bifurcation
Fixed point to limit cycle as a parameter crosses zero
Hydra Regeneration (Gierer-Meinhardt)
Activator-inhibitor Turing patterns explaining morphogenesis
Quantum Kicked Rotor
Standard map phase portraits from integrable to chaotic
Miura-Ori Origami Fold
Single-DOF Miura fold unfolding in 3D animation
Cyclic Cellular Automaton
Rotating spiral waves from Fisch-Gravner-Griffeath rules
Pendulum Clock Escapement
Animated anchor escapement mechanism tick-tock
Ant Mill Death Spiral
Army ants trapped in a pheromone-guided circular mill
Gravity Wave Ripples
2D wave interference from multiple point sources
FitzHugh-Nagumo Spirals
Spiral waves from excitable reaction-diffusion dynamics
Hodgkin-Huxley Neuron
Action potential spikes and ion channel gating dynamics
Turbulent Pipe Flow
Reynolds number visualization with laminar-to-turbulent transition, velocity profiles, and space-time chaos plots.
Stochastic SEIR Epidemic
Individual-based epidemic simulation on a grid with tunable R₀, incubation period, and real-time SEIR time series.
Boids Emergent Flocking
Craig Reynolds' three-rule flocking model — separation, alignment, cohesion — with predators and mouse scatter.
Double Pendulum Chaos
Two pendulums from nearly identical initial conditions diverge exponentially, visualizing the Lyapunov exponent.
Diffusion-Limited Aggregation
Random walkers stick to a seed to form a fractal cluster with Hausdorff dimension ≈ 1.71.
Kuramoto-Sivashinsky PDE
Spatiotemporal chaos in 1D — the KS equation governs flame fronts and thin films with a beautiful space-time scroll.
Ginzburg-Landau Vortex
Complex order parameter dynamics in a 2D superconductor — nucleate vortices and watch them annihilate.
Lenia Continuous CA
Bert Chan's smooth generalisation of Conway's Life — lifelike gliders emerge from a kernel convolution and bell growth function.
Turing Morphogenesis
Gierer-Meinhardt reaction-diffusion produces spots, stripes, and labyrinths — the math behind leopard coats and fish patterns.
Hodgepodge Machine
Martin-Odlyzko-Wolfram discrete CA generates rotating spiral waves — a cellular Belousov-Zhabotinsky reaction.
Langton's Ant Highway
Multiple ants with different rulesets emerge from chaos into periodic highways — a beautiful cellular automaton surprise.
Schrödinger Potential Well
Time-dependent wavepackets bounce through infinite wells, harmonic oscillators, and double-wells via split-operator method.
Navier-Stokes Lid-Driven Cavity
2D incompressible flow in a lid-driven cavity — Lattice Boltzmann simulation with streamlines and vorticity visualization.
Airy Function & Diffraction
Airy functions Ai(z) and Bi(z), single/double-slit interference, and the Fresnel-to-Fraunhofer diffraction transition.
Belousov-Zhabotinsky Scroll Waves
Oregonator model drives stunning spiral chemical waves — click to seed new spirals in this excitable reaction-diffusion medium.
Barchan Dune Formation
Wind-driven saltation, avalanche, and shadow zones sculpt crescent barchan dunes from a flat sandy bed.
Dendritic Solidification
Phase-field model grows snowflake-like crystal dendrites driven by anisotropy and thermal diffusion.
Topological Charge Defects
XY model spin field Monte Carlo — vortices and antivortices annihilate near the Kosterlitz-Thouless transition at T≈0.89.
Hamiltonian Chaos Billiard
Bunimovich stadium billiard transitions from integrable to fully chaotic — Poincaré section reveals the KAM torus breakdown.
Cellular Neural Oscillator
FitzHugh-Nagumo excitable neurons on a 2D lattice fire action potentials, spiral waves, and synchronization fronts.
Turbulence Energy Spectrum
Kolmogorov K41 cascade: E(k)~k⁻⁵/³ inertial subrange with synthetic multi-scale eddy field and vorticity visualization.
Vicsek Model Flocking
Self-propelled particles align with neighbors — watch the phase transition from disordered motion to coherent flocking as noise decreases.
Spinodal Decomposition
Cahn-Hilliard equation drives phase separation and coarsening dynamics: two immiscible phases spontaneously nucleate and grow.
Stokes & Oseen Flow
Creeping flow around a sphere — Stokes drag F=6πμaU with Oseen correction, pressure and vorticity field visualization.
3D Ising Model
Metropolis Monte Carlo on a 20³ lattice through the critical temperature T_c≈4.51 J/k_B with live magnetization and energy plots.
SIR Epidemic on Network
SIR spreading on Erdős-Rényi, scale-free, and small-world networks — R₀ threshold, herd immunity, and real-time SIR curves.
BTW Sandpile — SOC
Bak-Tang-Wiesenfeld sandpile self-organizes to criticality: avalanche sizes follow a power law P(s)~s⁻¹·² without tuning.
3D Quantum Harmonic Oscillator
Probability density |ψ_{nx,ny,nz}|² as a 3D point cloud — explore Hermite-Gaussian wavefunctions with quantum number controls.
Pair Creation & Annihilation
Vacuum fluctuations spawn particle-antiparticle pairs that annihilate into photons with Feynman diagram overlays.
Lorenz–Rössler Coupling
Two chaotic attractors coupled — tune ε to watch generalized synchronization emerge between the Lorenz butterfly and Rössler spiral.
Bouncing Ball Map
Period-doubling route to chaos in a bouncing ball under gravity — bifurcation diagram, phase portrait, and cobweb iteration map.
Chaotic Waterwheel
Malkus waterwheel with filling and draining buckets spontaneously reverses rotation — the physical incarnation of the Lorenz attractor.
Fluvial Erosion Landscape
River networks emerge from noise via stream-power-law erosion, divide migration, and drainage basin competition on a digital elevation model.
Polymer Chain Brownian Dynamics
Freely-jointed FENE chain in a thermal bath — watch Rouse-mode fluctuations, end-to-end distance statistics, and the random-coil scaling law.
Neural Field Wilson–Cowan
Excitatory-inhibitory neural field equations produce traveling waves, spiral waves, and Turing instabilities — click to stimulate.
Rician Fading Channel
Jakes multipath fading model with Rician K-factor, Doppler shifts, signal constellation, and BER vs SNR curves for QPSK and QAM.
Minority Game Market
El Farol Bar problem: adaptive agents using memory-based strategies self-organize toward the efficiency edge — emergent volatility clustering.
Three-Species Food Chain
Hastings-Powell model of chaotic prey-predator-top-predator dynamics — torus breakdown, phase portraits, and extinction thresholds.
Reaction Front Propagation
Fisher-KPP equation: pulled fronts travel at c*=2√(Dr), Allee effects push the front, and heterogeneous landscapes fracture invasions.
Fractal Basin Boundaries
Newton's method on degree-4 polynomials paints the complex plane with fractal boundaries between root-attraction basins — zoom to infinite detail.
Elastic Wave Phonon Dispersion
1D diatomic crystal lattice with acoustic and optical phonon branches — watch atom displacements, zone folding, and the band gap open at the Brillouin zone boundary.
DNA Transcription Dynamics
Stochastic gene expression with mRNA bursting, transcription factor binding, and protein-level fluctuations — Fano factor measures noise above Poisson.
Opinion Formation — Bounded Confidence
Deffuant model: agents with continuous opinions converge only when they agree within threshold ε, producing polarization or consensus depending on tolerance.
Sandstone Porosity Percolation
Correlated 2D porous rock with invasion percolation: fluid always enters the widest pore throat, tracing fractal breakthrough paths near the critical porosity.
SSH Model — Topological Edge States
Su-Schrieffer-Heeger chain with alternating hoppings: winding number ν flips at t₁=t₂, spawning zero-energy edge states protected by topology.
Climate Energy Balance Model
Budyko-Sellers 1D EBM with ice-albedo feedback: hysteresis loop between warm and Snowball Earth states, CO₂ forcing, and bifurcation diagram.
Hyperion Chaotic Tumbling
Saturn's moon Hyperion tumbles chaotically due to gravitational torque on its elongated shape — Poincaré sections reveal the chaotic sea and Lyapunov exponent.
Crystal Dislocation Glide
Edge dislocations glide through a crystal lattice driven by applied shear stress, resisted by the Peierls barrier and viscous drag — Hall-Petch effect visualized.
Atmospheric Jet Stream
Barotropic vorticity equation on a β-plane: Rossby waves propagate westward, jet stream meanders, and blocking events trap stationary high-pressure systems.
Thermal Convection — Mantle Dynamics
2D Rayleigh-Bénard convection with temperature-dependent viscosity: mantle plumes rise from the hot core, Nusselt number tracks convective heat flux.
Cell Chemotaxis
Keller-Segel model of cell migration up chemical gradients: receptor binding, run-and-tumble motility, and collective streaming toward chemoattractant sources.
Spiking Neural Network — STDP
Leaky integrate-and-fire neurons with spike-timing-dependent plasticity: Hebbian learning, raster plots, and the emergence of synchronization from noise.
Market Microstructure — Order Book
Live limit order book simulation with Poisson arrivals, price impact, bid-ask spread dynamics, and Kyle's lambda market depth.
Quantum Error Correction — Surface Code
Surface code stabilizer formalism: Z and X syndrome measurements reveal errors without collapsing qubit state, demonstrating the threshold theorem.
ASEP — Driven Diffusive System
Asymmetric simple exclusion process with three exact phases — low density, high density, maximum current — and real-time phase diagram tracking.
Gravitational Wave Ringdown
Binary black hole merger waveform from post-Newtonian inspiral chirp through merger to quasi-normal mode ringdown with frequency spectrogram.
Red Queen Coevolution
Host-parasite matching alleles model: frequency-dependent selection drives perpetual cycling, diversity maintenance, and the evolutionary advantage of sex.
Miura-ori Origami Mechanics
Rigid origami kinematics of the Miura-ori fold: single degree of freedom, auxetic negative Poisson ratio, deployable structure animation in 3D.
Arctic Sea Ice Thermodynamics
Stefan problem for sea ice growth and melt: albedo feedback, seasonal solar forcing, ocean heat flux, and ice-free Arctic bifurcation under CO2 warming.
Lipid Bilayer Phase Transition
2D Ising-like Monte Carlo model of lipid membranes: gel-to-liquid phase transition, raft formation, cholesterol effects, and temperature-dependent diffusion.
Variational Monte Carlo
VMC for the hydrogen atom: optimize a trial wavefunction via Metropolis sampling, watch local energy converge to the exact -0.5 Hartree ground state as α→1.
Active Nematics
Topological defects in active liquid crystals: ±½ disclinations nucleate, annihilate, and drive activity-powered turbulence in a Frank elasticity field.
Granular Gas Clustering
Inelastic hard-sphere gas with tunable restitution coefficient: watch the clustering instability emerge, track non-Maxwellian velocity distributions, and Haff's cooling law.
Yard-Sale Wealth Model
Random multiplicative exchange between agents spontaneously generates Pareto wealth concentration; adjust redistribution rate and watch the Gini coefficient and Lorenz curve respond.
Crystallization & Nucleation
Classical nucleation theory in action: supercritical nuclei form, grow dendritic arms, and impinge — follow Avrami kinetics X=1−exp(−kt^n) in real time.
Birdsong Learning Circuit
HVC-RA-LMAN neural circuit model of zebra finch song learning: LMAN-driven exploration, dopaminergic reinforcement, critical-period closure, and deafening effects.
Tropical Cyclone — WISHE
Emanuel's wind-induced surface heat exchange model of rapid intensification: warm-core development, potential intensity ceiling, and vertical shear suppression.
Quantum Dot Confinement
Particle-in-a-box quantization in 1D/2D/3D: tune dot size to shift energy levels, watch the absorption peak sweep the visible spectrum, and see size-dependent color.
Brownian Ratchet Motor
Flashing and tilting asymmetric ratchet potentials rectify thermal noise into directed particle flux — explore how temperature, asymmetry, and load approach the Carnot limit.
Forest Fire — Self-Organized Criticality
Drossel-Schwabl cellular automaton self-tunes to criticality: fire sizes follow a power law P(s)~s^{-1.5}, fractal burn patterns emerge without tuning any parameter.
L-System Plant Growth
Interactive fractal plant generator using Lindenmayer systems — edit axioms and rewriting rules to grow ferns, trees, and dragon curves with turtle graphics.
Mean-Field Ising & Spinodal
Watch ferromagnetic phase transitions unfold: the Ising model on a 2D lattice with live free-energy landscape and spinodal decomposition when quenched below Tc.
Multifractal Spectrum f(α)
Compute the singularity spectrum of binomial multifractals, logistic map chaos, and turbulence-like measures — visualizing the Legendre transform f(α) = qα − τ(q).
Navier-Stokes Lid-Driven Cavity
2D viscous flow in a square box with a moving lid, solved via vorticity-streamfunction — watch primary and corner vortices form as Reynolds number increases.
Network Resilience: Attacks
Compare random vs targeted hub-removal on Barabási-Albert scale-free networks — the giant component collapses catastrophically under targeted attack but is robust to random failure.
Kirkwood Gaps & Resonances
Animated asteroid belt simulation showing Jupiter's mean-motion resonances carving Kirkwood gaps at 3:1, 5:2, 7:3, and 2:1 commensurabilities.
Sierpiński Arrowhead Curve
Animated L-system construction of the Sierpiński arrowhead curve — a space-filling path that converges to the Sierpiński triangle with Hausdorff dimension log3/log2.
Parametric Surface Explorer
Explore 3D parametric surfaces — torus, Klein bottle, Boy's surface, Enneper minimal surface, trefoil knot tube — with drag-to-rotate and live parameter controls.
Diffusion-Limited Erosion
Random walkers eat away at a solid block in DLE mode or aggregate in DLA mode — both produce fractal structures with dimension ≈1.71 through strikingly different mechanisms.
Hyperbolic Tilings — Poincaré Disk
{p,q} tessellations of the hyperbolic plane rendered in the Poincaré disk via geodesic arcs and Möbius transforms — infinitely many tiles packed into a finite circle.
Voronoi Crystal Growth
Nucleation sites grow into polyhedral grains via Voronoi tessellation, simulating the polycrystalline microstructure of metals and ceramics.
Fourier Transform Visualizer
Draw any waveform and watch its frequency spectrum emerge in real time — includes square, sawtooth, AM, and noise presets with harmonic reconstruction.
Hamiltonian Chaos — Standard Map
The Chirikov-Taylor map reveals KAM tori, resonance islands, and the onset of global chaos as the perturbation parameter k crosses the critical threshold.
Information Bottleneck
Visualize the fundamental tradeoff between compressing X into T while preserving information about Y, tracing the IB curve with adjustable β parameter.
Knot Invariant Explorer
Spin 3D knot diagrams for trefoil, figure-eight, and torus knots while computing crossing number, writhe, genus, and Alexander and Jones polynomials.
Fibonacci Phyllotaxis
The golden angle spiral growth model for sunflower seeds and plant spirals — drag the divergence angle away from 137.5° to see why irrational numbers pack best.
Phase Space Reconstruction
Takens' theorem in action: recover the full attractor geometry from a single 1D time series using delay embedding with Lorenz, Rössler, and logistic map systems.
Turing Completeness — Rule 110
Rule 110 cellular automaton, proven Turing-complete by Matthew Cook in 2004 — explore all 256 elementary rules and watch gliders interact like logic gates.
Geodesic Sphere Builder
Project icosahedron subdivisions onto a sphere with interactive frequency controls, drag-to-rotate, and wireframe/solid/stress views revealing geometric distortion.
Feynman's Brownian Ratchet
Feynman's famous thermodynamics thought experiment: see why a ratchet-pawl cannot rectify Brownian motion at equilibrium, and what happens when T₁ ≠ T₂.
Ultrametric Tree Explorer
Hierarchical clustering via ultrametric spaces and p-adic trees: every triangle is isosceles, distances are heights of lowest common ancestors.
Weakly Interacting Bose Gas
Bogoliubov dispersion: watch free-particle parabola transform into linear phonon regime and roton minimum as interactions are tuned.
Birth-Death Process
Stochastic population dynamics with extinction probability, stationary geometric distribution, and animated sample paths through state space.
Devil's Staircase
The Cantor function: continuous, non-decreasing, yet derivative zero almost everywhere — it rises entirely on a set of measure zero.
Catastrophe Theory: Fold & Cusp
Thom's cusp catastrophe with bifurcation set, hysteresis loops, and interactive parameter sweep revealing bistability and sudden jumps.
Chern-Simons Knot Invariants
Wilson loops, linking numbers, and framing: visualize trefoil, figure-eight, Hopf link, and Borromean rings with Jones polynomial annotations.
Ostwald Ripening
Lifshitz-Slyozov coarsening: larger droplets grow as smaller ones dissolve, with mean radius growing as t^(1/3) and evolving size distribution.
Continued Fraction Explorer
Expand any real number into [a₀; a₁, a₂, ...], watch convergents approach exponentially, and trace the path through the Stern-Brocot tree.
Convex Hull Algorithms
Animated Jarvis march (gift wrapping) and Graham scan side by side: add points, choose distribution, watch the hull wrap step by step.
De Bruijn Sequence Generator
Every k-mer appears exactly once: explore the de Bruijn graph, animate the Eulerian circuit, and see applications from DNA sequencing to LFSRs.
Dyson Brownian Motion
Eigenvalue repulsion in random matrices: watch GUE/GOE particles repel logarithmically and compare empirical level spacing to the Wigner surmise.
Elastic Network Model
Protein normal modes from a Cα contact network: animate low-frequency collective motions and predict crystallographic B-factors with ENM/NMA.
Entropic Gravity (Verlinde)
Gravity as an emergent entropic force: explore holographic screens, Unruh temperature, and how Newton's law arises from information entropy.
Ergodic Billiards
Bunimovich stadium vs circle billiard: chaotic trajectories fill phase space ergodically while integrable systems trace closed curves forever.
TASEP
Totally asymmetric exclusion process: particles on a 1D lattice self-organize into low-density, high-density, and maximum-current phases.
2D Finite Element Stress
FEM stress analysis on a triangulated plate: apply loads, watch the von Mises stress colormap and deformation field update in real time.
Free Energy Landscape
Protein folding funnel: Langevin walkers navigate a 2D energy surface with metastable traps, barriers, and transition-state escape rates.
Gauge Field Visualization
U(1) lattice gauge theory: Metropolis Monte Carlo updates link phases, measuring Wilson loops, plaquette action, and magnetic flux quantization.
Hamming Code SECDED
Interactive Hamming(7,4) encoder: flip bits to inject errors, watch the syndrome pinpoint and correct single errors while detecting double errors.
Hodge Decomposition
Split any 2D vector field into curl-free, divergence-free, and harmonic parts — draw your own field or choose from fluid-dynamics presets.
Hyperboloid Model
Hyperbolic geometry realized on the upper sheet of a Minkowski hyperboloid — geodesics, isometries, and projection to the Poincaré disk.
Frustrated Ising Model
Antiferromagnetic Ising spins on a triangular lattice: geometric frustration creates an extensively degenerate spin-liquid ground state.
Maxwell-Boltzmann Gas
Hard-disk molecular dynamics thermalizes to the Maxwell-Boltzmann speed distribution — watch equipartition emerge from elastic collisions.
Kolmogorov-Smirnov Test
Visualize the KS statistic as the maximum gap between the empirical and theoretical CDF, with live p-value and two-sample comparison.
Lambda Calculus Reducer
Parse and β-reduce lambda expressions step by step — Church numerals, Y combinator, normal vs applicative order reduction.
Van der Pol Limit Cycle
Nonlinear damping drives every trajectory onto a unique stable limit cycle — phase portrait, Poincaré-Bendixson, and relaxation oscillations.
Lorenz Attractor 3D
Interactive 3D strange attractor with drag-to-rotate, parameter sliders, and two nearby trajectories diverging exponentially — deterministic chaos.
Mach-Zehnder Interferometer
Quantum path interference in a two-path interferometer: adjust phase, toggle which-way information, and watch complementarity destroy the fringes.
Markov Chain Mixing Time
Watch total variation distance decay to zero as a Markov chain converges — spectral gap, coupling time, and bottleneck chains visualized.
SO(3) Rotation Group
Euler angles, axis-angle, and quaternion representations of 3D rotations on a live spinning cube — plus gimbal lock demonstration.
Mean-Shift Clustering
Animated gradient ascent on a kernel density estimate — particles climb toward modes as bandwidth controls cluster granularity.
DNA Melting Curve
Nearest-neighbor thermodynamics predicts the sigmoidal helix-coil transition — tune GC content, salt, and length to shift Tm.
Hodgkin-Huxley Neuron
Real-time action potential simulation with voltage-gated Na⁺/K⁺ channel kinetics, refractory period, and firing threshold.
Metabolic Flux Analysis
Flux balance analysis on a simplified glycolysis/TCA network — stoichiometric constraints, optimal growth, arrow widths scale with flux.
Minimal Surfaces
Rotate and explore catenoid, helicoid, Scherk, Costa, and Enneper surfaces — mean curvature zero everywhere, soap-film geometry.
Modular Forms & j-Invariant
Phase portrait of the j-function over the upper half-plane — Eisenstein series, modular symmetry, and monstrous moonshine.
Critical Ising Model
Metropolis Monte Carlo at Tc reveals scale-free clusters, critical slowing down, and the second-order phase transition in real time.
Morse Theory Landscape
Height functions expose topology — count minima, saddles, and maxima on torus, sphere, and genus-2 surfaces to verify χ = M − S + m.
Mutual Information & Channel Capacity
Joint distribution heatmap and I(X;Y) curves for BSC, BEC, and AWGN channels — tune noise and input bias to approach Shannon capacity.
Nematic Liquid Crystal Defects
Click to place ±½ topological defects in a director field — Frank elastic energy, winding numbers, and defect annihilation dynamics.
Nonequilibrium Fluctuation Theorem
Jarzynski equality and Crooks theorem: sample work distributions from irreversible processes and recover free energy differences exactly.
Bounded Confidence Opinion Dynamics
Deffuant and Hegselmann-Krause models: watch society fragment into clusters or converge to consensus as the confidence threshold ε varies.
Optimal Stopping — Secretary Problem
The 1/e rule: skip the first N/e candidates to calibrate, then hire the next best — animated sequence with Monte Carlo success rates.
p-adic Number Arithmetic
Explore the p-adic norm, ultrametric inequality, and Ostrowski's theorem: every non-trivial absolute value on ℚ is either real or p-adic.
Sphere Packing & Kepler Conjecture
FCC, HCP, BCC, and random packings in 3D: compare densities and see why Hales's 1998 proof of π/(3√2) ≈ 74.05% required a computer.
Paramagnetism & Curie Law
Brillouin function B_J(x) for spin-J moments: Curie susceptibility χ=C/T, saturation magnetization, and the crossover from linear to saturated response.
Penrose Aperiodic Tiling
Kite-dart and rhombus tilings with 5-fold symmetry: inflation/deflation rules, fat-to-thin ratio converging to φ, and non-periodic coverage.
Persistent Homology Filtration
Vietoris-Rips filtration on a point cloud: watch connected components merge and loops form, tracked in barcode and persistence diagrams.
Potts Model Phase Transition
q-state generalization of Ising: q≤4 gives continuous transitions, q≥5 gives discontinuous ones — live Monte Carlo on a 2D lattice.
Phonon Dispersion — 1D Chain
Monatomic and diatomic chains: acoustic vs optical branches, Brillouin zone, group vs phase velocity, and animated normal modes.
Abelian Sandpile Model
Bak-Tang-Wiesenfeld self-organized criticality — drop grains and watch avalanching topples carve fractal patterns.
Belousov-Zhabotinsky Reaction
Spiral chemical waves emerge from a simple three-state reaction-diffusion automaton far from equilibrium.
Tropical Curve Amoeba
The amoeba of an algebraic curve under the Log map, alongside its piecewise-linear tropical limit skeleton.
Random Walk & Winding Number
A 2D lattice random walk colored by how many times the path winds around the origin — topology emerging from randomness.
Ramanujan Tau Function
The mysterious τ(n) — Fourier coefficients of the Ramanujan Δ-function, with Deligne's bound and complex Fourier visualization.
Chess on the Hyperbolic Plane
Chess pieces move along geodesics in the Poincaré disk — where parallel lines diverge and the board tiles infinitely.
Euler's Elastica
The curve minimizing bending energy ∫κ² ds — Euler's 1744 elastic rod solution animated across all modulus types via RK4.
Fibonacci Word Substitution
The fixed point of σ: 0→01, 1→0 grows into an infinite word whose turtle-walk traces a self-similar fractal curve.
Classical Hall Effect
Charged particles in crossed E and B fields follow cycloidal paths until the transverse Hall voltage balances the Lorentz force.
3D Lissajous Figures
Parametric space curves (cos at, cos bt, cos ct) — drag to rotate, tune frequencies and phases, watch rational ratios close.
p-adic Number Explorer
Visualize p-adic expansions of rationals, the ultrametric norm, and the fractal tree structure of the p-adic integers Z_p.
3D Ising Model Monte Carlo
Metropolis algorithm on a 3D lattice — sweep temperature through the ferromagnetic phase transition at T_c ≈ 4.51.
Fourier Epicycle Drawing
Draw any closed curve and watch the DFT decompose it into rotating epicycles that reconstruct the path in real time.
Algebraic K-Theory: K0
Visualize the Grothendieck group K0 from vector bundles over S², with rank and Chern class as the two generators of K0(S²) ≅ ℤ².
Optical Caustics
Rays reflecting off curved mirrors form nephroid and cardioid caustics — the luminous envelope tangent to every reflected ray.
Tropical Algebra
Min-plus semiring arithmetic: tropical matrix powers equal all-pairs shortest paths, making Floyd-Warshall a matrix exponentiation.
Kirchhoff Network Flow
Interactive resistor network solved by nodal analysis — drag nodes, edit resistances, and watch KVL/KCL currents update live.
Geodesic Sphere
Subdivide an icosahedron's 20 triangular faces n² times and project to sphere — Fuller's geodesic principle with live Euler characteristic.
Quantum Walk on 1D Lattice
Coin operator plus shift spreads probability ballistically as O(t) — quadratically faster than classical random walk O(√t).
Modular Origami Crease Patterns
Mountain/valley crease patterns for Sonobe, PHiZZ, and Waterbomb units with fold animation and assembly diagrams.
HP Lattice Protein Folding
Fold hydrophobic-polar sequences on a 2D square lattice to minimize energy — watch hill-climbing find compact structures via pivot moves.
Lid-Driven Cavity Flow
Finite-difference Navier-Stokes in a square cavity with a sliding lid — streamlines, vorticity, and pressure fields at tunable Reynolds number.
Penrose Tiling with Defects
P3 rhomb Penrose tiling with intentional matching-rule violations highlighted in red — explore local disorder in quasicrystalline order.
Continued Fraction Gauss Map
Iterate T(x)={1/x} to extract CF coefficients and see the Gauss-Kuzmin invariant measure emerge — compare phi, e, pi, and sqrt(2).
Fermat Spiral Phyllotaxis
Sunflower seed packing at the golden angle reveals Fibonacci spiral arms — nudge the angle to watch the pattern collapse.
Josephus Problem
Animate n people in a circle with every k-th eliminated — watch the survivor emerge and verify the closed-form formula.
Chromatic Polynomial
Draw any graph and compute its chromatic polynomial via deletion-contraction — then color it and detect conflicts in real time.
Cosmic Web Filaments
2D N-body gravity simulation shows how tiny density fluctuations collapse into filaments, voids, and dark-matter halos.
Duffing Oscillator Chaos
Explore the forced double-well oscillator's strange attractor, Poincare section, and period-doubling route to chaos.
Particle in a Box
Watch Gaussian wave packets evolve under the time-dependent Schrodinger equation — eigenstates, superpositions, and quantum revivals.
Quaternion Rotation Visualizer
Explore unit quaternions as points on S³, project to 3D rotations, and animate smooth SLERP interpolation.
Game of Life Rule Explorer
Tune Birth/Survival rules (B3/S23 and beyond), seed classic patterns, and watch emergent behaviors unfold.
Penrose Tiling Inflation
Watch P3 rhombus tiles subdivide via the golden ratio into infinitely complex aperiodic patterns.
Free Energy Landscape
Navigate metastable wells and transition states with Langevin particles on a tunable 2D potential surface.
Phonon Spectral Density
Compute phonon dispersion and density of states for 1D chains and 2D lattices, revealing van Hove singularities.
Zero-Crossing Rate & Audio
Generate tones and noise, compute ZCR, RMS, and spectral features, and visualize audio characteristics live.
Tensor Network Contraction
Visualize MPS bond dimensions, step through sequential index contractions, and watch SVD truncation in action.
Arnold Conjecture & Floer Theory
Find fixed points of Hamiltonian diffeomorphisms on the torus and trace Morse-Floer gradient flow lines.
Lagrangian Coherent Structures
Compute FTLE fields for the double-gyre flow and reveal invisible transport barriers that govern fluid mixing.
Hypergraph Coloring
Build hypergraphs interactively, run greedy coloring, and test Property B (2-colorability) with brute force.
Kauffman NK Fitness Landscape
Explore rugged epistatic fitness landscapes where K random gene interactions tune the number of local optima.
Quantum Wavefunction Collapse
Click to measure a superposition of Gaussian wavepackets and watch the Born-rule collapse in real time.
Riemannian Geodesics
Shoot geodesics on sphere, torus, and paraboloid — ODE integration reveals conjugate points and focal sets.
Hyperbolic Tilings H²
{p,q} tessellations of the Poincaré disk where (p−2)(q−2)>4 — infinitely many congruent tiles near the boundary.
Chladni Figures
Vibrational eigenmodes of rectangular and circular membranes — sand gathers at nodal lines of zero displacement.
Quantum Diode I–V Curve
Shockley equation with tunneling correction — explore forward bias, Zener breakdown, and differential resistance.
Crystallographic Point Groups
All 32 crystallographic point groups in 3D — symmetry operations, mirror planes, rotation axes, and inversion centers.
Fibonacci Sphere Lattice
Distribute N points uniformly on a sphere using the golden angle — nature's optimal phyllotaxis packing.
Spin Glass & Frustration
Random FM/AFM bonds on a triangular lattice create geometrically frustrated plaquettes and a glassy energy landscape.
Topological Charge Density
XY model vortices and antivortices — compute winding numbers on each plaquette and visualize the Kosterlitz-Thouless transition.
Power Voronoi Diagrams
Weighted Voronoi tessellations where each site's influence radius is modulated by a weight — click to place sites and watch power cells form.
Apollonian Circle Packing
Recursive tangent circles descend to fractal scales via Descartes' theorem (k₁+k₂+k₃+k₄)² = 2(k₁²+k₂²+k₃²+k₄²).
Fano Plane
The smallest projective plane — 7 points, 7 lines, symmetry group of order 168, with connections to octonions and Hamming codes.
Network Robustness Under Attack
Compare how scale-free, random, and lattice networks survive targeted versus random node removal — watch the giant component collapse.
Morse Theory Gradient Flow
2D height functions with minima, maxima, and saddle points — gradient flow lines, level sets, and topological handle decomposition.
Cornu Spiral & Fresnel Diffraction
The Euler spiral traces Fresnel integral phasors — watch how the clothoid encodes diffraction patterns from single slits and edges.
Graph Spectra
Compute and compare adjacency, Laplacian, and normalized Laplacian eigenvalue spectra for cycles, complete graphs, scale-free networks, and more.
Kuramoto Synchronization
N coupled oscillators with Lorentzian frequency spread — animate the sudden onset of synchronization as coupling K crosses the critical threshold.
Newton's Cradle
Elastic collision simulation demonstrating how momentum and energy conservation force the N-balls-in N-balls-out rule with realistic pendulum physics.
t-SNE Embedding
Watch t-SNE iteratively unfold high-dimensional Gaussian clusters into 2D — heavy-tailed t-distributions prevent the crowding that defeats PCA.
Optical Tweezer Trap
Gaussian beam gradient forces trap a dielectric particle — adjust beam power and watch Brownian motion compete with the trapping potential.
Catalan Number Lattice Paths
Visualize all Cₙ Dyck paths from (0,0) to (n,n) that never cross the diagonal — animate through every valid ballot sequence.
Ramsey Graph Coloring
Paint edges of K₆ red or blue and discover why R(3,3)=6 makes a monochromatic triangle unavoidable — an interactive proof by pigeonhole.
Wilsonian Renormalization
Integrate out high-momentum modes in φ⁴ theory and watch coupling constants flow toward fixed points in the Wilson RG phase portrait.
Ryu-Takayanagi Holographic Entropy
Drag boundary endpoints in AdS to move the minimal geodesic and see how entanglement entropy equals bulk area divided by 4G.
Neural Hopf Oscillator
Cross the Hopf bifurcation from fixed point to limit cycle, then drive the oscillator to see frequency entrainment and Arnold tongues emerge.
Z₂ Lattice Gauge Theory
Run Monte Carlo on a 2D Z₂ gauge lattice and watch the confinement-deconfinement transition as β crosses the critical coupling β_c ≈ 0.44.
Dehn Surgery on 3-Manifolds
Choose p/q surgery on the unknot, trefoil, or figure-8 knot and identify the resulting 3-manifold — lens spaces, Poincaré sphere, and more.
Turbulent Energy Cascade
GOY shell model sends energy from large to small scales, dynamically revealing Kolmogorov's k⁻⁵/³ inertial-range spectrum in real time.
Carbon Nanotube Dispersion
Select (n,m) chirality and watch zone folding of graphene's Brillouin zone determine whether your nanotube is metallic or semiconducting.
Moduli Space of Tori
Explore the fundamental domain of SL(2,ℤ) in the upper half-plane and watch τ shape the torus lattice.
Steane [7,1,3] Code
Visualize the 7-qubit quantum error-correcting code, inject errors, measure syndromes, and correct them.
Čech Nerve & Topology
Cover a point cloud with balls and watch the nerve simplicial complex reveal loops and connected components.
Epidemic Network Percolation
SIR epidemic spreading on a random network — the epidemic threshold is a bond percolation transition.
Acoustic Phonon Thermal Conductivity
Debye model phonon transport — compare materials, tune Landau levels, and see umklapp scattering suppress κ.
Graph Homomorphism Densities
Explore Razborov's flag algebra inequalities — triangle density vs edge density with the Kruskal-Katona bound.
Poincaré Map — Billiards
Trace billiard trajectories in circular, elliptical, and stadium tables — KAM tori vs chaotic orbits revealed.
Liouville Integrability
Toda lattice with N conserved quantities — watch the Lax pair isospectral flow trace invariant tori in phase space.
Morse Code & Entropy
Compare Morse, Huffman, and fixed-length codes against Shannon entropy for any message you type.
Quantum Hall Plateaus
Integer QHE — σ_xy = n·e²/h quantization, Landau levels, disorder, and animated chiral edge states.
Prime Gap Statistics
Distribution of gaps between primes and the Cramér conjecture
Riemann Surface
Branch surfaces of log(z) and sqrt(z) as 3D meshes
Double Pendulum Chaos
Butterfly effect: diverging trajectories from nearby initial conditions
Voter Model Consensus
Opinion dynamics on a 2D grid approaching consensus
Lagrange Points
Stability of L1-L5 in the restricted three-body problem
Rényi Entropy Spectrum
Multifractal spectrum via generalized entropy dimensions
Gauss-Kuzmin Distribution
Statistical distribution of continued fraction coefficients
Tensor Network MPS
Matrix product states and quantum entanglement visualization
Schnakenberg Patterns
Reaction-diffusion Turing patterns forming spots and stripes
Ramanujan Graphs
Spectral gap and eigenvalue distribution of optimal expanders
Spin Ice Frustration
Frustrated pyrochlore lattice with ice rules and monopole excitations
Laughlin Wavefunction
Fractional quantum Hall effect at ν=1/3 and composite fermions
Kolmogorov Cascade
Turbulent energy cascade and the k⁻⁵/³ inertial range spectrum
Persistent Homology
Vietoris-Rips filtration tracking topological features across scales
Flocking & Topology
Vicsek model flocking with topological defects in the velocity field
Hyperbolic Geodesics
Poincaré disk geodesics, triangles, and ideal tessellations
Mean-Field Ising
Phase diagram, order parameter, and free energy landscape near T_c
Euler Buckling
Elastic beam buckling, bifurcation diagram, and post-buckling shapes
Swift-Hohenberg Patterns
Stripes, hexagons, and labyrinthine patterns from a simple PDE
Adiabatic Evolution
Berry phase, adiabatic following, and Landau-Zener transitions
Kuramoto Chimeras
Coexisting synchronized and chaotic oscillator populations on a ring
Hyperbolic Knot Complements
Thurston's theorem: most knot complements carry hyperbolic geometry
Morse Theory & Floer
Gradient flows, critical points, and handle decomposition on surfaces
Loss Landscape
SGD and Adam navigating saddle points and basins of attraction
Opinion Coevolution
Adaptive network dynamics: edge rewiring drives polarization
Weak Wave Turbulence
Resonant triads and the Zakharov kinetic equation for waves
Chern Insulator
Berry curvature in k-space and topological Chern number of bands
RG Decimation
Real-space block-spin renormalization on the 1D Ising chain
Life Variants
Conway's Life, HighLife, Day&Night, Seeds — rule explorer
Gravitational Ringdown
Black hole quasinormal modes and gravitational wave chirp spectrogram
Logistic Bifurcation Diagram
Watch period-doubling cascades and the onset of chaos as you zoom into the logistic map's infinite self-similar structure.
SEIR Epidemic Waves
Animate epidemic curves and phase portraits for the SEIR model with tunable transmission, incubation, and recovery rates.
Möbius Band & Monodromy
Parallel-transport a fiber around the Möbius band's base loop and watch it return flipped — topology made visceral.
Power Iteration Eigenvector
Watch a random vector rotate toward the dominant eigenvector iteration by iteration, with live convergence tracking.
Padé vs Taylor Approximants
Compare how rational Padé approximants outrun Taylor series far from the expansion point, especially near poles.
Symplectic Euler Integrator
Standard Euler bleeds energy while symplectic Euler conserves it forever — same order, radically different long-term fate.
Mie Scattering & Sky Color
Explore how particle size and wavelength determine scattering efficiency — the physics behind blue skies and red sunsets.
Quantum Wavepacket Oscillator
A coherent Gaussian wavepacket oscillates in a quantum harmonic potential, its probability density sloshing like a classical particle.
SSH Topological Edge States
Toggle between trivial and topological phases of the SSH chain to see protected zero-energy edge states appear in the bulk gap.
Kuramoto Synchronization
N coupled oscillators spontaneously synchronize above a critical coupling K_c — a phase transition from chaos to collective rhythm.
Spin Glass: Frustrated Ising
Monte Carlo simulation of a frustrated Ising model on a triangular lattice with random ±J couplings — watch the energy landscape trap spins in disordered glass states.
Aharonov-Bohm Interference
Electron double-slit interference shifts as magnetic flux through a solenoid increases — even though B=0 outside, the topology of the vector potential is real.
Hyperbolic Tessellation
Navigate infinite {p,q} tilings in the Poincaré disk where every tile is geometrically identical yet the whole fits inside a circle.
Quantum Kicked Rotor
Classical chaos spreads momentum diffusively, but quantum mechanics halts it — Anderson localization in momentum space from a simple periodically kicked rotor.
BTW Sandpile & SOC
Add grains to a self-organized critical sandpile and watch avalanches cascade across all scales, building a power-law size distribution without any tuning.
Zero Sound in Fermi Liquid
Landau's Fermi liquid theory predicts a collisionless zero-sound mode that travels faster than ordinary sound — visualize both dispersion relations and density waves.
Alexandrov Topology on Posets
Finite posets are secretly topological spaces — click nodes in the Hasse diagram to reveal their open sets and explore the specialization order.
Pólya Urn Reinforcement
Watch reinforcement learning in its purest form: each draw begets more of the same, and the limiting fraction converges to a Beta-distributed random variable.
Hyperbolic Growth Singularity
When growth rate scales faster than linear, solutions blow up in finite time — compare the Malthusian, logistic, and hyperbolic hierarchy of growth laws.
Wigner Surmise & Level Spacing
Generate random matrix ensembles and watch eigenvalue spacings converge to the Wigner-Dyson distribution — a fingerprint of quantum chaos and level repulsion.
Stochastic Resonance
Noise helps: watch a weak subthreshold signal become detectable as SNR peaks at optimal noise amplitude.
Wavefunction Collapse
Click to measure a quantum superposition and watch Born-rule statistics emerge from repeated collapse events.
Tidal Locking
Watch a moon's spin slow under tidal torque until it locks into 1:1 resonance, always showing the same face.
Brusselator Limit Cycle
A chemical oscillator's phase portrait spirals onto a stable limit cycle as B crosses the Hopf bifurcation threshold.
Cayley Graph Explorer
See group structure made visible — colour-coded edges trace each generator's action across Z₆, D₄, Q₈, A₄ and more.
Newton Fractal Basins
Newton's method in ℂ carves the plane into fractal basins of attraction, coloured by which root each point converges to.
Jamming Transition
Pack 2D disks toward φ_c ≈ 0.84 and watch pressure diverge, contact number hit Z→4, and force chains crystallise.
Boids Flocking
Three simple rules — separation, alignment, cohesion — give rise to emergent flocking, splitting, and schooling.
Schelling Segregation
A mild 37% preference for similar neighbours is enough to drive dramatic large-scale residential segregation.
Free Probability
Add two random matrices and watch the empirical eigenvalue spectrum converge to the free convolution of semicircles.
El Farol Minority Game
N agents choose bar or home each week — those in the minority win, driving oscillating attendance and emergent strategy evolution.
Spring Network Rigidity
Watch a random bond network cross the rigidity percolation threshold as the Maxwell counting rule Z=2d is crossed.
Saddle Point Approximation
Visualize how e^{Nf(z)} concentrates at a saddle as N grows, deriving Stirling's formula from first principles.
KdV Soliton Collision
Two KdV solitons pass through each other completely unchanged — only a phase shift betrays the elastic collision.
Magnetic Pendulum Fractals
Release a pendulum over three magnets and watch fractal basin boundaries emerge as nearby trajectories diverge wildly.
Entropy Production
Animate free expansion vs piston compression and watch dS/dt ≥ 0 enforced in real time by the second law.
Riemann Zeta Zeros Spiral
Trace ζ(½+it) as a parametric curve in the complex plane and watch it pierce the origin at each non-trivial zero.
Langevin Brownian Dynamics
Overdamped particles in a harmonic trap build up a Boltzmann distribution, demonstrating equipartition and MSD ~ t.
Self-Avoiding Walk Polymer
Animate polymer growth on a lattice and measure the Flory exponent ν≈3/4 swelling past the ν=1/2 random walk baseline.
Persistence Diagrams TDA
Grow a Vietoris-Rips filtration on point clouds and watch H₀ components and H₁ loops born and die in the persistence diagram.
Gerchberg-Saxton Phase Retrieval
Watch iterative phase retrieval converge between image and Fourier domains as the GS algorithm recovers amplitude from intensity constraints.
Arctic Circle Theorem
Random domino tilings of the Aztec diamond reveal frozen corners and a disordered bulk bounded by an inscribed arctic circle.
Network SIR Epidemic
Animate infection spreading through Watts-Strogatz or Barabási-Albert networks and see how topology controls the epidemic threshold.
Lévy Stable Distributions
Explore α-stable PDFs with heavy tails and the generalized central limit theorem — Gaussian and Cauchy are special cases.
q-Analog Binomial Coefficients
Deform Pascal's triangle with a parameter q: Gaussian binomials count subspaces over finite fields and encode partition generating functions.
Active Matter MIPS
Self-propelled particles with run-and-tumble dynamics spontaneously cluster via motility-induced phase separation despite purely repulsive forces.
Kolmogorov Complexity Estimator
Approximate K(x) via LZ78 compression and compare random, structured, and natural strings on the algorithmic complexity spectrum.
Brachistochrone Race
Race beads down straight, parabolic, and cycloidal curves — the cycloid wins every time, as Euler-Lagrange theory demands.
Heat Kernel Spectral Geometry
Diffuse heat through graph Laplacian eigenmodes and hear the shape of a graph encoded in its eigenvalue spectrum.
Wallpaper Symmetry Groups
Tile the plane with all 17 crystallographic wallpaper groups — the complete classification of 2D periodic symmetry proved in 1891.
Langton's Ant
Watch chaotic early wandering give way to a perfectly ordered emergent "highway" after ~10,000 steps — with multi-ant and custom rule support.
Network Motifs: Feedforward Loops
Animate signal propagation through feedforward, bi-fan, and bi-parallel gene regulatory network motifs and see their sign-sensitive delay properties.
Stochastic Block Model
Generate networks with tunable community structure and watch spectral clustering recover the hidden communities with modularity Q scoring.
Zitterbewegung
Visualize the rapid trembling motion of a relativistic electron wavepacket at 2mc² frequency — the quantum interference between positive and negative energy components.
Beta-Skeleton & Gabriel Graph
Drag β from 0 to 3 and watch the graph continuously morph from a complete graph through the Gabriel graph and relative neighborhood graph.
Kleiber's Law: Metabolic Scaling
Explore the ¾-power law spanning 27 orders of magnitude of body mass, with the West-Brown-Enquist fractal vascular network derivation.
Detrended Fluctuation Analysis
Apply DFA to white noise, 1/f noise, and random walks to extract long-range correlation exponents from the log-log fluctuation scaling plot.
Hypercube & Gray Code
Animate a Hamiltonian path through the n-dimensional hypercube via Gray code — each step flips exactly one bit — projected from 4D and 5D.
Sphere Eversion
Animate turning a sphere inside out without tearing or creasing, passing through a Boy's surface-like halfway stage in this topological impossibility made possible.
Wilsonian Renormalization
Watch effective coupling constants flow under successive block-spin transformations of the Ising model, converging to fixed points that classify phase transitions.
Maximum Entropy Principle
Discover how entropy maximization under constraints uniquely selects exponential, Gaussian, and uniform distributions via Lagrange multipliers.
Chebyshev Approximation
Visualize the equioscillation theorem, Runge's phenomenon, and why Chebyshev nodes produce near-optimal polynomial approximations.
Dynamic Mode Decomposition
Extract spatiotemporal DMD modes from a synthetic vortex-shedding flow and see each frequency component as a distinct spatial pattern.
Penrose Inflation
Animate the golden-ratio deflation rules on kite-dart tiles to reveal the aperiodic 5-fold symmetry of Penrose tilings.
2D Drum Membrane
Click to pluck a finite-difference drum and watch standing waves, normal modes, and Chladni nodal figures emerge in real time.
Network Attack & Percolation
Compare targeted hub attacks versus random failures on scale-free and random networks, watching the giant component collapse.
Tropical Geometry
Explore the min-plus algebra of tropical polynomial curves as piecewise-linear varieties dual to Newton polygon subdivisions.
Lorentz Transformation
Animate Lorentz boosts on a Minkowski diagram and watch simultaneity shatter, time dilate, and lengths contract geometrically.
SAT Phase Transition
Probe the sharp satisfiability threshold of random 3-SAT at alpha near 4.267 where instances become hardest for DPLL solvers.
Mean-Field Ising Model
Solve the self-consistency equation graphically and watch spontaneous magnetization emerge as temperature drops below T_c.
Competitive Exclusion
Watch N-species Lotka-Volterra competition unfold via nullcline analysis and phase portraits, revealing when coexistence is possible and when one species inevitably wins.
Heisenberg Uncertainty Principle
Demonstrate Δx·Δk ≥ ½ interactively with Gaussian wavepackets, rectangle windows, and double-slit signals, watching the time-frequency trade-off in a live spectrogram.
Morse Theory & Gradient Flow
Animate gradient descent flow lines on a torus, sphere, and saddle surface, visualizing how critical points and their Morse indices sum to the Euler characteristic.
Network Synchronization
Explore the master stability function for coupled oscillator networks, showing how the Laplacian eigenvalue ratio governs whether synchrony is achievable.
Non-Equilibrium Steady State
Watch overdamped Brownian particles drift through a tilted washboard potential, revealing the steady-state probability current and stall force of a NESS heat engine.
Quasicrystal Diffraction
Compute the Fourier transform of Penrose tiling atom positions to reveal sharp Bragg peaks with 10-fold symmetry forbidden in any periodic crystal.
Delaunay Triangulation
Place and drag points to watch Graham scan build the convex hull, Bowyer-Watson compute the Delaunay triangulation, and circumcircles verify the empty-circle property.
Spectral Clustering
Apply normalized graph cuts to moons, rings, and spiral datasets by embedding them in Laplacian eigenvector space where k-means succeeds where Euclidean distance fails.
Continued Fraction Clock
Watch convergents of π, e, φ, and √2 step ever closer on the number line, with a live clock expressing the current time as a continued fraction.
Gravitational Slingshot
Design a gravity-assist flyby and watch a spacecraft steal momentum from a planet, gaining speed in the heliocentric frame via the Oberth effect.
Information Cascades
Bayesian agents observe private signals and public history, revealing how rational herding locks in early mistakes and makes group decisions fragile.
Contact Tracing Dynamics
SIR spreading on a network with quarantine shows how the tracing fraction p separates epidemic suppression from uncontrolled spread.
Gödel & Provability Logic
Explore GL modal frames, the provability operator □, and the fixed-point sentence G ↔ ¬□G at the heart of Gödel's incompleteness theorem.
Parametric Resonance
A pendulum with time-varying stiffness obeys the Mathieu equation — watch Arnold tongues emerge and amplitude explode inside unstable regions.
Arrow's Impossibility Theorem
Construct three-candidate elections to see how any voting rule must sacrifice IIA, Pareto efficiency, or non-dictatorship — no exceptions.
Boltzmann Machine Energy
A restricted Boltzmann machine learns patterns via Gibbs sampling — visualize the energy landscape, weight matrix, and contrastive divergence.
Vortex Reconnection
Two anti-parallel vortex tubes exchange topology in a reconnection event, radiating Kelvin waves through the vorticity field.
Kerr Effect & Optical Solitons
Split-step Fourier simulation of the nonlinear Schrödinger equation shows self-phase modulation, spectral broadening, and soliton formation.
Ant Colony Stigmergy
Pheromone-following ants spontaneously discover shortest paths between nest and food through evaporation-driven positive feedback.
Epidemic Threshold & Spectra
The SIS epidemic threshold β_c = δ/λ_max(A) reveals why scale-free networks are far more vulnerable than random or small-world topologies.
TQFT & Jones Polynomial
Visualize the Jones polynomial via Kauffman bracket skein relations for trefoil, figure-8, and Hopf link, with step-by-step q-deformation.
Protein Folding Energy Landscape
Explore the folding funnel, Levinthal paradox, and chaperone effects through an animated energy landscape and HP lattice model simulation.
Complex Ginzburg-Landau Solitons
Simulate dissipative solitons, spiral waves, and defect turbulence in the 2D complex Ginzburg-Landau equation across all Benjamin-Feir regimes.
Wilson-Cowan Neural Field
Watch traveling waves, Turing patterns, and gamma oscillations emerge from excitatory-inhibitory neural field dynamics with tunable coupling.
Adaptive Dynamics & ESS
Visualize evolutionary invasion fitness, pairwise invasibility plots, and evolutionary branching through logistic competition adaptive dynamics.
2D Conformal Bootstrap
Explore Virasoro algebra representations, Kac table operators, and crossing symmetry constraints for Ising, tricritical, and Potts minimal models.
FEM Heat Conduction
Click to set boundary conditions on a triangulated 2D domain and solve the Poisson equation via finite elements with live stiffness matrix sparsity.
Cellular Potts Model
Watch cells sort by differential adhesion in the Graner-Glazier CPM, demonstrating Steinberg's differential adhesion hypothesis through Monte Carlo dynamics.
Multifractal Spectrum f(α)
Compute the singularity spectrum of turbulent and synthetic multifractal signals, revealing intermittency through the width of the f(α) curve.
Birth-Death Process / M/M/1 Queue
Animate Poisson arrivals and exponential service in an M/M/1 queue, verifying Little's law L=λW and watching P(n) converge to the geometric distribution.
Effective Reproduction Number R(t)
Estimate the epidemic reproduction number from case data using the Cori method, watching R(t) cross the critical threshold of 1 as interventions take effect.
Geodesic on Ellipsoid
Numerically integrate geodesic equations on an oblate spheroid, revealing how geodesics precess and accumulate longitude advance compared to great circles on a sphere.
Symmetry Breaking & Goldstone Theorem
Explore the Mexican hat potential showing how spontaneous U(1) symmetry breaking produces a massless Goldstone boson and a massive Higgs mode with distinct dispersion relations.
Optimal Transport & Wasserstein Distance
Visualize the Wasserstein distance between 1D distributions via monotone rearrangement, and watch the Sinkhorn algorithm converge on the optimal discrete coupling matrix.
Hopfield Network Pattern Completion
Store letter patterns as attractors in a Hopfield associative memory, add noise, and watch asynchronous dynamics converge while energy decreases toward the stored pattern.
2D Turbulence Inverse Cascade
Simulate forced 2D Navier-Stokes and observe the dual cascade: energy accumulating at large scales with a k⁻⁵⁄³ spectrum while enstrophy flows to small scales as k⁻³.
Shannon Channel Capacity
Interactively explore C = B log₂(1+SNR) for AWGN, BSC, and BEC channels, visualizing the channel coding theorem's sharp boundary between achievable and impossible rates.
Differential Privacy & Laplace Mechanism
See how the Laplace mechanism adds calibrated noise to satisfy ε-differential privacy, trading accuracy for privacy budget across sequential query composition.
Complex Contagion Threshold Model
Watch social cascades on random networks where adoption requires a threshold fraction of neighbors, revealing the vulnerable window in the threshold-degree phase diagram.
Airy Function & Caustics
Visualize the Airy function bridging oscillatory and evanescent WKB regions at a turning point, and see how it governs the intensity pattern of optical caustics and rainbow arcs.
Fractional Calculus & Anomalous Diffusion
Explore fractional derivatives D^α f and how subdiffusion (MSD ~ t^α, α<1) and Lévy walks arise from power-law waiting-time distributions in disordered media.
Ising Model Transfer Matrix
Solve the 1D Ising model exactly via transfer matrix eigenvalues, visualize the partition function Z = Tr(T^N), correlation length, and Onsager's 2D exact solution.
Knapsack Dynamic Programming
Watch the DP table fill cell-by-cell as the 0/1 knapsack is solved, then trace the optimal item selection and contrast exponential brute force against polynomial DP scaling.
Quantum Many-Body Scars
Animate the anomalous revivals of the Néel state in the PXP model, revealing ETH violation and slow entanglement growth amid an otherwise thermalizing spectrum.
Rayleigh-Taylor Instability
See how heavy fluid over light is linearly unstable with growth rate σ ~ √(Akg), then watch nonlinear spike and bubble formation in a live 2D simulation.
Neural ODE — Continuous Depth
Visualize a neural network as a continuous ODE dh/dt = f(h,t,θ) and watch a normalizing flow learn to transform a Gaussian into a complex target distribution.
SSH Model — Topological Band Theory
Explore the Su-Schrieffer-Heeger chain's topological phase transition via winding number, see bulk-boundary correspondence produce zero-energy edge states, and animate an adiabatic Thouless pump.
Oregonator — Belousov-Zhabotinsky
Simulate the Field-Körös-Noyes Oregonator model of the BZ reaction, including the limit-cycle attractor, Hopf bifurcation diagram, and self-organizing 2D spiral waves.
Polymer Reptation Dynamics
Watch a polymer chain snake through its entanglement tube in de Gennes' reptation model and compare the four MSD regimes distinguishing Rouse from reptation dynamics.
Random Energy Model — Spin Glass
Explore Derrida's exactly solvable spin glass: energy level statistics, the freezing transition at T_c, and one-step replica symmetry breaking in the glass phase.
Cell Polarity (Turing)
Activator-inhibitor symmetry breaking on a cell membrane
Quantum Discord
Geometry of quantum vs classical correlations in two-qubit states
Lorenz Bifurcation
From fixed points to chaos as the Rayleigh number increases
FitzHugh-Nagumo Neuron
Phase plane and spiking dynamics of a minimal neuron model
SSH Topological Chain
Edge states and topological phase transition in 1D
Fluid Mixing & Advection
Dye advection in vortical flow and the Péclet number
Brusselator 2D
Turing pattern formation: spots and stripes from the Brusselator
Bell State Correlations
Measurement correlations and entanglement in the four Bell states
Vegetation Patterns
Klausmeier model: tiger bush and spatial self-organization
Spin Glass
Random frustration, replica overlap, and the glass transition
Kuramoto Order Parameter
Phase synchronization of coupled oscillators and the K_c transition
Laughlin Wavefunction
Fractional quantum Hall state at ν=1/3 and topological correlations
Ecological Succession
Patch occupancy and coexistence via colonization-extinction tradeoffs
Vortex Reconnection
Counter-rotating vortices reconnecting in 2D Navier-Stokes flow
Chimera States
Coexisting synchronized and chaotic regions in coupled oscillators
Contact Process
Absorbing-state phase transition and directed percolation universality
Hyperbolic Geodesics
Poincaré disk model, geodesics, and hyperbolic triangles
Langevin Dynamics
Brownian motion in a double well and Kramers escape rate
BTW Sandpile
Self-organized criticality and power-law avalanches
Wave Function Collapse
Procedural generation via entropy-minimizing constraint propagation
Rule 110 & Beyond
Turing-complete elementary cellular automata and all 256 rules
Wright-Fisher Drift
Genetic drift, fixation, and selection in finite populations
Abrikosov Vortex Lattice
Magnetic flux tubes and triangular lattice in Type II superconductors
Information Bottleneck
Optimal compression-relevance tradeoff and the IB curve
Helfrich Membrane
Thermal fluctuations and bending rigidity of elastic membranes
Cascade Failures
Load redistribution and blackout cascades in power grid networks
Voter Model
Coarsening, consensus, and interface dynamics in opinion spreading
Vicsek Flocking
Self-propelled particles and the order-disorder transition in flocking
Polymer Chain (FJC)
Random walk conformations, end-to-end distributions, and entropic elasticity
OGY Chaos Control
Stabilizing unstable periodic orbits in a chaotic attractor
Extreme Value Theory
Gumbel, Fréchet, and Weibull attractors for block maxima
Network Robustness
Random failure vs targeted attack: giant component percolation
Positional Information
Morphogen gradients and the French flag model of cell fate
Age-Structured SEIRD
Age contact matrices and stratified epidemic dynamics
RG Decimation
Block-spin renormalization and coupling flow on the 1D Ising chain
Geodesic Flow on S²
Parallel transport, holonomy, and Gauss-Bonnet on the sphere
BZ Reaction Waves
Spiral and target waves in the Belousov-Zhabotinsky Oregonator model
Stochastic Thermodynamics
Entropy production, fluctuation theorem, and the driven ratchet
Grover's Algorithm
Amplitude amplification and quadratic speedup for unstructured search
Replicator Dynamics
Evolutionary game theory on the simplex: Nash equilibria and ESS
Hypergraph Contagion
Higher-order interactions and discontinuous spreading on hypergraphs
Coulomb Blockade
Single-electron tunneling and Coulomb diamonds in quantum dots
Itô vs Stratonovich
Two interpretations of multiplicative noise and their physical consequences
TDA Mapper
Topological shape of data via the Mapper nerve construction
Mode-Coupling Glass
Two-step relaxation, caging, and the MCT glass transition
Jarzynski Equality
Free energy from nonequilibrium work and the fluctuation theorem
Contact Tracing
Critical tracing coverage and epidemic containment on networks
Jaynes-Cummings
Vacuum Rabi oscillations and collapse-revival in cavity QED
Spatial Prisoner's Dilemma
Cooperation via spatial clustering and the temptation phase transition
Colloidal Self-Assembly
Hard-sphere crystallization, hexatic order, and the fluid-crystal transition
Echo Chambers
Bounded confidence, homophily, and opinion polarization on networks
Quantum Quench
Loschmidt echo and dynamical phase transitions in the TFIM
Adaptive Networks
Coevolution of opinions and topology via rewiring dynamics
Quasispecies & Error Threshold
Eigen's quasispecies equation and the collapse of genetic information
Potts Model
q-state Potts transitions: continuous vs first-order at q=4
Amplitude Death
Coupled oscillators silenced by diffusion and frequency mismatch
Information Cascades
Bayesian herding and the fragility of social learning
Dicke Superradiance
Superradiant phase transition and bistability in open cavity QED
Lévy Flights
Heavy-tailed random walks, anomalous diffusion, and optimal search
Turing Test Game Theory
The imitation game as strategic interaction and its Nash equilibria
Coupled Oscillator Bifurcation
Van der Pol oscillators and Hopf bifurcations — phase portrait transitions with coupling and damping
Cayley Graph Symmetry
Visualize finite group structure — nodes colored by coset, edges by generator
Neural Synchrony & Gamma
Leaky integrate-and-fire network synchronizing into gamma-band oscillations
Adiabatic Quantum Computation
Quantum annealing and energy gap dynamics across a quantum phase transition
Entropy Production & Fluctuation
Gallavotti-Cohen theorem: P(ΔS)/P(-ΔS) = e^ΔS for stochastic trajectories
Sandpile Group Identity
The abelian sandpile identity element — a stunning self-similar fractal from toppling rules
Feigenbaum Period Doubling
Logistic map bifurcation diagram, δ≈4.669 universal constant, and Lyapunov exponents
Ricci Flow Visualization
Discrete Ricci flow on 2D meshes — curvature evolving toward uniformity, Poincaré conjecture
Continued Fractions & Stern-Brocot
Navigate the Stern-Brocot tree to find best rational approximations to any real number
Ising Exact Solution
2D Ising Monte Carlo with Onsager's exact 1944 solution and critical temperature Tc
Algebraic Topology & Euler
Build simplicial complexes interactively — compute χ = V−E+F and Betti numbers β₀, β₁
Minority Game Theory
N agents choose 0 or 1 — minority wins. Watch herding dynamics and phase transition at memory length.
Boids Flocking Behavior
Reynolds' three rules — separation, alignment, cohesion — produce emergent flocking with predator avoidance.
Logistic Growth with Delay
Hutchinson's delay equation dx/dt = rx(1−x(t−τ)/K): delay drives oscillations and chaos.
Galton Board — Bean Machine
Balls fall through pegs building binomial → normal distribution in real time.
Apollonian Gasket
Recursive circle packing via Descartes' theorem. Fractal dimension ≈ 1.3057. Zoom and color by generation.
Hyperbolic Fibonacci Spiral
Fibonacci spirals in the Poincaré disk — hyperbolic geodesics and the golden ratio in curved space.
Watts-Strogatz Small-World
Rewire a ring lattice toward a random graph — watch clustering and path length reveal the small-world regime.
Directed Percolation
Absorbing-state phase transition: probabilistic CA near p_c ≈ 0.6447 with DP universality exponents.
Kac Ring Model
Spins on a ring flip at marked bonds — Poincaré recurrence, H-theorem, and the reversibility paradox.
Cahn-Hilliard Spinodal Decomposition
Binary mixture phase separation: coarsening domains grow as L(t) ~ t^(1/3) via Lifshitz-Slyozov-Wagner.
Evolutionary Snowdrift Game
Cooperators and defectors on a spatial grid — spatial structure sustains cooperation above mean-field prediction.
Barnsley Fern
Iterated function system chaos game — complex fern geometry from four random affine transforms applied with fixed probabilities.
Kuramoto Oscillators
N coupled phase oscillators synchronizing — watch the order parameter jump at the critical coupling K_c (phase transition).
Double Pendulum + Poincaré
Hamiltonian chaos in the double pendulum with live Poincaré section — add trajectories to see exponential divergence.
Gray-Scott Reaction-Diffusion
Turing instability in two reacting chemicals — spots, stripes, worms, and mazes from a single PDE with different F/k parameters.
Three-Body Figure-Eight
The famous periodic figure-eight orbit (Chenciner & Montgomery 2000) plus chaotic and Lagrange configurations.
Hopf Fibration
S³→S² fiber bundle visualized via stereographic projection — linked Villarceau circles, π₃(S²)=ℤ made visible.
2D Navier-Stokes Fluid
Incompressible fluid simulation (Stam stable fluids) — click to inject vortices, switch between vorticity, speed, and dye views.
Heat Equation Solver
Draw heat sources and watch them diffuse — ∂u/∂t = α∇²u solved with explicit finite differences, with live cross-section profile.
Fibonacci Sunflower
Golden angle phyllotaxis — 137.508° packs seeds optimally because φ is the most irrational number. Drag the angle to break the symmetry.
Penrose Tiling
Aperiodic kite-and-dart tiling via substitution inflation rules — 5-fold symmetry, golden ratio tile counts, quasicrystal structure.
L-System Fractal Plants
Lindenmayer system plant generator — edit axiom, rewriting rules, and angle to grow branching trees, ferns, and space-filling fractals in real time.
Sierpinski Pyramid 3D
Rotatable 3D Sierpiński tetrahedron (Tetrix) — IFS subdivision to depth 5, solid + wireframe toggle, Hausdorff dimension log(4)/log(2) = 2.
Collatz Conjecture
Visualize the 3n+1 mystery — trajectory on log scale, stopping-time heatmap across thousands of starting numbers, and the Collatz tree.
Voronoi Shatter
Animated Voronoi diagram with moving sites — click to fracture, drag sites to reshape cells, watch the plane partition continuously reform.
ODE Solver: Euler vs RK4
Compare Euler, Heun (RK2), and Runge-Kutta 4 on classic ODEs — harmonic oscillator, logistic growth, Van der Pol. Error divergence made visible.
Geodesic Sphere
Icosahedron subdivision into geodesic spheres — frequency 1–6, wireframe/solid, dual fullerene overlay, Euler characteristic V−E+F = 2.
SPH Fluid Simulation
Smoothed particle hydrodynamics — 2D fluid with pressure, viscosity, and gravity. Pour, drain, splash, and push particles with your cursor.
Hyperbolic Plane Tiling
Poincaré disk {p,q} tilings — infinite hyperbolic tessellations rendered as geodesic arcs. Navigate by dragging; choose from 7 Schläfli symbols.
Toroidal Flow
3D torus with (p,q)-torus knot flow lines on its surface — tune meridian and longitude winding numbers, watch knots and links emerge.
Knot Theory
Interactive 3D knots — trefoil, figure-8, cinquefoil, Hopf link and more. Shows crossing number, writhe, Jones polynomial, and Frenet framing.
Small-World Network
Watts-Strogatz model: rewiring probability transitions from ring lattice to random graph, with live clustering coefficient and average path length.
Optical Caustics
Light ray caustics through curved surfaces — refraction, focal points, and rainbow caustic patterns from circle, ellipse, and water-wave lenses.
Gibbs Phenomenon
Partial Fourier series of square, sawtooth, and triangle waves — the ~9% overshoot at discontinuities that never vanishes as N grows.
Sierpinski Carpet
Recursive subdivision at depth 0–6 with colorful themes — fractal dimension log8/log3 approx 1.893, area remaining shrinks to zero.
Prime Sieve Visual
Sieve of Eratosthenes animated, Ulam spiral with diagonal alignments, prime gap histogram, and pi(x) vs x/ln(x) vs Li(x) comparison.
Telegraph Equation
Real-time 1D wave with damping and dispersion — compare pure wave, Heaviside telegraph, and diffusive regimes on a transmission line.
Braid Group
Click generators to compose n-strand braids — visualize crossings, build braid words, and explore the Artin braid relations interactively.
Bloch Sphere NMR
3D Bloch sphere with T1/T2 relaxation and precession — apply RF pulses, run spin echo sequences, and watch magnetization trajectories.
Turing Morphogenesis
Gray-Scott reaction-diffusion: tune feed and kill rates to switch between spots, stripes, labyrinths, and mitosis patterns in real time.
Halvorsen Attractor
Cyclically symmetric Z3 strange attractor as a rotating 3D point cloud — colored by velocity magnitude, drag to rotate, tune parameter a.
Hat Monotile
The 2023 aperiodic monotile discovery: a single shape that tiles the plane but never repeats — pan and zoom to explore the infinite non-repeating tiling.
Sphere Eversion
Animate turning a sphere inside-out through smooth deformations — no tearing, no pinching, but self-intersections allowed. Smale's 1957 paradox made visible.
Continued Fractions
Enter any number to see its continued fraction expansion, convergents table, Stern-Brocot tree path, and error decay — compare π, e, φ, and √2.
SIR Bifurcation
Drag R₀ across the critical threshold of 1 and watch the epidemic bifurcation: phase portrait, time series, bifurcation diagram, and final size relation simultaneously.
Apollonian Packing
Recursive kissing circles via Descartes' theorem — zoom into the fractal Apollonian gasket with Hausdorff dimension ≈ 1.3057 and integer curvature labels.
Surfaces by Genus
Every closed orientable surface is a sphere with handles — explore the classification theorem with spinning 3D models of sphere, tori, Klein bottle, and projective plane.
Symplectic Integrators
Leapfrog/Verlet vs Euler vs RK4: compare energy drift over long orbits and see why symplectic methods preserve phase space structure where others fail.
Shannon Entropy
Edit a probability distribution with sliders and watch entropy, channel capacity, and mutual information update live — from uniform (maximum entropy) to peaked.
Brownian Motion
Multiple Wiener process paths with √t envelope, quadratic variation [W]_t = t, 2D random walk, and distribution at fixed time — animate growth from scratch.
Graph Laplacian
Draw any graph, compute Laplacian eigenvalues, and visualize the Fiedler vector as node colors — the second eigenvector reveals the optimal graph bisection.
Clifford Attractor
Iterate (x,y)→(sin(ay)+c·cos(ax), sin(bx)+d·cos(by)) — millions of points accumulate into intricate fractal strange attractors that shift dramatically with parameters a/b/c/d.
Lattice QCD Plaquette
2D U(1) lattice gauge theory: link variables on a grid, Wilson plaquette action, Monte Carlo Metropolis updates thermalize the system from ordered or disordered starts.
Spatial Prisoner's Dilemma
Agents on a grid copy the most successful neighbor strategy each generation. Spatial structure lets cooperators form clusters that resist defector invasion — watch phase transitions.
Laplace Equation Solver
Draw Dirichlet boundary conditions on a grid, then watch Gauss-Seidel relaxation converge to the unique harmonic solution ∇²φ=0 with equipotential and gradient field visualization.
Nonlinear Pendulum Phase Portrait
Full nonlinear pendulum phase space: separatrix dividing libration from rotation, energy contours, Poincaré map under periodic driving, and onset of chaos at large amplitude.
Logistic Growth & Allee Effect
Compare logistic growth, weak Allee effect, and strong Allee bistability. A critical population threshold determines survival vs extinction — explore bifurcations and harvesting effects.
Möbius Transformation
Animate f(z)=(az+b)/(cz+d) from the identity — watch circles and lines map to circles and lines. Explore fixed points, Cayley transform, inversion, rotation, and loxodromic maps.
Quantum Harmonic Oscillator
Visualize energy eigenstates ψₙ(x) and probability densities, apply ladder operators to raise and lower quantum number n, and explore minimum-uncertainty coherent states.
Adaptive RK45 Step Control
Dormand-Prince adaptive stepping: watch step size shrink through stiff regions and expand through smooth ones. Compare acceptance rate, rejected steps, and tolerance vs accuracy tradeoffs.
Lotka-Volterra Predator-Prey
Classic and extended predator-prey dynamics: neutral cycles, logistic prey, Holling type II response, and 3-species chaotic attractors. Phase portrait with nullclines and bifurcations.
Spring-Mass Normal Modes
1D chain of coupled oscillators: visualize normal mode frequencies and shapes. Slide between modes to see standing waves emerge from the tridiagonal eigenvalue problem.
Quantum Harmonic Oscillator — Wigner
Wigner quasi-probability distribution W(x,p) for energy eigenstates and superpositions. Negative regions reveal genuine quantum non-classicality in phase space.
SK Spin Glass
Sherrington-Kirkpatrick model: random Gaussian couplings, frustration, and rugged energy landscape. Watch the system freeze into a glassy state as temperature drops.
Hyperbolic Tiling — Poincaré Disk
Regular {p,q} tilings of the hyperbolic plane rendered in the Poincaré disk model. Drag to pan through infinite hyperbolic geometry with geodesic arcs.
Small-World Network — Watts-Strogatz
Rewiring probability beta sweeps from regular ring lattice to random graph. Watch path length collapse while clustering coefficient stays high.
Hegselmann-Krause Opinion Dynamics
Bounded-confidence opinion dynamics: agents average only with neighbors within distance epsilon. See consensus, polarization, and fragmentation emerge.
Einstein Ring Gravitational Lensing
Drag the background source to see Einstein rings, arcs, and double images form. Interactive mass and distance controls show how the lens geometry shapes light paths.
Airy Disk Diffraction
Fraunhofer diffraction through a circular aperture: 2D Airy pattern with wavelength, aperture, and focal length controls. Rayleigh resolution limit shown in real time.
Lattice Boltzmann Fluid
D2Q9 LBM simulation of 2D viscous flow past a circular obstacle. Visualize vorticity, speed, or pressure with velocity vector field overlay.
Double Pendulum Chaos
Two double pendulums with nearly identical initial conditions diverge exponentially. Watch sensitive dependence on initial conditions and track the Lyapunov exponent estimate.
Sierpiński Triangle — Chaos Game
Random IFS iteration builds the fractal in real-time. Tune contraction ratio and vertex count.
Boids — Emergent Flocking
Craig Reynolds' 1987 algorithm: separation, alignment, cohesion produce collective behavior from local rules.
2D Ising Model — Glauber Dynamics
Spin lattice phase transition. Sweep through T_c ≈ 2.269 and watch magnetization collapse.
Mandelbrot ↔ Julia Correspondence
Hover the Mandelbrot set to see the live Julia set for each c. Connected ↔ in M.
Schrödinger's Cat — Decoherence
Quantum superposition decays as the system entangles with the environment. Watch coherence |ρ₀₁| fall.
SIS Epidemic on Networks
Infection spreading on ER, Barabási–Albert, and Watts–Strogatz graphs. Threshold behavior at R₀=1.
Benford's Law — First Digit
Fibonacci, primes, factorials, Pareto: first-digit histograms vs log₁₀(1+1/d). The law that catches fraud.
Diffusion-Limited Aggregation
Random walkers stick to a growing cluster, forming fractal dendrites with dimension ≈ 1.71.
Fibonacci Phyllotaxis Spiral
Golden angle packing: why sunflowers have Fibonacci spiral counts. Drag the angle off golden to see gaps appear.
Game of Life Variants
Life, HighLife, Seeds, Day & Night, Replicator — switch rules and watch wildly different emergent behaviors.
Lorenz Attractor 3D
Interactive butterfly attractor with σ/ρ/β sliders, multiple trajectories, and 3D rotation — chaos from deterministic equations.
Geodesic Sphere
Buckminster Fuller's icosahedral dome: frequency slider showing triangle subdivision, pentagon highlights, and 3D rotation.
Vietoris-Rips Filtration
TDA: grow ε-balls around a point cloud, watch connected components merge and loops (β₀, β₁) appear in the persistence barcode.
Young's Double-Slit
Wave interference with wavelength and slit-spacing controls — fringe patterns, path differences, and single/double slit modes.
Network Motifs
Detect feed-forward loops, bi-fans, and bi-parallels in random vs bio-inspired networks — the recurring subgraph building blocks.
Hardy-Weinberg Genetics
Allele frequency dynamics with drift, selection, mutation, and migration — watch populations evolve away from equilibrium.
Quantum Measurement
Qubit superposition on the Bloch sphere: choose a measurement basis, apply the Born rule, and collapse the wavefunction.
Möbius Strip
3D parametric non-orientable surface with ant walk, normal vectors, twist controls — one side, one edge, infinite strangeness.
Wireworld Computer
Turing-complete cellular automaton: draw wires, inject electrons, and simulate AND/OR/NOT gates and signal clocks.
Entropy and Time's Arrow
Gas particles expand irreversibly — watch entropy rise, reverse velocities to test Boltzmann's H-theorem, open the Maxwell's demon wall.
Penrose Impossible Objects
Impossible triangle, stairs, devil's fork — isometric figures that violate 3D consistency
Klein Bottle Surface
4D→3D projection of a non-orientable surface with no inside or outside — parametric rendering with cross-section slider
Fractal Coastline — Richardson Effect
Measured coastline length diverges as ruler shrinks — log-log Richardson plot reveals fractal dimension
Kuramoto Synchronization
Ring of coupled oscillators — spontaneous synchrony emerges above the critical coupling threshold
Simulated Annealing — TSP
Probabilistic optimization finds near-optimal tours — live 2-opt moves with energy and temperature plots
Butterfly Effect
Two nearly-identical Lorenz trajectories diverge exponentially — sensitive dependence made visible
Power Grid Cascade Failure
Trigger a node failure and watch overloads cascade across an interconnected grid
Red Queen Dynamics
Predator-prey co-evolution — trait distributions shift in endless arms-race cycles
Ant Colony Stigmergy
Pheromone trails enable collective shortest-path finding — emergent intelligence without central control
Coupled Map Lattice
1D lattice of logistic maps with diffusive coupling — spatiotemporal chaos, waves, and synchrony
Rule 90 Cellular Automaton
XOR-based elementary CA generates Sierpinski triangle patterns from a single cell.
FitzHugh–Nagumo Oscillator
Simplified neuron model showing excitability and limit cycle oscillations in the phase plane.
Topological Soliton
Sine-Gordon kink solution — a topologically stable wave packet that travels without dispersion.
Viscous Fingering (Saffman–Taylor)
DLA simulation of branching instability when low-viscosity fluid displaces high-viscosity fluid.
Arnold Tongues & Mode Locking
Circle map parameter space revealing rational resonance tongues and quasiperiodic regions.
De Bruijn Sequences
Sequences visiting every k-ary string of length n exactly once, traced as Eulerian paths on De Bruijn graphs.
Winding Number Explorer
Topological invariant counting how many times a closed curve winds around a point.
Scale-Free Networks (Barabási–Albert)
Preferential attachment generates power-law degree distributions — the rich-get-richer network model.
Rényi Entropy & Multifractals
Generalized entropy parameterized by q, interpolating between min-entropy and Hartley entropy.
Supercritical Hopf Bifurcation
Stable limit cycle emerges from an equilibrium as the bifurcation parameter crosses zero.
Quantum Billiards (Stadium)
Wave function scarring in a chaotic stadium billiard — quantum signatures of classical chaos.
Nonlinear Pendulum Bifurcations
Driven damped pendulum Poincaré sections revealing the route to chaos through period-doubling.
Rabinovich–Fabrikant Attractor
Three-dimensional strange attractor arising from plasma wave interactions and nonlinear oscillators.
Josephson Junction Phase Dynamics
Superconducting tunnel junction — phase slips, IV curves, and Shapiro steps under AC drive.
Riemann Surface & Branch Cuts
Multi-sheeted complex surface of √z or log(z), visualizing monodromy around branch points.
Sandpile on a Critical Slope
BTW sandpile model on a slope — self-organized criticality with power-law avalanche distributions.
Spectral Graph Coloring
Graph partitioning via the Fiedler vector of the graph Laplacian — algebraic connectivity made visual.
Spatial Poisson Process
Random point patterns, Voronoi tessellations, and nearest-neighbor statistics from a 2D Poisson field.
Optical Caustics & Wavefronts
Ray tracing reveals caustic envelopes — the bright folds of light from curved reflectors.
BZ Reaction Oscillator
Oregonator model on a 2D grid generates spiral waves and target patterns of the Belousov–Zhabotinsky reaction.
Turbulent Energy Cascade
Kolmogorov's k⁻⁵/³ inertial range and the Richardson cascade from large eddies to viscous dissipation.
Spontaneous Symmetry Breaking
Mexican hat potential — Goldstone modes and the Higgs mechanism visualized in field space.
Prime Gap Distribution
Statistics of gaps between consecutive primes, Cramér's conjecture, and the Ulam spiral.
Contact Process & Epidemic Threshold
Directed percolation universality class — infection spreading on a lattice near the critical threshold.
Liouville's Theorem
Phase space volume is conserved under Hamiltonian flow — incompressible fluid in phase space.
Entanglement Entropy
Bipartite entanglement entropy revealing area-law vs volume-law scaling in quantum systems.
Breather Solutions & Rogue Waves
Peregrine soliton and Akhmediev breathers — exact rogue wave solutions of the nonlinear Schrödinger equation.
Riemann Zeta Zeros & GUE Statistics
Zeros of ζ(½+it) along the critical line show random matrix (GUE) level spacing — Montgomery–Odlyzko law.
Wave Packet Dispersion
Group vs phase velocity under linear, quadratic, relativistic, and anomalous dispersion relations.
Langevin Dynamics & Kramers Escape
Overdamped Brownian motion in a double-well potential — noise-driven escape and stationary distributions.
Cellular Potts Model
Monte Carlo cell sorting via differential adhesion — watch tissues self-organize and boundaries form.
Poincaré Disk
Hyperbolic geometry: geodesics as circular arcs, {7,3} tessellations, and K = −1 curvature.
Morse Code Binary Tree
Morse code as a binary tree — encode text, decode dots and dashes, watch signal waveforms.
Gravitational Lensing
Einstein ring formation by a point mass lens — drag lens and source to see image distortion.
Dissipative Structures
Gray-Scott reaction-diffusion: Prigogine's non-equilibrium self-organization into spots and stripes.
Period-Three Implies Chaos
Li-Yorke theorem: period-3 orbit in the logistic map implies orbits of all periods — chaos guaranteed.
3-SAT Energy Landscape
Phase transition at α ≈ 4.27 — explore the satisfiability cliff and WalkSAT energy trajectories.
Euler Buckling
Elastic instability: animate column buckling modes as load exceeds P_cr = π²EI/(KL)².
Majorana Zero Modes
Kitaev chain topological superconductor — phase transition at |μ|=2t, zero-energy Majorana end modes.
Liquid Crystal Defects
XY model nematic defects: ±1/2 winding number topological defects, Kibble-Zurek quench, Frank energy.
Feigenbaum Universality Constant
Bifurcation diagram of the logistic map — watch period-doubling ratios converge to δ ≈ 4.6692, universal across all unimodal maps.
Geodesic Deviation & Curvature
Tidal forces as geometry: nearby geodesics focus (positive curvature) or diverge (negative curvature) via the Riemann tensor equation D²ξ/dτ² = −Rξ.
Double Pendulum — Phase Portrait
Launch nearby trajectories and watch them diverge exponentially — a live demonstration of Lyapunov sensitivity and the θ₁–θ̇₁ phase portrait.
Holographic Entropy — Ryu-Takayanagi
Entanglement entropy equals minimal geodesic length in the AdS bulk. Visualize the RT formula S = Area(γ)/4G_N on the Poincaré disk.
Jarzynski Equality — Nonequilibrium Work
Free energy from irreversible processes: ⟨e^{−βW}⟩ = e^{−βΔF}. Watch work distributions shift with protocol speed and Jarzynski estimates converge.
SSH Chain — Topological Band Structure
The Su-Schrieffer-Heeger model: tune hopping ratio t'/t to cross the topological phase transition and watch protected edge states appear and vanish.
Heisenberg Uncertainty — Wavepacket
Squeeze a Gaussian in position space and watch its Fourier transform broaden in momentum — Δx·Δp ≥ ℏ/2 made vivid with time evolution.
Newton Fractal Basin Boundaries
Newton's method for z^n = 1 produces fractal basin boundaries. Click to zoom into infinite self-similar complexity at the edge of convergence.
Pauli Exclusion — Fermi Gas
Fermi-Dirac distribution f(E) at finite temperature — animate the thermal smearing of the Fermi surface and explore zero-point pressure.
Correlated Random Walk — Persistent Motion
Ballistic-to-diffusive crossover: tune step correlation ρ and watch ⟨r²⟩ scale as t² then t, with crossover at the persistence time τ.
Wigner Crystal Formation
Electrons crystallize at low density — watch Monte Carlo relaxation reveal triangular lattice order as r_s crosses the quantum phase transition.
Bogdanov-Takens Bifurcation
Codimension-2 normal form with saddle-node, Hopf, and homoclinic curves — click in parameter space to explore the full phase portrait.
SK Spin Glass — Mean Field
Sherrington-Kirkpatrick model with random J_ij couplings — watch aging, non-ergodicity, and the Edwards-Anderson order parameter q_EA emerge.
Simulated Annealing — TSP
Metropolis algorithm solves the traveling salesman problem — watch 2-opt swaps and cooling schedules beat greedy solutions.
Life Variants — CA Zoo
Five cellular automaton rule sets — Conway Life, HighLife, Day and Night, Seeds, and Maze — draw cells and switch rules live.
Kuramoto Chimera State
Nonlocal coupling on a ring spontaneously splits oscillators into synchronized and incoherent populations.
Diffraction Grating — N Slits
From Young's double slit to a diffraction grating — peaks sharpen as N increases, with wavelength-controlled color and sinc squared envelope.
Replica Symmetry Breaking
Parisi order parameter q(x), ultrametric tree of states, and hierarchical block matrix — the mathematical heart of spin glass theory.
Directed Polymer — KPZ
Free energy fluctuations scale as t to the 1/3 — KPZ universality class with Tracy-Widom distribution and anomalous roughness exponents.
Kolmogorov Turbulence
Eddy cascade from integral scale L to microscale eta — animated scale hierarchy and E(k) proportional to k to the minus 5/3 energy spectrum.
Ising Model Monte Carlo
Phase transitions in a 2D spin lattice via Metropolis sampling.
Lorenz 96 Atmospheric Chaos
Chaotic dynamics of N=40 atmospheric variables on a cyclic ring.
Fractional Brownian Motion
fBm paths with adjustable Hurst exponent — from rough to persistent.
Random Sequential Adsorption
Circles pack randomly until jamming at the RSA limit ~54.7%.
Minority Game
101 agents compete to be in the minority — emergent coordination.
Diffusion-Limited Aggregation
Random walkers cluster into fractal branching structures.
Sierpiński Carpet
Recursive fractal subdivision with dimension D≈1.893.
Smoluchowski Coagulation
Particles merge and grow — coagulation kinetics in real time.
KPZ Surface Growth
Kardar-Parisi-Zhang roughening — interface width grows as t^(1/3).
Poisson-Voronoi Tessellation
Random points generate a dynamic Voronoi partition of the plane.
Glauber Dynamics
Kinetic Ising model with single spin-flip rates — domain coarsening in time.
Information Bottleneck
Tishby's IB principle: optimal compression trading relevance for conciseness.
Extreme Value Theory
Block maxima converge to Gumbel/Fréchet/Weibull distributions.
Cell Cycle Network
Tyson's Cdk1/APC oscillator — limit cycles driving mitosis.
Chaotic Maps
Tent and logistic maps — bifurcation diagrams and Lyapunov exponents.
Cramér-Lundberg Ruin
Insurance surplus process with compound Poisson claims — ruin probability.
DFA Long-Range Correlations
Detrended Fluctuation Analysis reveals the Hurst exponent of time series.
Lotka-Volterra Competition
Two-species competition: coexistence, exclusion, and bistability in phase space.
Gierer-Meinhardt Patterns
Activator-inhibitor reaction-diffusion — Turing spots and stripes.
Network Cascade Threshold
Watts model: social cascades spreading through threshold networks.
Mean-Field Percolation
Giant component phase transition on a Bethe lattice.
Schelling Segregation
Tipping-point segregation from mild local preferences.
Zero-Range Condensation
Bose-Einstein-like condensation in an interacting particle hop model.
Hyperbolic Discounting
Time preference reversals — why we're impatient in ways that hurt us.
Fokker-Planck Flow
Probability density evolving in a double-well potential.
Vicsek Active Matter
Self-propelled particles align to flock — order from noise.
Giant Component Transition
Erdős-Rényi random graph — sudden emergence of the giant component.
Turbulent Energy Cascade
Kolmogorov's k⁻⁵/³ spectrum from inertial-range energy cascade.
Mackey-Glass Chaos
Delay differential equation with period-doubling route to chaos.
Kuramoto Synchronization
N=100 phase oscillators snap into collective rhythm above K_c.
Sandpile Avalanche Statistics
Power-law avalanche sizes in the BTW sandpile — self-organized criticality.
Sznajd Opinion Model
Social influence dynamics: neighbor pairs spreading opinions on a ring.
Allee Effect
Logistic growth with an extinction threshold — bistability and population collapse.
Duffing Oscillator
Nonlinear resonance, jump phenomena, and chaos in the forced Duffing equation.
Galton-Watson Branching
Branching process extinction probability — critical at mean offspring μ=1.
Anderson Localization
Disorder localizes quantum wavefunctions — no transport in 1D for any W>0.
MAPK Signaling Cascade
Three-tier kinase cascade with Michaelis-Menten ultrasensitivity.
Random Energy Model
Derrida's REM — glass transition as entropy vanishes at T_c.
Brownian Ratchet
Asymmetric flashing potential drives net particle current from noise.
Fiber Bundle Fracture
Equal-load-sharing fracture cascade — stress-strain curve and avalanches.
Stochastic Predator-Prey
Individual-based Lotka-Volterra — spatial spirals and stochastic extinction.
Cavity Method
Belief propagation on a Bethe lattice computing Ising marginals.
Adaptive Network Epidemic
SIS spread with rewiring — bistability and discontinuous transitions.
Rock-Paper-Scissors Spirals
Cyclic dominance in spatial RPS — self-organizing spiral waves.
Bayesian Social Learning
Information cascades — rational herding overrides private signals.
Swift-Hohenberg Patterns
Labyrinthine stripes and hexagons from a pattern-forming PDE.
TASEP Exclusion Process
Totally asymmetric exclusion — three density phases and boundary control.
Driven Diffusive System
Lattice gas with external drive — nonequilibrium steady states and currents.
Network Robustness
Scale-free networks: robust to random failures, fragile to targeted attack.
Eden Cluster Growth
Perimeter-growth cluster with KPZ roughness exponent β=1/3.
Chirikov Standard Map
KAM tori dissolve as K grows — chaos spreading via resonance overlap.
Self-Replicating Automaton
Rule 110 universality and self-replication in cellular automata.
KdV Solitons
Solitons pass through each other unchanged — elastic nonlinear wave collision.
Clonal Selection
Immune affinity maturation — hypermutation drives antibody improvement.
Oslo Sandpile Model
Stochastic self-organized criticality with power-law avalanche statistics.
Carnot Heat Engine
Carnot cycle on a PV diagram — ideal efficiency η = 1 − T_C/T_H.
Binary Fragmentation
Recursive random splitting — power-law fragment size distributions.
Directed Percolation
Contact process absorbing transition — activity lives or dies at λ_c.
Glass Energy Landscape
Trap model aging — mean escape time grows with waiting time.
Neural Avalanche Criticality
Branching ratio σ=1 produces P(s)~s⁻³/² neural avalanche statistics.
Stochastic Resonance
How noise can enhance weak signal detection in threshold systems
Lévy Flight
Anomalous diffusion with heavy-tailed power-law step lengths
Quantum Chaos Billiards
Stadium billiard revealing chaotic trajectories and ergodicity
Contact Process SIS
Lattice epidemic model near the absorbing-state phase transition
Bootstrap Percolation
Cascade activation when neighbors exceed a threshold k
Cahn-Hilliard Spinodal
Phase separation and domain coarsening in binary fluids
Josephson Junction Chaos
Driven RCSJ junction showing periodic and chaotic phase dynamics
Lorenz-96 Atmosphere
N-variable coupled atmospheric model and its chaotic attractor
Julia Set
Escape-time Julia sets for z²+c with interactive c parameter
Sierpiński Triangle IFS
Chaos game construction of the Sierpiński triangle via IFS
Mandelbrot Explorer
Zoom into the Mandelbrot set with escape-time coloring
Lissajous Curves
Parametric figures from coupled harmonic oscillators
Gray-Scott Reaction
Spots, stripes, and solitons from reaction-diffusion dynamics
Boids Flocking
Emergent flocking from separation, alignment, and cohesion rules
Noisy Voter Model
Opinion dynamics with noise on a lattice near the critical point
Zipf's Law
Power-law rank-frequency distributions and Zipf's law
Ornstein-Uhlenbeck
Mean-reverting stochastic process and its stationary distribution
Riemann Domain Coloring
Complex function visualization via domain coloring
Magnetic Field Lines
Field lines and vectors for dipole and multipole configurations
Coupled Map Lattice
Spatiotemporal chaos in a 1D coupled logistic map lattice
Perrin Brownian Motion
Pollen particle trajectories and mean-squared displacement
Self-Avoiding Walk
Lattice polymer model and the Flory exponent
Abelian Sandpile
Self-organized criticality via grain toppling avalanches
Poisson Process
Random events in time with exponential inter-arrivals
Dragon Curve
Paper-folding fractal via iterated L-system rewriting
Three-Body Problem
Chaotic gravitational dynamics and sensitive dependence
1/f Pink Noise
Colored noise spectra from white to Brownian on log-log scale
Rényi Entropy
Parameterized entropy family interpolating between min and Shannon
Taylor-Couette Flow
Vortex instability between rotating concentric cylinders
Kapitza Pendulum
Inverted pendulum stabilized by rapid vertical oscillation
Percolation Crossing
Spanning cluster probability at the bond percolation threshold
Penrose Tiling
Quasiperiodic tiling via inflation and deflation rules
Frustrated Kuramoto
Phase frustration breaking synchronization in coupled oscillators
ODE Solver Comparison
Error scaling of Euler, RK2, and RK4 integrators
Pendulum Phase Portrait
Exact phase space with separatrix, libration, and rotation
Arnold's Cat Map
Chaotic torus automorphism and Poincaré recurrence
Lyapunov Spectrum
Lyapunov exponent vs parameter alongside the bifurcation diagram
Gaussian Mixture EM
Expectation-maximization fitting animated step by step
Birth-Death Chain
Stationary distribution of birth-death Markov chains
Ising Exact Solution
Onsager's exact phase transition alongside Monte Carlo
Poincaré Map
Period-doubling cascade revealed on the Poincaré section
Community Detection
Modularity optimization revealing planted graph communities
Information Bottleneck
Optimal compression tradeoff on the information plane
Bose-Einstein Condensation
Macroscopic ground-state occupation in the ideal Bose gas
Wavelet Transform
Time-frequency scalogram via continuous wavelet transform
Maxwell-Boltzmann
Speed distribution of an ideal gas at thermal equilibrium
NLS Bright Soliton
Soliton solutions of the nonlinear Schrödinger equation
Eigenvector Centrality
Power iteration converging to the leading network eigenvector
Point Vortex Dynamics
Hamiltonian vortex motion in 2D ideal fluid
Feynman Path Integral
Quantum amplitude as sum over interfering classical paths
RMT Level Repulsion
Wigner surmise for eigenvalue spacing in GOE/GUE ensembles
Gibbs Sampling Ising
Single-spin-flip Gibbs sampler for the 2D Ising model
Lanczos Algorithm
Iterative eigenvalue convergence via Krylov subspace projection
Zipf-Mandelbrot Law
Generalized power-law rank-frequency with shift parameter
WireWorld
Electron-propagating CA supporting logic gates and circuits
Ulam-Warburton CA
Fractal growth via the one-neighbor activation rule
Lenz's Law
Induced EMF opposing flux change as magnet enters a loop
Morse Theory
Critical points, level sets, and topology of height functions
Erdős-Rényi Threshold
Giant component emergence at the percolation threshold p=1/n
2D Coulomb Gas
Log-repulsion equilibrium and the Wigner semicircle law
Goldstone Boson & Symmetry Breaking
Mexican hat potential, order parameter dynamics, and massless Goldstone modes from spontaneous symmetry breaking.
Gray-Scott Reaction-Diffusion
Two chemical species form stunning Turing patterns — spots, stripes, and worms — in this classic reaction-diffusion system.
SSH Model & Topological Edge States
Su-Schrieffer-Heeger chain: winding number, band topology, and zero-energy edge modes at the boundary.
Freely Jointed Chain Polymer
Random-walk polymer conformations, end-to-end distance distributions, and the Gaussian chain approximation.
Giant Component Transition
Erdős–Rényi random graphs: watch a giant connected component emerge as edge density crosses the percolation threshold.
Acoustic Cavity Resonance
Standing wave pressure modes in a 2D rectangular cavity — the mathematics of musical instruments and room acoustics.
Kolmogorov-Arnold Networks
The KAN theorem: any multivariate function decomposes into univariate functions on a network of learnable splines.
Saddle-Node Bifurcation
Normal form ẋ = r + x²: stable and unstable fixed points collide and annihilate at the critical parameter value.
Loschmidt Echo & Quantum Fidelity
How quickly does a quantum state lose memory of its past? Fidelity decay reveals chaos and decoherence regimes.
Jackiw-Rebbi Zero Modes
A topological mass kink in 1D Dirac theory traps a zero-energy bound state — the birth of topological protection.
Entropy Production & Clausius Inequality
Irreversible heat flow, thermodynamic forces, and the entropy production rate — the arrow of time in action.
Kerr Effect & Nonlinear Optics
Intensity-dependent refractive index drives self-phase modulation, self-focusing, and optical soliton formation.
Quantum Eraser: Which-Path
Double-slit interference vanishes with which-path information — and returns when that information is erased.
Fokker-Planck Equation
Probability density of a stochastic particle drifting and diffusing through a potential landscape.
Adiabatic Theorem & Berry Phase
A slowly rotating magnetic field guides a quantum spin, accruing a geometric Berry phase equal to half the solid angle swept.
Mott Insulator & Hubbard Model
Strong electron-electron repulsion U opens a gap — a conductor becomes an insulator through correlation, not band structure.
Jamming Transition in Soft Matter
Soft disks transition from a freely flowing state to a rigid jammed solid at a critical packing fraction φ_J.
Critical Exponents & Scaling Collapse
Near a phase transition, divergences follow power laws — and data from different temperatures collapse onto a universal curve.
Quantum Graph Spectrum
Schrödinger equation on a network of 1D wires: eigenvalues, eigenstates, and level statistics from star to complete graphs.
Kelvin-Helmholtz Instability
A shear flow between fluid layers rolls up into billowing vortices — the instability behind clouds, oceans, and Jupiter's bands.
Holonomy & Parallel Transport
A vector transported around a closed loop on a sphere returns rotated — geometric phase as pure curvature.
Spherical Harmonics Y_lm
Eigenfunctions of the angular Laplacian on a sphere: the basis for atomic orbitals and multipole expansions.
Quantum Annealing
Solve an Ising optimization problem by adiabatically reducing quantum fluctuations — quantum tunneling finds the ground state.
Elasticity: Stress & Strain
Tensor Hooke's law in 2D: deformation fields, von Mises stress, and principal stress trajectories under applied load.
Cavity QED: Jaynes-Cummings
A two-level atom in a photon cavity: vacuum Rabi oscillations, quantum collapse and revival, and Schrödinger cat states.
Huygens Principle & Diffraction
Every wavefront point is a wavelet source — their superposition builds up single and double-slit diffraction patterns.
Hall Viscosity & Topology
Non-dissipative Hall viscosity in 2D quantum fluids: a topological transport coefficient tied to the Chern number.
Spectral Flow & Index Theorem
Eigenvalues crossing zero as a parameter varies — the net spectral flow equals the topological index.
Riemannian Geodesics
Geodesic equations on a torus and sphere: shortest paths, Gaussian curvature, and Jacobi field focusing.
Maxwell Construction & van der Waals
Equal-area rule resolves the unphysical van der Waals isotherm, revealing liquid-vapor coexistence and the critical point.
Voronoi Tessellation & Lloyd's Algorithm
Partition the plane by nearest seed; Lloyd relaxation drives seeds to centroids, producing centroidal Voronoi diagrams.
Penrose Tiling via Inflation
Kites and darts subdivide recursively, filling the plane aperiodically with perfect 5-fold symmetry and golden-ratio scaling.
Chern Insulator & Berry Curvature
A 2D lattice with broken time-reversal symmetry: Berry curvature integral gives the Chern number and topological edge states.
Mean First Passage Time
How long does a biased random walker take to first reach a target? MFPT solves a boundary-value problem on the drift-diffusion equation.
Fraunhofer Diffraction is a Fourier Transform
Far-field diffraction patterns are the 2D Fourier transform of the aperture — draw any shape and watch its spectrum appear.
Anderson Localization
Disorder localizes quantum wavefunctions exponentially in 1D — all states are localized, no matter how weak the randomness.
Continued Fractions & Best Approximations
The golden ratio is the hardest number to approximate by rationals — its continued fraction [1;1,1,1,...] converges most slowly.
Swarmalators: Sync in Space
Oscillators that swarm and synchronize simultaneously — five collective states emerge from the interplay of spatial and phase coupling.
Feynman Path Integral
The quantum propagator sums over all paths weighted by exp(iS/ℏ) — stationary phase picks out the classical trajectory.
Kramers-Wannier Duality
The 2D Ising model maps to itself under duality — the self-dual point predicts the exact critical temperature.
Gibbs Paradox & Entropy of Mixing
Why does mixing identical gases produce no entropy? The paradox reveals the quantum origins of extensivity and indistinguishability.
Euler Characteristic & Topology
V−E+F = 2 for any polyhedron homeomorphic to a sphere — a topological invariant connecting geometry to genus and curvature.
Metapopulation Epidemic Dynamics
SIR waves in patchy landscapes: travel between patches synchronizes outbreaks and sustains epidemics below local extinction thresholds.
Mutualistic Network Nestedness
Plant-pollinator networks show nested structure — generalists connect to specialists' partners, stabilizing the ecological community.
Cayley Diagrams of Finite Groups
Directed graphs encoding group multiplication: visualize cyclic, dihedral, symmetric, and quaternion groups through their generators.
Bayesian Updating & Conjugate Priors
Watch a prior sharpen into a posterior as data arrives — conjugate pairs (Beta-Binomial, Gamma-Poisson) make Bayes exact.
Global Carbon Cycle Flux Model
Box model of atmosphere, biosphere, and ocean carbon reservoirs — simulate anthropogenic perturbation, airborne fraction, and warming.
Alfvén Waves in MHD
Transverse plasma oscillations propagate along magnetic field lines at the Alfvén speed v_A = B/√(μ₀ρ).
Archimedean Solids
All 13 convex polyhedra with regular faces and identical vertices — from cuboctahedron to truncated icosidodecahedron, rotating in 3D.
Breit-Wigner Resonance Scattering
Cross-section peaks sharply at resonance energy E₀ with Lorentzian width Γ = ℏ/τ — quantum scattering meets complex poles.
Osmotic Pressure & van't Hoff Law
Water crosses membranes to equalize chemical potential — van't Hoff's Π = iMRT and its cellular consequences.
Quantum Chaos & Level Statistics
Chaotic billiards show Wigner-Dyson level repulsion; integrable systems follow Poisson statistics — the fingerprint of classical chaos in quantum spectra.
Brachistochrone: Fastest Descent
The cycloid beats all other curves for fastest descent under gravity — Euler-Lagrange calculus of variations in action.
Polymer Excluded Volume & Flory Exponent
Self-avoiding walks swell beyond Gaussian: R ~ N^0.588 in 3D, with ν = 3/(d+2) from Flory's elegant mean-field theory.
Adaptive Dynamics & Evolutionary Branching
Invasion fitness and pairwise invasibility plots reveal evolutionary singular strategies — including branching points where one species splits into two.
Polaron & Self-Trapping
An electron distorts its surrounding lattice, becoming heavier — Fröhlich coupling and the self-trapping transition at strong electron-phonon interaction.
Geodesic Dome Frequency
Icosahedron subdivision at frequency ν: strut lengths, vertex counts V=10ν²+2, and the geometry behind Buckminster Fuller's revolutionary domes.
Lyapunov Stability
A Lyapunov function certifies stability without solving differential equations — visualize energy-like functions draining toward fixed points.
Arnold's Cat Map & Ergodic Mixing
A linear toral automorphism stretches and folds phase space chaotically, yet an image returns near its original after enough iterations.
Euler Buckling & Elastica
Slender columns buckle at P_cr = π²EI/L² — the exact post-buckled shape (elastica) follows elliptic integrals in a classic bifurcation.
Kuramoto-Sakaguchi Phase Oscillators
N=50 phase oscillators with frustration parameter α — coupling above K_c drives spontaneous synchronization, frustration shifts the transition.
Off-Lattice DLA
Diffusion-limited aggregation without lattice bias — random-walking particles stick on contact, growing fractal clusters with dimension D≈1.71.
Zipf's Law — Rank-Frequency Power Law
Live log-log sampling from a Zipf distribution — empirical word frequencies converge to the power-law slope −s as samples accumulate.
Minority Game — Grand Canonical
101 adaptive agents each choose 0 or 1; the minority wins. Strategy learning drives a phase transition in volatility and market efficiency.
Stochastic Block Model
Random network with 3 communities — tune p_in and p_out to explore community detectability near the Kesten-Stigum information-theoretic threshold.
Flux Balance Analysis — Metabolic Network
Animated flux flow through a 6-node glycolytic pathway — O₂ availability controls aerobic vs. anaerobic (Warburg) modes and ATP yield.
Hebbian Learning — Hopfield Attractor
10×10 Hopfield network stores 3 patterns via Hebbian learning — add noise and watch synchronous updates converge to the nearest memory.
Quantum Tunneling — WKB Barrier
Gaussian wave packet incident on a rectangular barrier — split-operator evolution shows |ψ|² tunneling through with WKB-predicted probability.
Directed Percolation — Absorbing State
Active cells propagate with probability p on a 120×80 grid — below p_c≈0.6445 activity dies out; above it a steady active phase persists.
Social Influence — Threshold Cascade
Granovetter's threshold model on a random network — small seeds cascade through when neighbor activation fractions exceed individual thresholds.
Ising Mean-Field Transition
Bifurcation of magnetization at T_c; self-consistency equation m = tanh(βJzm) and the pitchfork transition
Bond Percolation
Square lattice bond percolation with cluster coloring; critical threshold p_c = 0.5 (Kesten 1980)
Abelian Sandpile
BTW self-organized criticality: cells topple at 4 grains, avalanche sizes follow a power law
Oregonator Reaction-Diffusion
Belousov-Zhabotinsky spiral waves from the three-variable Oregonator model
Boids Flocking
Reynolds' three rules — separation, alignment, cohesion — produce emergent flocking behavior
Langton's Ant
Turing-complete two-state automaton: chaos for 10,000 steps then a periodic highway emerges
Hodgepodge Machine
Dewdney's excitable-medium cellular automaton producing rotating spiral waves
Telegraph Equation
Wave propagation with damping u_tt + 2γu_t = c²u_xx; compare damped vs pure wave solutions
Spectral Clustering
Graph Laplacian eigenvectors reveal cluster structure; Fiedler value measures connectivity
Lorentz Gas Billiard
Hard disk billiard with triangular obstacle lattice; chaotic trajectories and Lyapunov instability
Spin Glass — Edwards-Anderson
2D ±J Ising spin glass: frustrated bonds, Metropolis dynamics, and frozen glassy order at low T
Double Pendulum Chaos
Two double pendulums with nearly identical ICs diverge exponentially; live Lyapunov exponent estimate
Fibonacci Quasicrystal Diffraction
Cut-and-project quasicrystal from 5D lattice; 10-fold diffraction symmetry forbidden in periodic crystals
Game of Life — Garden of Eden
Conway's Life on 80×60; Garden of Eden patterns have no predecessor — provably unreachable states
Abelian Sandpile Identity
The fractal identity element of the sandpile group emerges from iterated toppling on an n×n grid
Random Energy Model
Derrida's REM: 2^N Gaussian energies, exact glass transition at T_g = 1/√(2 ln 2), replica symmetry breaking
FitzHugh-Nagumo Reaction-Diffusion
Excitable media on 120×80 grid: rotating spiral waves and target patterns modelling cardiac tissue
Coupled Van der Pol Oscillators
Two nonlinear oscillators synchronize in-phase or anti-phase depending on coupling strength k
Arnold Tongues — Circle Map
Phase-locked vs quasiperiodic regions in (Ω,K) space; devil's staircase winding number structure
Ornstein-Uhlenbeck Process
Mean-reverting diffusion dX=−θX dt+σ dW with stationary Gaussian distribution and exponential autocorrelation
Feed-Forward Loop Motif
Signal propagation through the C1-FFL — sign-sensitive delay filter in gene networks
SSH Topological Insulator
1D Su-Schrieffer-Heeger chain — topological edge states and winding number
Quantum Walk: Discrete Coin
Ballistic quantum spreading vs classical diffusion — interference fringes and coin bias
Rule 110 (Turing Complete CA)
The only elementary CA proven computationally universal — gliders from simple rules
SIR Epidemic on Networks
Power-law degree distribution lowers epidemic threshold — R₀ and herd immunity
Anomalous Diffusion
MSD ∝ tᵅ — subdiffusion (CTRW) and superdiffusion (Lévy flights) vs normal
Frustrated Antiferromagnet
Geometric frustration on triangular lattice — bonds that cannot all be satisfied
Critical Slowing Down
τ ∝ |r|^(−½) diverges near bifurcation — rising AC(1) as early warning signal
Kolmogorov Complexity
LZ76 complexity and compression ratio as proxies for algorithmic information content
Maximum Entropy (MaxEnt)
Jaynes' principle — constraints select the least-informative distribution (Gaussian)
SGD Loss Landscape
Navigate a 2D loss landscape with SGD, Momentum, and Adam — learning rate and noise sliders
Real-Space Renormalization Group
1D Ising block-spin decimation — coupling K flows to zero under RG with cobweb diagram
Small-World Network Rewiring
Watts-Strogatz model — rewiring probability p interpolates between lattice and random graph
Geodesic Flow — Hyperbolic Plane
Poincaré disk: geodesics as circular arcs, ideal triangles, and hyperbolic tessellations
Worm-Like Chain Polymer
Semi-flexible polymer Langevin dynamics — persistence length, WLC formula, force-extension curve
Flashing Brownian Ratchet
Asymmetric sawtooth potential flashes on/off — net drift from thermal noise resolves Feynman paradox
Wireworld Logic Gates
Four-state CA with electron signals — OR, AND, XOR gates and full adder circuits
Goldstone Modes — 2D XY Model
Spin waves, S(k)∝1/k² Goldstone spectrum, and BKT vortex unbinding transition
Spatial Predator-Prey Turing Patterns
Reaction-diffusion Lotka-Volterra — diffusion-driven Turing instability generates spots and stripes
Monte Carlo Integration
Estimating π with crude MC, stratified sampling, and importance sampling — variance reduction compared
XY Model Vortices
Topological defects in 2D: vortex/antivortex pairs and BKT transition
Neural Field Bump Attractor
1D neural field with Mexican-hat coupling: stable localized activity bump
Hopfield Network Capacity
Store and retrieve patterns; observe the capacity cliff at P/N ≈ 0.14
3D Ising Model
Metropolis MC on 20³ cubic lattice; phase transition at T_c ≈ 4.51
Chimera States
256 identical oscillators: one half synchronized, the other incoherent
Continuum Percolation
Random disk model: giant cluster and percolation threshold ρ_c ≈ 1.128
Erdős-Rényi Giant Component
G(n,p) phase transition: giant component emerges sharply at ⟨k⟩ = 1
HPP Lattice Gas
Hardy-Pomeau-Pazzis automaton: head-on collisions produce fluid-like flow
Heat Equation Green's Function
Gaussian spreading from δ-function IC; log σ² vs log t slope = 1
Wigner Function
Quantum phase space quasi-probability; negative regions in Fock and cat states
Genetic Drift & Founder Effect
Watch allele frequencies wander under random drift — bottleneck and founder effects in small populations
Fourier Heat Equation
See how heat diffuses along a rod — Fourier series decomposition solving the PDE in real time
Rule 110 — Turing Complete CA
The elementary cellular automaton proven Turing complete — complex patterns from a 3-cell rule
HP Lattice Protein Folding
Explore the hydrophobic-polar model of protein folding on a 2D lattice — NP-hard optimization
Abelian Sandpile — Self-Organized Criticality
Drop grains of sand and watch avalanches distribute in a power law — Bak-Tang-Wiesenfeld 1987
Boids Flocking Simulation
Three rules — separation, alignment, cohesion — produce lifelike flocking from local interactions
Continued Fractions & Irrationals
Every irrational has a unique infinite continued fraction — visualize convergents approaching the target
Cantor Set & Fractal Dimension
Remove the middle third forever — a set with measure zero but uncountably many points, dim = log2/log3
Butterfly Effect — Lorenz Sensitivity
Two nearly identical trajectories diverge exponentially — the geometric meaning of chaos
Maxwell-Boltzmann Gas Speeds
Watch a gas of particles reach thermal equilibrium — speed distribution converges to Maxwell-Boltzmann
2D Ising Model — Monte Carlo
Spins flip by Metropolis sampling — watch spontaneous magnetization emerge below the Curie temperature
Logistic Map — Period Doubling
x → rx(1-x) bifurcates to chaos through period-doubling — Feigenbaum's universal constant δ ≈ 4.669
Kuramoto Oscillators — Synchronization
Coupled oscillators with random frequencies spontaneously synchronize above a critical coupling strength
Random Boolean Network — Kauffman NK
N nodes, K inputs each — tune K to find the phase transition between order and chaos
Brusselator — Turing Patterns
A two-species reaction-diffusion system that spontaneously breaks spatial symmetry into stripes and spots
Fourier Transform — Live Spectrum
Draw any waveform and watch its frequency spectrum appear — sums of sines decompose everything
Mandelbrot Set — Deep Zoom
Click to zoom into the boundary of the Mandelbrot set — infinite complexity at every scale
Barabási-Albert — Preferential Attachment
Rich get richer: new nodes attach to high-degree nodes, producing a power-law degree distribution
FitzHugh-Nagumo — Excitable Media
Spiral waves and target patterns in an excitable medium — the math behind cardiac arrhythmias
Small-World Network — Watts-Strogatz
Rewire a ring lattice with probability p — average path length drops fast, clustering stays high
Brownian Motion — Zero Crossings
How often does a random walk return to zero? Arc-sine law and the fractal structure of recurrence times
Boltzmann Entropy — Gas Mixing
Two gases separated by a partition — watch entropy increase irreversibly as they mix
Kleinberg Navigation — Geographic Small World
Long-range links with P(r) ∝ r^(-α) — only α=2 allows efficient decentralized routing in 2D
Voter Model — Opinion Coarsening
Each agent copies a random neighbor — domains grow slowly, consensus reached only in 1D and 2D
Polymer as Random Walk — End-to-End Distance
A polymer chain is a random walk — end-to-end distance scales as N^ν where ν=3/5 in 3D (Flory)
Cascade Failure — Network Robustness
Remove a hub and watch the network fragment — targeted vs random attack on scale-free graphs
Diffusion Maps — Geometric Data Structure
Eigenvalues of a diffusion operator reveal the intrinsic geometry of high-dimensional data
Random Matrix — Eigenvalue Repulsion
Eigenvalues of random Hermitian matrices repel each other — GUE spacing follows Wigner surmise
Chimera States — Synchrony and Chaos Coexist
Identical coupled oscillators where some synchronize and others remain chaotic — discovered 2002
Network Resilience — Percolation on Graphs
Remove nodes randomly and find the percolation threshold — scale-free networks have p_c = 0
Persistent Homology — Loop Detection
Topological data analysis: find loops in point clouds by tracking when holes appear and disappear
Hamiltonian Monte Carlo — Leapfrog
MCMC using Hamiltonian dynamics — leapfrog integration proposes distant moves with high acceptance
Cayley Graphs — Group Structure as Geometry
Every group becomes a graph when you connect elements by generators — symmetry becomes topology
Rock-Paper-Scissors — Cyclic Dominance
Three species in cyclic competition produce rotating spirals — biodiversity through intransitivity
DLA — Diffusion-Limited Aggregation
Particles diffuse randomly and stick on contact — a fractal cluster with dimension ≈ 1.71 grows
Kinetic Theory — Effusion & Graham's Law
Gas escaping through a tiny hole — lighter molecules escape faster, rate ∝ 1/√mass (Graham 1848)
Jacobi Iteration — Solving Poisson's Equation
Repeated local averaging converges to the solution — watch the electrostatic potential relax
Firefly Synchronization — Pulse-Coupled Oscillators
Fireflies adjust their flash timing when they see neighbors — global sync emerges from local rules
Anomalous Diffusion — Sub and Superdiffusion
Normal diffusion: ⟨x²⟩ ∝ t. Anomalous: ⟨x²⟩ ∝ t^α — from trapped particles to Lévy flights
Bistable Switch — Multistability & Hysteresis
Two stable states, one unstable — slow parameter change causes sudden irreversible jumps (catastrophe)
Sandpile Identity — Abelian Structure
The identity element of the abelian sandpile group has fractal structure — a mathematical surprise
Mean Field Game — Crowd Navigation
Rational agents navigate a crowd — Nash equilibrium from a PDE system, not individual optimization
Ramsey Theory — Unavoidable Patterns
Color edges of K_n red/blue — for n≥6 you always get a monochromatic triangle (R(3,3)=6)
Laplacian Growth — Saffman-Taylor Fingers
Viscous fingering: a less viscous fluid displaces a more viscous one — unstable interface grows fractal fingers
Fibonacci Quasicrystal — 1D Penrose
A and B tiles arranged by substitution rule — quasiperiodic order with 5-fold diffraction
Master Equation — Birth-Death Process
Stochastic evolution of population probabilities — exact solution via generating functions
Vicsek Model — Flocking Phase Transition
Self-propelled particles align with neighbors — order parameter jumps discontinuously at noise threshold
Chaotic Scattering — Sinai Billiard
A billiard ball bouncing between circular obstacles — exponential sensitivity and fractal escape basins
Crumpled Membrane — Surface Roughness
Elastic sheets crumple into fractal ridges — roughness exponent H ≈ 0.64 from scaling theory
Chip-Firing — Toppling Waves on Graphs
Add chips to a graph — when a node overflows it fires to neighbors, producing a toppling wave
Van Hove Singularities
Kinks in the density of states where group velocity vanishes — animate 1D/2D DOS with interactive band parameters
Dielectric Breakdown & DLA
Diffusion-limited aggregation grows fractal lightning and mineral dendrites — tune sticking probability and watch the fractal dimension emerge
Majority Rule Dynamics
Agents adopt the local majority opinion — interactive grid showing consensus vs disorder phase transition with noise parameter
Catastrophe Theory
Fold and cusp catastrophes — interactive control sliders reveal bifurcation, hysteresis, and sudden jumps in a potential landscape
BTW Sandpile
Self-organized criticality — drop grains, watch avalanches cascade with power-law size distribution and no parameter tuning
Hopf Fibration S³→S²
Each point on S² lifts to a linked circle in S³ — visualize Villarceau fibers on nested tori projected to 3D, drag to rotate
Hutchinson's Ecological Niche
N-dimensional hypervolume concept — interactive 2D niche overlap with Schoener's D, competitive exclusion, and coexistence
Sznajd Model
"United we stand" opinion dynamics — agreeing pairs convince their neighbors; watch herding and consensus on an interactive grid
Pólya Urn Process
Reinforcement learning model — draw a ball, add more of that color; explore path dependence and Beta distribution convergence
Iterated Prisoner's Dilemma
Axelrod tournament — round-robin competition between TFT, AllD, AllC, Pavlov, Grudger, and Random with payoff matrix visualization
Network Motifs
Milo 2002: subgraph patterns over-represented in real networks vs random; interactive directed graph with Z-score motif counter
Spectral Clustering
Normalized cut via graph Laplacian eigenvectors; interactive 2D point cloud partitioned into clusters with eigenspectrum display
Kramers Escape Problem
Thermal activation over a double-well potential; Brownian particle with Arrhenius rate, adjustable temperature and barrier height
Kuramoto Model (Finite-N)
Finite-size fluctuations in the synchronization transition; N coupled oscillators with order parameter r(t) and phase circle
Swift-Hohenberg Equation
Pattern formation PDE generating stripes, hexagons, and spirals; real-time simulation with ε and nonlinearity controls
Transition State Theory
Eyring-Evans-Polanyi activated rate theory; interactive energy landscape with temperature-dependent rate constants and Arrhenius plot
Hill Equation & Cooperativity
Sigmoidal binding curves; interactive Hill coefficient, hemoglobin vs myoglobin O₂ saturation, cooperativity Hill plot
Chaos & Shadowing Lemma
Pseudo-orbits in the logistic map shadowed by true orbits; visualize Lyapunov divergence and shadow length vs perturbation
Virial Theorem (N-body)
2K + V = 0 at gravitational equilibrium; interactive N-body simulation showing kinetic and potential energy oscillating around virial ratio
Non-equilibrium Entropy Production
σ = J·F in driven systems; Brusselator oscillator with entropy production rate, phase portrait, and bifurcation diagram
GOE Eigenvalue Repulsion
Level repulsion in the Gaussian Orthogonal Ensemble; animate nearest-neighbor spacing statistics vs Poisson (integrable) and Wigner surmise (chaotic)
Deformable Elastic Surface
Helfrich membrane energy F = κ∫(2H)²dA + σ∫dA; interactive mesh deformation with bending rigidity, tension, pressure, and thermal noise
Worm Algorithm (Ising)
Prokof'ev-Svistunov worm Monte Carlo near criticality; visualize the worm path updating the Ising configuration, magnetization, and energy
Jarzynski Equality
Free energy from non-equilibrium work: ⟨e^{−βW}⟩ = e^{−βΔF}; interactive pulling experiment showing convergence vs direct averaging
Crooks Fluctuation Theorem
P_F(W)/P_R(−W) = e^{(W−ΔF)/kT}; visualize forward and reverse work distributions crossing at W = ΔF with log-ratio test
Landau-Ginzburg Theory
Free energy F = am² + bm⁴ + cm⁶ with interactive potential landscape; first/second order transitions, tricritical point, and symmetry breaking
Mode-Coupling Theory
α and β relaxation in supercooled liquids; visualize F(q,t) intermediate scattering function with two-step decay and dynamical arrest near T_c
Glass Transition
Supercooled liquid → glass; 2D particle simulation with Voronoi tessellation, non-affine displacement maps, and dynamic heterogeneity as T decreases
Roughening Transition
Crystal interface transitions from smooth to rough at T_R; SOS model with height fluctuations, roughness W² = ⟨h²⟩, and temperature scan
Reptation & Polymer Dynamics
Entangled polymer chain diffusing through a tube (de Gennes); animate primitive path, Rouse modes, and MSD ~ t^{1/2} → t crossover
DMRG & Matrix Product States
Variational MPS ansatz for 1D quantum chains; entanglement entropy S(x), bond dimension χ, energy convergence
Thouless Pump
Adiabatic charge pumping — one quantum of charge per cycle; Chern number, band structure, quantized transport
Kitaev Chain & Majorana Modes
1D p-wave superconductor hosting Majorana zero modes; interactive phase diagram, edge states, BdG spectrum
Goldstone Theorem & SSB
Mexican hat potential, massless Goldstone bosons from continuous symmetry breaking, Higgs mode
Frenkel-Kontorova Model
Atoms in a periodic potential; commensurate/incommensurate phases, Aubry transition, kink solitons
SSH Model
Paradigmatic 1D topological insulator; winding number, bulk-boundary correspondence, zero-energy edge states
Luttinger Liquid & Bosonization
Interacting 1D fermions; power-law correlations, spin-charge separation, spectral function A(k,ω)
SYK Model
Random all-to-all interacting fermions; maximal quantum chaos, OTOC growth, spectral form factor, holography
OTOCs & Quantum Chaos
Out-of-time-order correlators measure scrambling; butterfly effect, Lyapunov exponent, level statistics
Floquet Topological Phases
Periodically driven systems host anomalous π-mode edge states impossible in equilibrium; Floquet SSH bands
Mean Field Games
Nash equilibria for large agent populations — interactive traffic flow showing individual best-response vs. collective equilibrium
Replicator Dynamics
Evolutionary game theory ODEs on the 3-strategy simplex — rock-paper-scissors and hawk-dove with trajectory visualization
Kirman Herding Model
Ant colony recruitment gives bimodal opinion distributions — tune α and ε to switch between unimodal and bimodal regimes
Watts Cascade Model
Global cascades on networks via local threshold rules — interactive threshold distribution and cascade size visualization
DeGroot Social Learning
Weighted belief averaging on a social network — watch convergence to consensus and wisdom vs. foolishness of crowds
Signal Detection Theory
d' separates sensitivity from bias — interactive ROC curve with hit/false-alarm rates and criterion adjustment
Wisdom of Crowds & Diversity
Hong-Page theorem: diverse teams beat expert teams — simulation of diverse vs. homogeneous agents on a fitness landscape
Collective Memory Reconstruction
Bartlett's schema theory — network simulation of story retelling with systematic drift toward familiar patterns
Schelling Tipping Points
Thomas Schelling's neighborhood model — hysteresis and irreversible tipping from mild tolerance thresholds
Network Centrality & Power
Bonacich power centrality — being connected to powerful nodes multiplies your power, interactive network bargaining
Markov Decision Processes
Value iteration & policy iteration on an interactive grid world with animated policy arrows and value heatmap.
Multi-Armed Bandit — UCB
Upper Confidence Bound vs ε-greedy vs Thompson Sampling on a k-arm bandit with live regret curves.
TD(0) Learning
Bootstrapped value estimation in a random walk: TD errors, convergence vs Monte Carlo, animated learning.
Q-Learning Gridworld
Model-free off-policy control with an editable gridworld, Q-table heatmap, and ε-greedy exploration paths.
REINFORCE Policy Gradient
Monte Carlo policy gradient on a pendulum task: reward history, policy parameter evolution, softmax policy.
Sequential Bayesian Updating
Beta posterior narrows as observations stream in. Interactive prior selection, 95% CI, and posterior mean tracking.
Actor-Critic with Advantage
Separate actor (policy) and critic (value) in a gridworld. Advantage A(s,a) = Q − V drives policy updates.
Monte Carlo Tree Search
UCT selection, expansion, simulation, backpropagation. Interactive tree with visit counts and win rates.
Exploration-Exploitation
ε-greedy, UCB, Thompson Sampling, and Boltzmann compared live on a bandit — cumulative reward and regret charts.
Inverse Reinforcement Learning
Recover a hidden reward function from expert trajectories. Side-by-side true vs IRL-inferred reward heatmaps.
Quantum Tunneling Barrier
Wavepacket meets a potential barrier: watch it split into reflected and tunneled parts, defying classical physics.
Network Motifs & Feedback Loops
Feedforward and feedback loop motifs in gene regulation — the recurring circuit patterns that give cells their dynamics.
Convex Hull — Quickhull
Step through the Quickhull algorithm: recursively find the farthest point from each edge to build the convex hull.
Stochastic Resonance
Adding noise to a weak signal can make it detectable — a counterintuitive phenomenon found in neurons and sensory systems.
Diffusion-Limited Aggregation
Random-walking particles stick on contact, building branching fractals with dimension approximately 1.71 — DLA in real time.
Dendritic Crystal Growth
Snowflake-like dendrites emerge from the Reiter cellular automaton — freezing, diffusion, and six-fold symmetry.
QHO Coherent States
Coherent states of the quantum harmonic oscillator oscillate without spreading — the most classical quantum state.
Bifurcation & Cobweb Map
Explore the logistic map route to chaos via period-doubling bifurcations. Feigenbaum constant delta approximately 4.669.
Laplacian Growth
Random walkers aggregate according to the harmonic measure — Laplace equation governs where the cluster grows.
Geodesic Sphere Triangulation
Subdivide an icosahedron, project to the sphere. Drag to rotate the 3D wireframe and explore Euler formula V minus E plus F equals 2.
Random Geometric Graph
Points in unit square connect if distance < r; watch the connectivity phase transition.
Pendulum Phase Space
Phase portrait and Poincaré section of the driven damped pendulum; regular orbits vs chaos.
Kármán Vortex Street
Alternating vortex shedding behind a cylinder simulated with point vortex dynamics.
SIR Epidemic on Networks
Susceptible-Infected-Recovered spreading on ER, BA, and lattice networks.
Bloch vs Néel Domain Wall
Spin texture inside magnetic domain walls: out-of-plane Bloch vs in-plane Néel rotation.
Julia Set Parameter Sweep
Interactive Julia set J_c with smooth coloring; click the Mandelbrot mini-map to pick c.
Elementary CA Rule Space
All 256 elementary CA rules: from fixed points to Turing-complete Rule 110.
SK Spin Glass
Sherrington-Kirkpatrick model: Metropolis MC on random couplings, Parisi RSB order parameter.
Kauffman NK Network
Boolean networks at the edge of chaos: K=2 critical phase, attractors and basin structure.
Wave Equation d'Alembert
1D wave equation: d'Alembert's solution splits initial conditions into left- and right-traveling waves.
Lorenz Attractor Parameter Space
Explore σ, ρ, β parameters of the Lorenz system; watch bifurcations as ρ crosses chaos onset.
Power Law & Pareto Distribution
Heavy tails and scale invariance: compare power laws to exponentials across linear, log-log, and CCDF plots.
Young-Laplace Equation
Pressure jump across curved interfaces, contact angle, and capillary rise via the Young-Laplace equation.
Moran Process
Genetic drift and selection in finite populations: fixation probability and selection-drift balance.
Hypercube Graph & Gray Code
Q₂ to Q₅ hypercube graphs with interactive 3D rotation; animate the Gray code Hamiltonian path.
Freely-Jointed Polymer Chain
Random walk model of polymer physics: end-to-end distance, radius of gyration, and P(R) distribution.
Complex Ginzburg-Landau Equation
Spiral waves, dissipative solitons, and Benjamin-Feir chaos in the CGLE reaction-diffusion PDE.
Granular Packing & Jamming
Bidisperse disk simulation: jamming transition near φ≈0.84, force chain visualization, and compression.
Kicked Rotor — Hamiltonian Chaos
Standard map Poincaré section: KAM tori, Chirikov threshold K≈0.97, and Arnold diffusion.
Multiplex Network Cascades
Two interdependent network layers: trigger cascading failures and watch systemic collapse propagate.
Sierpiński Gasket — Chaos Game
Random IFS iteration converges to the fractal: pick a vertex, jump halfway, repeat. Dimension log2/log3 ≈ 1.585.
Lid-Driven Cavity Flow
2D Navier-Stokes finite-difference simulation. Moving top lid drives recirculation vortices — tune Reynolds number for complex flow.
Threshold Model of Social Contagion
Watts (2002): cascade size vs. threshold heterogeneity on random networks. Tiny seed can trigger global adoption.
Ising Correlation Function
C(r) = ⟨σ₀σᵣ⟩ at different temperatures. At T_c the correlation length diverges — a hallmark of criticality.
Game of Life — Methuselahs
R-pentomino (1103 gen), Acorn (5206 gen), Diehard (130 gen). Small patterns with enormous lifespans.
Geodesic Billiards
Circle and square give integrable orbits; the Bunimovich stadium and cardioid are fully chaotic — nearby trajectories diverge exponentially.
Bell Inequality (CHSH)
Drag detector angles to explore quantum vs. classical correlations. Quantum max 2√2 ≈ 2.828 violates the classical bound of 2.
Cantor Set & Devil's Staircase
Iterative middle-third removal: measure → 0, yet uncountably infinite. The Cantor function is continuous but has derivative 0 almost everywhere.
Autocatalytic Reaction Networks
A + X → 2X: substrate drives runaway catalyst growth. Explore bistability, mutual autocatalysis, and logistic saturation.
Forest Gap Dynamics
Disturbance → pioneer → mid-succession → climax. Spatial cellular automaton of gap-phase succession with tunable disturbance rate.
Heat Equation Fourier Modes
Decompose initial conditions into Fourier modes; watch each decay as e^(−k²t) under the diffusion equation.
Airy Disk & Rayleigh Criterion
Far-field diffraction through a circular aperture; explore the Rayleigh resolution limit and wavelength dependence.
Neural Field Bump Attractors
Amari neural field equation with Mexican-hat connectivity; localized activity bumps as spatial working memory.
2D Totalistic Cellular Automata
Explore the full rule space of 2D neighbor-sum automata including Conway Life, HighLife, Seeds, and Day & Night.
Einstein Brownian Motion & MSD
Simulate Perrin's colloid tracks; measure mean squared displacement growing as 4Dt to extract the diffusion coefficient.
Quantum Chaos: Stadium Billiard
Bunimovich stadium wavefunctions, eigenstate scarring, and Wigner-Dyson level spacing statistics from quantum chaos.
RG Block-Spin Decimation
Block-spin renormalization group on the 1D Ising model; RG flow, fixed points, and cobweb diagram of K to K prime.
Fractal Basin Boundaries
Newton's method basins of attraction for polynomial roots; fractal structure at all scales along basin boundaries.
Nonlinear Pendulum & Elliptic Integrals
Exact pendulum period via complete elliptic integral K(k); compare to small-angle approximation as amplitude grows.
FitzHugh-Nagumo Neuron
Excitable neuron dynamics with fast membrane potential and slow recovery; phase plane, limit cycles, and spike trains.
Lissajous Figures & Resonance
Parametric x=A sin(at+δ), y=B sin(bt) — sliders for frequency ratio and phase, preset rational resonances
2D Ising & Onsager Solution
Metropolis MC simulation with Onsager's exact magnetization overlay — mark T_c = 2/ln(1+√2) ≈ 2.269
Abelian Sandpile (BTW)
Bak–Tang–Wiesenfeld model: drop grains, watch avalanches topple, fractal boundary emerges from self-organized criticality
Jacobi Elliptic Pendulum
Exact pendulum via sn(u,k) vs small-angle approximation — phase portrait, period divergence near π
Spin Glass — TAP & RSB
Frustrated random bonds, simulated annealing, replica overlap P(q) distribution — Parisi's replica symmetry breaking
Bose–Einstein Condensate
Occupation numbers vs temperature, macroscopic ground-state condensation, momentum-space delta peak below T_c
Continuum Percolation
Boolean model: random discs, giant cluster colored blue, percolation transition at η_c ≈ 0.676
Josephson Junction
DC & AC Josephson effects: phase dynamics φ(t), I=Ic sin(φ), I-V characteristic, RCSJ model
Kuramoto Model with Noise
N noisy oscillators synchronize above K_c = 2Δ — order parameter r(t), theory vs simulation, phase circle
Kármán Vortex Street
Alternating vortex shedding behind a cylinder — streamline tracers, Strouhal number St ≈ 0.2
Stochastic Resonance
Optimal noise enhances weak signal in a bistable double-well — SNR peak and particle animation
Stokes Flow
Creeping flow around a sphere at Re→0: streamlines, drag F=6πηRU, vs potential flow
Wiener-Khinchin Theorem
Autocorrelation C(τ) and power spectral density S(f) are Fourier pairs — interactive signal builder
Hénon-Heiles Chaos
Poincaré section of Hénon-Heiles Hamiltonian — KAM tori at low energy, chaos at high energy
Min Protein Oscillations
MinD/MinE reaction-diffusion in E. coli finds cell midpoint for division via Turing-like instability
Crystal Dislocation
Edge dislocation dynamics: Burgers vector, stress field σ(x,y), glide under applied shear stress
Quantum Density Matrix
Bloch sphere — pure vs mixed states, decoherence animation, purity Tr(ρ²) and von Neumann entropy
Spin Ice Monopoles
Pyrochlore spin ice with 2-in 2-out ice rules — Pauling entropy, magnetic monopole defects, Monte Carlo
Contact Process
Directed percolation universality: active-absorbing phase transition at λ_c ≈ 3.298 in 1+1D
Barabási-Albert Network
Preferential attachment grows scale-free network — P(k)~k^{-3} degree distribution live log-log plot
Crooks Fluctuation Theorem
Work distributions for forward/reverse protocols; Jarzynski equality recovering ΔF from non-equilibrium trajectories.
Kramers Escape Rate
Thermal escape from a double-well potential. Langevin dynamics, Arrhenius plot, and τ ∝ e^{ΔU/kT}.
Quantum Error Correction
3-qubit bit-flip stabilizer code. Syndrome measurement detects errors without revealing the logical state.
Allometric Metabolic Scaling
Kleiber's Law B ∝ M^{3/4} from bacteria to blue whales. Fractal vascular branching (West-Brown-Enquist).
Arctic Circle Theorem
Random domino tilings of Aztec diamonds. Frozen corners, chaotic bulk, and the inscribed arctic circle boundary.
Vicsek Flocking Model
Self-propelled particles aligning with neighbors. Order parameter φ vs noise η — a collective motion phase transition.
Burgers Equation Shocks
u_t + u·u_x = ν·u_xx. Watch smooth initial conditions steepen into shocks; viscosity ν→0 limit.
Eden Model & KPZ Growth
Random cluster growth with rough interfaces. Interface width W(t) ~ t^{1/3} — KPZ universality class.
Vibrating Drum Eigenmodes
Bessel function modes J_m(j_{mn}r)cos(mθ) on a disk. Animated nodal lines, color, and Web Audio tones.
Spinodal Decomposition
Cahn-Hilliard equation: phase separation from noise. Spinodal ring in Fourier space; coarsening dynamics.
Amplitude Death
Stuart-Landau oscillators with coupling and frequency mismatch — phase transition to fixed point silence
Hawking Radiation
Thermal spectrum T_H = ℏc³/8πGMk — blackbody glow shifting as a black hole evaporates
Majority Vote Model
Spin flips to local majority on 2D lattice — phase transition at noise q_c ≈ 0.075, Ising universality
Topological Solitons
φ⁴ kink φ(x)=tanh((x−x₀)/√2) — kink-antikink collision, annihilation, sine-Gordon breather
Polymer Flory Exponent
Self-avoiding walk on 2D lattice — Flory ν≈3/4 vs ideal ν=1/2, log-log scaling
Galton-Watson Branching
Poisson offspring branching trees — extinction probability phase transition at mean μ=1
Nematic Liquid Crystal
Director field relaxation — ±½ topological defects, Frank elastic energy, order parameter vs temperature
Holographic Entropy
Ryu-Takayanagi: S = Length(γ)/4G — minimal geodesic in AdS₃ bulk, S=(c/3)ln(l/ε)
Markov Cutoff Phenomenon
TV distance drops sharply at mixing time t*~n log n — hypercube random walk, cutoff visualization
Swift-Hohenberg Patterns
2D pattern formation: stripes, hexagons, spots via ∂u/∂t=(ε−(1+∇²)²)u−u³. Interactive ε slider.
Virial Equation of State
Z=PV/NkT virial expansion vs van der Waals. Boyle temperature where B₂=0. Interactive T slider.
Glassy Dynamics & Energy Landscape
Rugged energy landscape, cage effect, aging — thermal particle exploration and MSD plateau.
Quantum Phase Estimation
QPE circuit: n-qubit register estimating eigenphase φ of unitary U. Probability distribution and IQFT.
BTW Sandpile Avalanches
Self-organized criticality: avalanche size distribution P(s)~s^{-τ} with τ≈1.2. Log-log histogram and power-law fit.
Hysteresis & First-Order Transition
Landau free energy with metastable states, spinodal limits, and hysteresis loop area vs sweep rate.
Microcanonical Entropy
Count microstates Ω(E,N), derive S=k ln Ω, extensivity, and temperature 1/T=∂S/∂E interactively.
Ergodic Billiard
Bunimovich stadium (ergodic) vs rectangle (non-ergodic). Phase space filling, invariant tori, coverage.
Turing Patterns 2D
Activator-inhibitor reaction-diffusion: Gray-Scott model producing spots, labyrinths, and stripes.
Chimera State
Non-locally coupled Kuramoto oscillators: coherent and incoherent regions coexist spontaneously on a ring.
Voronoi Diagram Dynamics
Click to place seeds; Lloyd's algorithm relaxes cells toward centroids in real time.
Max-Flow / Min-Cut
Ford-Fulkerson augmenting paths on a layered network; visualize the min-cut theorem.
Minority Game Dynamics
Adaptive agents compete to be in the minority; watch volatility collapse near criticality.
Magnetic Reconnection X-Point
Field-line topology near a magnetic X-point; particles trace reconnection flows.
Spectral Clustering & Fiedler Vector
Partition random graphs via the Laplacian's second eigenvector (Fiedler vector).
Kramers Escape Rate
Brownian particles escape a double-well potential; measure Arrhenius thermal activation.
Herding & Threshold Cascades
Granovetter's threshold model: see how distribution shape triggers or prevents cascades.
Contact Process / SIR Lattice
Epidemic spreading on 2D lattice near the percolation critical point; fractal clusters emerge.
Topological Charge & Vortices
XY-model spins; vortex/antivortex pairs reveal the BKT topological phase transition.
Cantor Set & Fractal Dust
Animated Cantor construction with variable removal; devil's staircase and Hausdorff dimension.
Reaction Network Dynamics
Mass-action kinetics on Lotka-Volterra, Schlögl, and brusselator networks with live phase portraits.
Polymer Chain / Rouse Model
Bead-spring chain in thermal noise; watch end-to-end distance fluctuate and Rouse modes relax.
Phase Retrieval — Gerchberg-Saxton
Recover complex phase from amplitude measurements via alternating Fourier projections.
Crystal Nucleation Theory
Classical nucleation: ΔG barrier, critical radius, and Arrhenius rate; nuclei grow or dissolve.
Entropy Production & Irreversibility
Gas expansion into vacuum; Boltzmann H-theorem in action — entropy only increases.
Telegraph Equation
Interpolate between wave and diffusion with adjustable damping; click to inject pulses.
Lagrange Multipliers
Constrained optimization visualized: level curves, constraint, gradient parallelism at optima.
Active Matter — Vicsek Model
Self-propelled particles align and flock; tune noise past the polar order phase transition.
Jamming Transition
Compress soft disks to φ_J≈0.64; watch pressure diverge and the force network percolate.
Cahn-Hilliard Phase Separation
Spinodal decomposition with coarsening; domain growth follows t^(1/3) Lifshitz-Slyozov law.
Rényi & Tsallis Entropy
Drag a probability distribution and compare Shannon, Rényi, and Tsallis entropies across all orders q.
Hyperbolic Billiard
Geodesic trajectories in the Poincaré disk; chaotic reflections reveal positive Lyapunov exponent.
Wigner-Seitz Cell & Lattices
Voronoi cells of 2D Bravais lattices; compare real-space WS cell with reciprocal Brillouin zone.
Huffman Coding & Entropy
Build optimal prefix-free codes for any text; see how code length approaches the Shannon entropy bound.
Random Geometric Graph
Giant component appears at percolation threshold r_c; sweep radius to watch the transition.
Advection-Diffusion Tracer
Passive dye stirred by gyre and vortex flows; tune Péclet number from diffusive to chaotic mixing.
Stochastic Integrate-and-Fire
LIF neuron with noise; ISI histogram shows transition from regular to Poisson-like firing.
Boids Flocking
Three local rules — separation, alignment, cohesion — produce emergent murmuration-like flocks.
Bifurcation Zoo
Saddle-node, transcritical, pitchfork, and Hopf bifurcations; animate parameter sweep to watch topology change.
Turing Patterns — 3-Species
Three-component reaction-diffusion on 2D grid; substrate species enables spirals, hexagons, and labyrinths.
Riemann Zeta — Critical Strip
Phase-color plot of ζ(s); zeros on Re=½ appear as phase vortices — the Riemann Hypothesis visualized.
Mixed-Strategy Nash Equilibrium
Edit the payoff matrix; replicator dynamics finds Nash equilibria with live PIP and phase portrait.
Viscosity & Reynolds Number
Lattice-Boltzmann lid-driven cavity: tune Re from laminar recirculation to unsteady turbulent flow.
Maxwell Construction
van der Waals isotherms with equal-area Maxwell construction; sweep temperature through the critical point.
Evolutionary Branching
Adaptive dynamics with competition; watch populations branch at fitness minima — origin of diversity.
Chemical Potential & Fugacity
Two containers equilibrate by diffusion; μ-equalizing visualized for ideal and van der Waals gases.
Neural Avalanches & SOC
Neuronal network self-tunes to criticality; avalanche size distribution follows P(s)∝s⁻¹·⁵ power law.
First-Passage Times
Many walkers race to an absorbing boundary; FPT distribution compared to Lévy-Smirnov theory.
Kolmogorov Complexity
Approximate K(x) via LZW compression; compare random, periodic, Fibonacci, and Mandelbrot strings.
Logistic Growth & Allee Effects
Population dynamics with Allee threshold and harvesting; click phase portrait to launch trajectories.
Topological Soliton / Kink
φ⁴ and sine-Gordon kink-antikink collisions, breather formation, topological charge conservation.
Spectral Density & Eigenvalues
Wigner semicircle, Marchenko-Pastur, and circular law from random matrix ensembles GOE/Wishart/Ginibre.
Fokker-Planck Equation
Probability density evolution under drift and diffusion. Compare PDE solution with Langevin particle simulations.
Lattice Walk & Polymer Statistics
Self-avoiding walks vs random walks. Flory exponent ν=3/4 in 2D. Scaling R~N^ν for polymer chains.
Cosmic Web & Filament Formation
Gravitational N-body collapse producing sheets, filaments, and halos — the large-scale cosmic web structure.
Ising Transfer Matrix
Exact 1D Ising solution via transfer matrix. Free energy, magnetization, and correlation length from eigenvalues.
Mean-Field Self-Consistent
Graphical solution of m=tanh(βJzm+βh). Bifurcation at Tc=Jz and order parameter β=1/2 critical exponent.
Kuramoto Synchrony Phase
N coupled oscillators with disorder. Phase transition to synchrony at Kc=2/(πg(0)). Ott-Antonsen exact theory.
Escape Basin Fractal Structure
Newton's method basins for z^n=1 and Hamiltonian scattering. Wada basins with fractal boundary dimension.
Dilation Symmetry & Scaling
Scale invariance: Weierstrass self-similar functions, power laws, RG flow diagrams, and 1/f^α noise spectra.
Percolation Crossing Probability
Bond percolation crossing transition at p_c=0.5. Cardy's exact formula and sharp threshold visualization.
Matrix Product States
MPS tensor networks, Schmidt decomposition, entanglement entropy, and area law vs bond dimension.
Quantum Walk (Coined)
Hadamard coin quantum walk spreads ballistically (σ∝t) vs classical diffusion (σ∝√t). Twin-peak interference pattern.
Chern Insulator & Edge States
Haldane model Berry curvature, Chern number C=±1, and topological phase diagram in parameter space.
Epidemic Extinction Threshold
Stochastic SIS extinction times. Below R₀=1 epidemics collapse; above, quasi-stationary state persists exponentially long.
Virial Expansion & Real Gas
Second virial coefficient B₂(T) from Lennard-Jones potential. Compare ideal, vdW, and virial equation of state.
Stochastic Resonance
Optimal noise enhances signal detection in a bistable system. SNR peaks at resonant noise strength.
Debye Phonon Heat Capacity
Debye T³ law, Einstein model, and Dulong-Petit limit. Compare C_v(T) for diamond, copper, lead, silicon.
Adiabatic Quantum Annealing
Quantum annealing on Ising problems. Energy gap, adiabatic condition, and tunneling success probability.
Saddle Points & Instantons
Instanton paths in Euclidean time mediate quantum tunneling. Euclidean action, energy splitting, and one-loop prefactors.
CA Rule Space Explorer
All 256 Wolfram elementary CA rules. Four complexity classes, Turing-complete rule 110, and class navigation.
Swift-Hohenberg Patterns
Canonical pattern formation PDE near pitchfork bifurcation. Stripes, hexagons, and labyrinthine patterns emerge.
Lattice Field Theory Continuum
Scalar field on lattice: dispersion, propagator, species doubling artifact, and continuum limit a→0.
Bayesian Model Selection
Evidence and BIC for polynomial regression. Occam's razor naturally penalizes complexity as data arrives.
Cavity Method & Belief Propagation
Message-passing on sparse Ising graphs. BP marginals vs exact values. Loopy convergence and factor graph.
Abelian Sandpile & Chip Firing
Deterministic chip-firing on a grid. Abelian group structure, fractal identity element, SOC avalanches.
Hamiltonian Monte Carlo
HMC leapfrog proposals vs random walk MH. Energy conservation, acceptance rates, and 2D target comparison.
Community Detection & Modularity
Louvain greedy modularity on planted partition graphs. Detectability threshold and community structure.
Thermodynamic Integration
Free energy differences via TI, Jarzynski equality, and BAR. Hysteresis, work distributions, and alchemical transforms.
Poisson & Renewal Processes
Poisson, regular, bursty, and Hawkes point processes. Inspection paradox, renewal theorem, and inter-event statistics.
Entanglement Entropy & Area Law
S(l) scaling for area, logarithmic (CFT), and volume law states. Calabrese-Cardy formula and Ryu-Takayanagi holographic geodesic.
Graph Neural Network: Message Passing
Visualize GNN layer-by-layer feature aggregation on a graph. Mean, sum, and max pooling with Weisfeiler-Lehman expressiveness.
Lattice Gas Cellular Automaton
HPP lattice gas with 4-bit per cell collision rules. Poiseuille flow and obstacle scattering emerge from local particle interactions.
Geodesic Flow in Hyperbolic Space
Poincaré disk geodesics as circular arcs perpendicular to boundary. Click to place points, measure hyperbolic distance, explore chaotic flow.
Entropy & Large Deviation Theory
Cramér's theorem: tail probabilities decay as exp(-n·I(x)). Rate function as Legendre transform of log-MGF, with Monte Carlo verification.
Optimal Stopping & Wald's Identity
Secretary problem 1/e rule, threshold stopping, and Wald's identity E[S_τ]=E[X]·E[τ]. Random walk hitting times and renewal theory.
The Secretary Problem
Interactive secretary problem: reject first 37% then hire the next best. Converges to 1/e success probability. Analyse the optimal strategy.
Large Deviation: Cramér's Theorem
CGF, Legendre-Fenchel transform, and exponential tail decay. Monte Carlo verification of P(S_n/n≥x) ~ exp(-n·I(x)) across distributions.
Loschmidt Echo & Quantum Fidelity
M(t)=|⟨ψ|e^{iH't}e^{-iHt}|ψ⟩|² under perturbation. Fermi golden rule vs Lyapunov regime — quantum chaos and OTOC diagnostics.
Passive Scalar Mixing
Advection-diffusion of dye by 2D chaotic flows. Batchelor k⁻¹ spectrum, Péclet number effects, and scale-resolved variance.
Stochastic Block Model
SBM random graphs with community structure. Kesten-McKay detectability threshold and label propagation inference.
Kolmogorov Complexity & Compression
Approximating algorithmic complexity via compression. LZ77, run-length, entropy bounds on π, Thue-Morse, and random strings.
Multifractal Cascade & Turbulence
Multiplicative energy cascade: intermittency, structure function scaling ζ_p, She-Leveque model, and multifractal spectrum f(α).
TASEP Exclusion Process
Totally asymmetric simple exclusion process: phase diagram (LD/HD/MC), density profiles, and KPZ fluctuations.
Quantum Measurement Back-Action
Continuous weak measurement of a qubit on the Bloch sphere. Quantum Zeno effect, purity decay, and measurement-induced phase transitions.
Hopfield Network & Associative Memory
Store patterns as energy minima, corrupt and recall. Hebb rule, capacity limit 0.138N, spurious attractors, and Modern Hopfield connection.
Kinetic Ising: Glauber Dynamics
2D Ising model with heat-bath dynamics. Critical slowing down at T_c, domain coarsening, and dynamic critical exponent z≈2.17.
Telegraph Process & Dichotomous Noise
Two-state Markov process with Lorentzian power spectrum. Exact autocorrelation, driven oscillator, and CLT for colored noise.
Skorokhod Embedding & Martingales
Embed any zero-mean distribution as stopped Brownian motion. Optional stopping theorem verification and Wald's identity for stopping times.
Schrödinger Bridge
Entropy-regularized optimal transport via Sinkhorn iterations. Particle geodesics between distributions — foundation of diffusion generative models.
Nonequilibrium Work & Fluctuations
Jarzynski equality and Crooks theorem: recover free energy from fast pulling. Work distributions and dissipation vs protocol speed.
Continuum Percolation
Boolean model: discs at Poisson process points. Giant component at λ_c·πr²≈1.128 — same universality class as lattice percolation.
Voronoi Diagrams & Lloyd's Algorithm
Interactive Voronoi tessellation with centroidal relaxation. Delaunay duality, optimal quantization, and hexagonal convergence.
Protein Folding: HP Lattice Model
Hydrophobic-polar model on square lattice. Simulated annealing finds native state energy minimum — funnel energy landscape.
KdV Soliton Collisions
Exact N-soliton solutions of the Korteweg-de Vries equation. Elastic collisions, phase shifts, and space-time diagrams.
Quantum Spin Chain: XXZ Model
Bethe ansatz exact solution. Phase diagram: ferromagnet, critical XY (c=1), and Néel antiferromagnet at Δ=±1 transitions.
Symbolic Dynamics & Shift Spaces
Encode chaos as symbol sequences. Golden mean shift, topological entropy, Perron-Frobenius eigenvalue, and de Bruijn graphs.
Tropical Geometry
Tropical semiring (min,+): piecewise-linear curves, Legendre transforms, and shortest-path algorithms as tropical matrix multiplication.
Mode-Coupling Theory & Glass Transition
MCT self-consistent equations: two-step relaxation, non-ergodicity parameter f_q, and diverging τ_α ~ ε^{-γ} at the MCT transition.
Goldstone Boson
Spontaneous symmetry breaking and massless Goldstone modes in the Mexican hat potential
Network Motif: FFL
Feed-forward loop motif dynamics and sign-sensitive delay in gene regulatory networks
Magnetic Monopole
Hedgehog field configuration of a magnetic monopole with Dirac string visualization
Ising Glauber Dynamics
2D Ising model evolving via Glauber single-spin flips near the critical temperature
Chladni Figures
Standing wave patterns on a vibrating plate — modes where sand collects at nodal lines
Bose-Einstein Condensate
Gross-Pitaevskii equation dynamics showing BEC formation and quantum solitons
Point Vortex Dynamics
Interacting point vortices in ideal 2D fluid via Biot-Savart — chaotic trajectories
Quantum Harmonic Oscillator
Wavefunctions and probability densities for energy eigenstates of the quantum oscillator
Mean-Field Ising Spinodal
Mean-field magnetization dynamics and spinodal decomposition through the critical temperature
Coupled Oscillator Normal Modes
Normal mode decomposition of coupled harmonic oscillators — in-phase and out-of-phase motion
Wigner-Seitz Cell
Voronoi primitive cell of crystal lattices — square, hex, oblique & more. Animate lattice distortion and watch 2–6-fold symmetry.
Poincaré Section
Double pendulum Poincaré section at θ₁=0. Closed KAM curves for regular orbits, scattered chaos for high energy. Add orbits live.
Escape Rate
Chaotic billiard with a hole — particles leak exponentially fast. N(t)~e^{-γt}. Fractal repeller set of non-escaping orbits.
Ammann-Beenker Tiling
Octagonal aperiodic quasicrystal tiling via substitution rules. Squares and 45° rhombi, silver ratio 1+√2, 8-fold local symmetry.
Penrose P3 Tiling
Thick and thin rhombi with 5-fold symmetry. Inflate or deflate through generations; ratio of tile types converges to φ².
Minimal Surfaces
Helicoid, catenoid, Enneper, Scherk surfaces — mean curvature H=0 everywhere. Rotate in 3D, color by Gaussian curvature or soap iridescence.
Fractal Basin Boundaries
Newton's method on z³−1 in the complex plane. Three basins of attraction meet at a fractal boundary. Click to zoom into infinite detail.
Percolation Cluster Hull
Bond percolation near p_c=0.5 on a square lattice. Spanning cluster highlighted with its external hull. Sweep p to see phase transition.
Chladni Nodal Lines
Mathematical nodal geometry of vibrating plates: cos(mπx)cos(nπy)−cos(nπx)cos(mπy)=0. Explore modes (m,n) and animate vibration.
Fibonacci Heap
Animated Fibonacci heap operations — insert, extract-min, decrease-key. Forest of heap-ordered trees with amortized O(1) magic.
Crystal Momentum & Bloch
Bloch theorem and energy bands in periodic potentials — crystal momentum and band gap formation
Capillary Wave Dispersion
Surface wave dispersion relation showing crossover from gravity waves to capillary waves
Optical Molasses
Laser cooling of atoms in 1D optical molasses — Doppler cooling toward the recoil limit
Hausdorff Dimension
Box-counting dimension of fractal sets — Cantor set, Koch snowflake, and Sierpiński triangle
Morse Potential
Anharmonic diatomic vibration — Morse potential energy levels and dissociation dynamics
Abelian Sandpile
BTW sandpile toppling cascades and the fractal identity element of the abelian group
Coriolis Effect
Trajectories in rotating frames — Coriolis deflection, Foucault pendulum, and cyclone rotation
Rayleigh-Taylor Instability
Heavy fluid falling through light fluid — finger formation and mushroom cap structures
Arrow of Time
Entropy production and irreversibility — Boltzmann H-function rise as a partition is removed
Soap Bubble Pressure
Young-Laplace equation — the counterintuitive deflation of a small bubble into a large one
Fröhlich Polaron
Electron dresses with phonon cloud in a polar crystal, gaining effective mass. Watch lattice distortion, explore weak/strong coupling regimes and Pekar scaling m*~α⁴.
Stochastic SIR Epidemic
Exact stochastic SIR simulation via Gillespie algorithm. Individual infection/recovery events; fluctuations visible around ODE mean field. R₀ and herd immunity threshold.
Adaptive Network Voter Model
Opinions and network links coevolve — nodes copy neighbors or rewire to avoid disagreement. Phase transition between consensus and fragmentation at critical rewiring rate φ_c.
Kink Soliton — φ⁴ Theory
Topological kinks in φ⁴ scalar field theory interpolating between double-well vacua. Animate kink-antikink collisions, bion formation, and the double-well potential landscape.
Contractile Active Matter
Model of actomyosin cytoskeleton: self-propelled polar particles with alignment and contractile stress. Director field, ±½ defect dynamics, and activity-driven flow patterns.
Topological Band Winding
SSH model winding number: H(k) loops around the origin as k sweeps the Brillouin zone. W=1 hosts edge states; W=0 is trivial. Tune t'/t across the topological transition.
Firefly Synchronization
Kuramoto model: N oscillators with random natural frequencies coupled on a circle. Watch the order parameter r leap from incoherence to synchrony as K crosses K_c.
Lyapunov Exponent — Chaos
Two initially-close Lorenz or Rössler trajectories diverge exponentially. Plot log|δ(t)| and extract λ₁ from the slope — the fingerprint of chaos.
Density Matrix & Purity
Qubit density matrix on the Bloch ball. Pure states on the surface (Tr ρ²=1), mixed states inside. Animate dephasing, amplitude damping, and depolarizing channels.
Quantum Decoherence
Qubit Bloch vector shrinks as system-environment coupling destroys quantum coherence. Off-diagonal density matrix elements decay at rate γ; decoherence time τ_D=1/γ.
Holographic Entanglement
Ryu-Takayanagi formula — minimal geodesics in AdS bulk computing boundary entanglement entropy
Matrix Product States
MPS tensor network representation of 1D quantum states — bond dimension and entanglement entropy
Thermodynamic Geometry
Ruppeiner metric on state space — curvature diverges at the critical point of a van der Waals gas
Network Renormalization
Real-space coarse-graining of networks — scale-free graphs as renormalization group fixed points
Persistent Homology
Vietoris-Rips filtration on point clouds — birth and death of topological loops in the persistence diagram
Variational Quantum Eigensolver
VQE energy landscape — parameterized quantum circuit finding ground state via gradient descent
Winding Number & Vortices
Topological vortices in 2D vector fields — winding numbers, phase singularities, and annihilation
Replica Wormhole & Page Curve
Black hole information paradox — Hawking entropy vs the island formula Page curve
Spectral Form Factor
Dip-ramp-plateau structure of K(t) distinguishing integrable Poisson from chaotic GUE statistics
Quantum Chaos & OTOC
Out-of-time-order correlator exponential growth — Lyapunov exponent and quantum information scrambling
Amplitude Estimation
Quantum amplitude estimation via QPE — quadratic speedup over classical Monte Carlo sampling
Grover Search
Grover's algorithm amplitude amplification — quadratic speedup for unstructured database search
Variational Monte Carlo
VMC ground state energy minimization — Monte Carlo sampling of trial wavefunctions
Quantum Repeater
Entanglement swapping through a repeater chain — extending quantum communication range
BB84 Key Distribution
BB84 quantum cryptography protocol — basis reconciliation and eavesdropper detection via QBER
Bell Inequality (CHSH)
CHSH test violating the classical bound S≤2 — quantum correlations reaching the Tsirelson bound 2√2
Pauli Group Algebra
Cayley table and commutation structure of the Pauli group — the foundation of quantum error correction
Stabilizer Formalism
Clifford gates updating Pauli stabilizer generators — efficient classical simulation of stabilizer circuits
Surface Code Threshold
Logical vs physical error rate curves crossing at p_th≈1% — fault-tolerant quantum computing threshold
Toric Code
Kitaev's toric code on a torus — syndrome measurement and error correction for topological qubits
Proximity Effect
Cooper pair penetration into a normal metal — induced pair amplitude and minigap from proximity to a superconductor
Andreev Reflection
Electron-to-hole retroreflection at a normal-superconductor interface — subgap conductance enhancement
Mesoscopic Conductance
Universal conductance fluctuations and weak localization — quantum interference in disordered wires
Landauer-Büttiker Transport
Conductance quantization in a quantum point contact — steps of 2e²/h as channels open
Ward Identity & Noether
Gauge symmetry → current conservation — Noether's theorem and Ward identities in quantum field theory
Schwinger-Dyson Equation
Self-consistent self-energy iteration — spectral function broadening and quasiparticle lifetime
Keldysh Formalism
Forward-backward time contour for nonequilibrium quantum field theory — retarded and Keldysh Green's functions
NEGF Quantum Transport
Nonequilibrium Green's functions for a biased quantum chain — transmission, current, and spectral function
Caldeira-Leggett Model
Quantum Brownian motion — Wigner function decoherence as a particle couples to a thermal bath
Spin-Boson Model
Dissipative two-level system in an oscillator bath — coherent to incoherent crossover with Ohmic coupling
Topological Semimetal Arcs
Fermi arcs connecting Weyl node projections on the surface Brillouin zone of a topological semimetal
Weyl Semimetal
Weyl nodes with chiral charge and surface Fermi arcs — chiral anomaly pumps charge between nodes
Coulomb Blockade
Single-electron transistor Coulomb diamonds — one electron at a time through a quantum dot
Quantum Hall Edge States
Chiral edge modes immune to backscattering — Landau levels bending at the boundary
Fractional Quantum Hall
Laughlin wavefunction and ν=1/3 plateau — quasiparticles with fractional charge e/3
Quantum Critical Point
Hertz-Millis quantum criticality — non-Fermi liquid fan above a T=0 phase transition
Heavy Fermion Lattice
Kondo lattice hybridization — flat f-band and conduction band forming heavy quasiparticles
Kondo Effect
Magnetic impurity resonance at the Fermi level — Kondo minimum in resistivity and T_K scaling
BCS Cooper Pairing
Phonon-mediated Cooper pairs and the BCS gap — Δ(T) opening below T_c with DOS divergence
Hubbard Mott Transition
Metal-insulator transition as U/t grows — quasiparticle peak splitting into Hubbard bands
Lévy Flight Diffusion
Anomalous superdiffusion with power-law jump lengths — heavy tails that let the walker escape traps
Fisher-KPP Traveling Wave
Reaction-diffusion front invading unstable state — universal wave speed c*=2√(rD)
Onsager Exact 2D Ising
The exact Onsager solution vs Monte Carlo — critical temperature Tc=2/ln(1+√2)
Crumpling & Elastic Disorder
Random elastic network under compression — stress focusing, ridge formation, energy localization
Jamming Transition
Granular packings transition from flowing to rigid at critical density — the J-point
Rouse Polymer Dynamics
Bead-spring chain — normal mode relaxation, Rouse time τ∝N², mean square displacement
Ginzburg-Landau Vortices
Type-II superconductor: Abrikosov vortex lattice, flux quantization, order parameter field
Active Brownian Particles
Self-propelled particles with rotational diffusion — motility-induced phase separation (MIPS)
Szilard Engine & Information
Maxwell's demon cycles: measurement + feedback extracts kT ln2 work. Information is physical.
Leidenfrost Droplet
Water droplet levitates above a hot surface on a vapor cushion — inverse Leidenfrost too
Classical Nucleation Theory
Droplet nucleation: competition between bulk free energy gain and surface energy cost — critical nucleus radius
Ostwald Ripening
Coarsening dynamics: large droplets grow at the expense of small ones — LSW theory, R∝t^(1/3)
Charge Density Wave
Peierls instability: 1D metal with nesting at 2k_F develops periodic charge modulation and gap
Spin Density Wave
SDW order from Fermi surface nesting: Q=2k_F connects parallel segments, diverging χ_spin
Magnon Spin Waves
Collective spin precession in ferromagnet — Bloch's law M(T)∝T^(3/2), quadratic dispersion ω∝k²
Phonon Umklapp Scattering
Normal vs Umklapp phonon processes — thermal resistance from momentum transfer to lattice
Kondo Cloud & Screening
Magnetic impurity screened by conduction electrons — logarithmic resistance, Kondo temperature TK
Efimov Three-Body States
Universal three-body bound states with geometric spectrum — towers of states when two-body has resonance
Berezinskii-KT Transition
2D XY model: vortex-antivortex unbinding at T_BKT — topological phase transition without order parameter
Transfer Matrix — 1D Ising
Exact transfer matrix solution for 1D Ising model — correlation length, free energy, partition function
Wigner Crystal
Electrons crystallize at low density when Coulomb repulsion dominates kinetic energy — r_s parameter
Exciton-Polariton Condensate
Light-matter hybrid quasiparticles — non-equilibrium BEC with pumping and decay
Majorana Fermion in a Wire
Topological superconductor: Kitaev chain with Majorana zero modes at ends — non-Abelian anyons
Frustrated Magnets & Spin Liquids
Frustrated antiferromagnet on kagome lattice — extensive degeneracy, no magnetic order down to T=0
Floquet Time Crystal
Discrete time translation symmetry breaking in a periodically driven system — subharmonic response
Non-Hermitian Exceptional Points
Where eigenvalues and eigenvectors coalesce — square-root topology, enhanced sensing, exceptional rings
Measurement-Induced Phase Transition
Entanglement transition from volume law to area law as measurement rate increases — monitored circuits
Quantum Error Threshold
Fault-tolerant quantum computing: below error threshold p_th, logical error rate suppressed exponentially
Planckian Strange Metal
Linear-in-T resistivity in cuprates — scattering rate saturates at ℏ/k_BT, violating Fermi liquid theory
AdS/CFT Strange Metal
Holographic RN black hole gives linear-T resistivity — emergent infrared geometry codes non-Fermi liquid
Rydberg Blockade
Dipole-dipole interaction between highly excited atoms prevents double excitation — quantum gate in free space
Optical Lattice Mott Transition
Superfluid to Mott insulator: cold atoms in optical lattice at commensurate filling
Cold Atoms & Hubbard Model
Fermions in optical lattice: metal-Mott-insulator phase diagram, Néel AFM order, pseudogap
Atom Interferometry
Matter-wave interferometer measures gravity with precision — Kasevich-Chu sequence, gravitational phase
Squeezed Light & Heisenberg
Quantum noise below shot noise: ΔX·ΔP≥1/2, squeezing trades amplitude for phase noise
Dicke Superradiance
N atoms decay collectively as N² — quantum coherence turns independent emitters into a phase-locked ensemble
VQE Energy Landscape
Variational quantum eigensolver: cost function landscape, barren plateaus, local minima trap
Quantum Fisher Information
Cramér-Rao bound: QFI sets ultimate precision limit. Heisenberg scaling 1/N² beats shot noise 1/N
NV-Center Quantum Sensor
Nitrogen-vacancy diamond: single spin magnetometry, sub-nanotesla sensitivity at room temperature
Spin Squeezing
Wineland spin squeezing reduces collective spin noise — metrological advantage scales as ξ²=N·(ΔJ_⊥)²/⟨J⟩²
Axion Insulator
Topological magnetoelectric effect: θ=π axion angle, surface half-quantum Hall, P·B coupling
Quantum Anomalous Hall Effect
Chern insulator with spontaneous magnetization — quantized Hall conductance σ_xy=Ce²/h without external B
Magic Angle Flat Bands
Twisted bilayer graphene at θ≈1.1°: bandwidth collapses, Coulomb interactions dominate
Moiré Hubbard Physics
Twisted TMD homobilayer: emergent triangular Hubbard model, Wigner crystal, superconductivity
Photonic Topological Edge Modes
Optical analog of quantum Hall: photonic crystal with broken time-reversal, unidirectional edge photons
Auxetic Mechanical Metamaterial
Negative Poisson ratio: pulls in x, expands in y — re-entrant honeycombs, rotating units
Acoustic Black Hole (Dumb Hole)
Unruh 1981: supersonic flow traps sound like gravity traps light — Hawking radiation analog
Chern-Simons Theory
Topological field theory in 2+1D: knot invariants, anyons, FQHE effective theory, no local degrees of freedom
Fractional Chern Insulator
Lattice analog of FQHE: flat Chern band + interactions → fractional Hall without Landau levels
Higher-Order Topological Insulator
HOTI: bulk and surface gapped but conducting hinge/corner modes — beyond the bulk-boundary correspondence
Riemann Zeta Zeros
Explore zeros of ζ(s) on the critical line Re(s)=1/2.
Lévy Stable Superdiffusion
α-stable random walks vs Gaussian: anomalous MSD ~ t^(2/α).
SIR Herd Immunity
Epidemic dynamics with R₀ slider; herd immunity threshold 1−1/R₀.
BTW Sandpile
Self-organized criticality: add grains, watch avalanche power laws emerge.
Edwards-Anderson Spin Glass
Random ±J couplings, frustration, and Glauber dynamics on a 2D lattice.
Genetic Drift & Fixation
Wright-Fisher model: allele frequency trajectories and fixation probability.
Eden Growth & KPZ
Random cluster growth; surface roughness W(t) ~ t^(1/3) KPZ universality.
WKB Quantum Tunneling
Semiclassical tunneling through adjustable barriers: T = exp(−2∫κ dx).
Information Bottleneck
IB curve: compression I(X;T) vs relevance I(T;Y) trade-off plane.
Ecological Stoichiometry
Liebig's law of minimum: nutrient ratios limit consumer growth.
Polymer Chain & Flory Theory
Random walk vs self-avoiding walk: end-to-end distance R ~ N^ν, Flory exponent ν≈3/5 in 3D.
Abelian Sandpile
Chip-firing on a grid: deterministic toppling, identity element, self-organized criticality.
Gray-Scott Reaction-Diffusion
F/k parameter space: spots, stripes, worms, mazes, and chaos from two reacting chemicals.
Logistic Map Bifurcation
Period-doubling cascade to chaos; zoom into Feigenbaum constant δ≈4.6692 universality.
Carnot Cycle Engine
P-V and T-S diagrams for the ideal reversible engine; efficiency η = 1 − T_C/T_H.
Fitness Landscape Evolution
Adaptive walks on rugged NK landscapes; epistasis K controls local optima density.
Neural Population Coding
Gaussian tuning curves, population vector decoding, and Fisher information in sensory neurons.
Heteroclinic Cycle Dynamics
Rock-paper-scissors Lotka-Volterra: saddle connections, winnerless competition, exponential dwell times.
Levins Metapopulation
Patch occupancy dynamics: colonization/extinction balance, persistence threshold c/e > 1.
Voronoi & Lloyd's Algorithm
Centroidal Voronoi tessellation via Lloyd's algorithm; crystal grain growth and foam structure.
Dendritic Solidification
Diffusion-limited crystal growth: snowflake-like dendrites emerge from a solidifying melt.
Lifshitz-Slyozov Coarsening
Ostwald ripening: large droplets grow at small ones' expense, mean radius ~ t^(1/3).
2D Percolation Hull
Fractal external perimeter of the spanning percolation cluster, dimension d_f ≈ 7/4.
Traffic Jam Nucleation
Nagel-Schreckenberg model: phantom jams nucleate spontaneously in free-flowing traffic.
Pulse-Coupled Oscillators
Mirollo-Strogatz theorem: any initial phases almost surely synchronize via pulse coupling.
Ballistic Deposition & KPZ
Columnar surface growth with lateral sticking; roughness W ~ t^β, KPZ universality class.
Abelian Sandpile
Self-organized criticality: grains topple in avalanches with power-law size distributions.
Network Motifs
Regulatory circuit motifs: autoregulation, feed-forward loops, and their signal response dynamics.
Deffuant-Weisbuch Opinions
Bounded confidence: agents only interact if opinions differ by less than ε, forming clusters.
GOY Turbulence Shell Model
Energy cascade in the GOY shell model: K41 spectrum E(k) ~ k^(−5/3) and intermittency.
Spectral Clustering
Graph Laplacian eigenvectors, Fiedler vector, and community detection via spectral embedding.
Stochastic Block Model
Planted partition graph model, detectability threshold, and the Kesten-McKay phase transition.
Pedestrian Flow
Mean-field crowd dynamics: fundamental diagram, stop-and-go waves, and Greenshields model.
Fractional Brownian Motion
Hurst exponent H controls long-range correlations; MSD ~ t^{2H} from anti-persistent to persistent.
Voter Model Coarsening
Opinion dynamics on a lattice: interface density decay, consensus time, and dimension dependence.
SIR on Random Networks
Epidemic spreading on Erdos-Renyi and scale-free graphs; threshold 1/<k²/k>.
Cascading Power Grid Failures
Load redistribution triggers avalanches; capacity parameter alpha controls robustness.
Self-Replicating Patterns
Langton's loops and self-replicating cellular automata; information copying in 2D grids.
Wolfram 2D Totalistic CA
Outer totalistic rule explorer: Game of Life, HighLife, and the vast space of 2D CA rules.
Jerne Idiotypic Network
Antibody stimulation and suppression form attractor states; memory and immune response dynamics.
Excitable Media: Spiral Waves
FitzHugh-Nagumo reaction-diffusion on a 2D grid — spiral wave formation, propagation, and breakup.
Phase Ordering After Quench
Ising model coarsening dynamics: domain growth L(t) ~ t^½ after a sudden quench below T_c.
HP Lattice Protein Folding
Hydrophobic-polar model on a 2D lattice — Monte Carlo search through the rugged energy landscape.
Discrete Quantum Walk
Quantum walker with Hadamard coin: ballistic σ ~ t spread vs classical diffusive σ ~ t^½.
BKT Transition: Vortex Unbinding
Berezinskii-Kosterlitz-Thouless transition in the 2D XY model — vortex-antivortex pairs unbind above T_BKT.
Host-Parasite Coevolution
Red Queen dynamics: matching-allele model drives perpetual allele frequency cycling — endless arms race.
SIR on Heterogeneous Networks
Epidemic threshold λ_c = ⟨k⟩/⟨k²⟩ — scale-free networks have near-zero threshold; hubs accelerate spread.
Complex Contagion Cascades
Watts threshold model: adoption requires fractional neighbor reinforcement — clustered networks block global cascades.
Rössler Attractor: Chemical Chaos
Period-doubling route to chaos in the Rössler system — bifurcation diagram and live Lyapunov exponent.
Lyapunov Spectrum: Coupled Maps
Full Lyapunov spectrum of coupled logistic maps — KS entropy, Kaplan-Yorke dimension, synchronization transition.
Reactive Spreading Epidemics
Contagion on a random network where each node's reactivity determines infection probability — revealing super-spreader dynamics.
BTW Sandpile — Self-Organized Criticality
Bak-Tang-Wiesenfeld sandpile where grain additions trigger avalanches with power-law size distributions — no parameter tuning needed.
Bell Inequality & Quantum Entanglement
CHSH test of quantum entanglement: vary Alice and Bob's measurement angles to see violations of S≤2 up to the Tsirelson bound 2√2.
Dendrite Solidification — Snowflake Growth
Phase-field model of diffusion-limited solidification producing snowflake-like branching morphologies from supercooling and anisotropy.
Active Matter — Self-Propelled Particles
Vicsek model of flocking: tune noise and density to cross the non-equilibrium phase transition from disordered gas to ordered flock.
Kinetic Theory — Hard-Sphere Gas
2D hard-disk elastic collisions relaxing to the Maxwell-Boltzmann speed distribution, with live temperature and pressure readouts.
Firefly Synchrony — Pulse-Coupled Oscillators
Mirollo-Strogatz integrate-and-fire fireflies that inevitably synchronize through excitatory phase kicks — a finite-time absorption theorem.
Schelling Segregation
How mild neighbor-preference thresholds (even 33%) produce dramatic spatial segregation — Schelling's 1971 micro-to-macro paradox.
Turbulence Intermittency — Multifractal Cascade
Multiplicative energy cascade with anomalous scaling exponents ζ_p: compare K41, log-normal K62, and She-Leveque's exact formula.
Ocean Gravity Waves — Dispersion
Surface gravity waves with ω²=gk·tanh(kh): animated wave packets showing group velocity lagging phase velocity, with a live dispersion curve.
Activator-Inhibitor Reaction-Diffusion
Gierer-Meinhardt model generates Turing patterns — spots, stripes, and labyrinths — from local activation and long-range inhibition.
Spectral Radius & Epidemic Threshold
The largest eigenvalue of the adjacency matrix sets the SIS epidemic threshold — network topology determines whether disease persists or dies out.
Kinesin Molecular Motor Walk
Kinesin steps hand-over-hand along microtubules in 8 nm increments powered by ATP, with a stochastic force-velocity relationship and load-dependent detachment.
Extreme Value Theory: Gumbel & GEV
Block maxima from any parent distribution converge to the Gumbel, Fréchet, or Weibull family — the Fisher-Tippett-Gnedenko theorem made visible.
Social Learning & Bayesian Updating
Agents update beliefs via private signals and neighbor observations, producing wisdom of crowds or information cascades depending on network structure.
Elastic Network Model: Protein Modes
Harmonic springs between nearby Cα atoms yield normal modes via Kirchhoff matrix diagonalization — low-frequency modes capture functional collective motions.
Double Pendulum: Chaotic Sensitivity
Two nearly identical double pendulums diverge exponentially, demonstrating sensitive dependence on initial conditions with live Poincaré section.
2D Quantum Walk: Interference
Quantum interference during a 2D lattice walk produces ballistic spreading and corner-peaked distributions, starkly different from classical diffusion.
Network Motifs: Gene Circuit Dynamics
Feedforward loops, negative autoregulation, and toggle switches are overrepresented gene network motifs with distinct dynamical functions.
Stress Propagation in Granular Media
Force chains carry stress through heterogeneous granular packings while most grains remain load-free — the origin of jamming and arch formation.
RG Flow in 2D
Wilson's renormalization group flow in coupling space: fixed points, relevant/irrelevant operators, and the Wilson-Fisher universality class.
SIRS Epidemic Oscillations
Waning immunity drives persistent epidemic cycles — explore the Hopf bifurcation between damped spirals and sustained oscillations.
Cellular Potts Model
Tissue organization via energy minimization: cells sort by differential adhesion, deform, and self-organize through Monte Carlo dynamics.
Echo State Networks
Reservoir computing with a fixed random recurrent network — only the readout is trained, and the edge of chaos maximizes memory capacity.
SSH Topological Band Gap
The Su-Schrieffer-Heeger model: alternating hoppings open a band gap, and the topological phase hosts protected zero-energy edge states.
Smoluchowski Coagulation
Cluster growth and gelation: the product kernel drives a sol-gel transition where an infinite cluster forms in finite time.
Spatial Predator-Prey Turing Patterns
Diffusion-driven Turing instability in Lotka-Volterra: slow predator diffusion destabilizes uniform equilibria into spatial patches.
Minority Game
Agents compete for the scarce resource of being in the minority — a phase transition separates herding from diverse market ecology.
Bipartite Network Projection
Project two-mode networks onto one mode: weighted co-occurrence, nestedness, and the hidden structure of author-paper or plant-pollinator webs.
Lattice Gas Automaton
HPP and FHP lattice gases: emergent fluid dynamics from Boolean particle collisions on a grid, the microscopic ancestor of lattice Boltzmann.
Rayleigh-Taylor Instability
Heavy fluid over light fluid develops fingering instabilities driven by gravity and Atwood number density contrast.
Epidemic Spatial Heterogeneity
SIR epidemic spreading across a fragmented landscape, showing how habitat structure shapes outbreak size and speed.
Hierarchical Modular Network
Networks with nested community structure at multiple scales, with modularity Q quantifying the strength of communities.
Fisher-KPP Traveling Wave
Reaction-diffusion front propagation with minimum wave speed c* = 2√(Dr) selected from steep initial conditions.
FitzHugh-Nagumo Spiral Waves
Excitable medium generating spiral waves and reentrant excitation, a model of cardiac arrhythmias.
Mutualism Coevolution
Plant-pollinator trait matching coevolution comparing obligate vs facultative symbiosis and evolutionary dynamics.
Critical Branching Process
Galton-Watson branching at criticality produces power-law avalanche sizes P(s) ~ s⁻³/² and extinction transitions.
Josephson Junction Fluxon
Sine-Gordon soliton (fluxon) propagating in a long Josephson junction with IV characteristics and Shapiro steps.
Competitive Exclusion Principle
Gause's R* rule: the species with lowest minimum resource requirement wins; niche differentiation enables coexistence.
Delay Differential Equations
Mackey-Glass chaos: a Hopf bifurcation at critical delay τ_c transitions from stable equilibrium to chaos.
XY Model Vortex Defects
Topological vortex-antivortex pairs in the 2D XY model, with interactive Kosterlitz-Thouless transition and charge visualization.
Age-Structured SEIR Epidemic
Age-stratified epidemic model with realistic contact matrices, age-specific mortality, and herd immunity threshold calculations.
Spatial Prisoner's Dilemma
Cooperators and defectors compete on a spatial lattice, forming clusters through Nowak-May dynamics of evolutionary game theory.
Simulated Annealing
Stochastic optimization on a rugged energy landscape with configurable cooling schedules and live acceptance probability tracking.
Entropy and Mixing
Random tile swaps drive a solved puzzle toward maximum disorder, visualizing Shannon entropy growth and the Second Law of Thermodynamics.
Neutrino Oscillations
Quantum flavor oscillations via the PMNS mixing matrix — explore how mixing angles and mass-squared differences shape flavor probabilities.
Climate Energy Balance Model
Budyko-Sellers EBM with ice-albedo and water-vapor feedbacks reveals bistability between warm Earth and Snowball Earth states.
Bak-Sneppen Coevolution
Species fitness evolves through punctuated avalanches in the Bak-Sneppen model, producing self-organized criticality and power-law distributions.
Driven Nonequilibrium Transition
Contact process on a 2D lattice transitions from absorbing to active phase at the Directed Percolation critical point λ_c ≈ 1.65.
Ecological Succession
Pioneer to climax community dynamics on a spatial grid, with facilitation, inhibition, and disturbance shaping Markov state transitions.
Logistic Map Bifurcation Diagram
Period-doubling route to chaos in the logistic map — drag to zoom into self-similar bifurcation structure and the Feigenbaum constant δ ≈ 4.669.
Tragedy of the Commons
Cooperators and defectors compete for a shared renewable resource on a spatial grid, revealing how rational self-interest leads to collective collapse.
Magnetic Skyrmion Dynamics
Topologically protected spin whirlpools in a 2D magnetic film, stabilized by the Dzyaloshinskii-Moriya interaction — click to nucleate skyrmions.
Spatial Epidemic Spreading
SIR model on a 2D lattice with long-range jumps — watch wave fronts propagate, herd immunity emerge, and R₀ control epidemic fate.
KPZ Surface Roughening
Kardar-Parisi-Zhang stochastic interface growth — tune noise, diffusion, and nonlinearity to measure the universal β = 1/3 growth exponent.
Minority Game (El Farol)
Adaptive agents compete to be in the minority using historical patterns — emergent coordination drives attendance near capacity without any communication.
Adaptive Network Rewiring
Voter model on a co-evolving network — opinions and topology reshape each other, producing fragmentation into echo chambers as rewiring probability rises.
Information Cascades (Bayesian)
Rational Bayesian agents sequentially decide and observe others — once a cascade forms, private signals are rationally ignored and herding locks in.
Boltzmann Wealth Distribution
Gas-like random trades produce the Maxwell-Boltzmann exponential wealth distribution — explore inequality via Gini coefficient and the Lorenz curve.
Zipf Rank-Frequency Law
Power-law rank-frequency distributions in language, cities, and more — fit a log-log slope and paste your own text to watch Zipf's Law emerge.
SGD Loss Landscape Explorer
Navigate rugged loss landscapes with SGD, Momentum, and Adam optimizers — watch gradient descent escape local minima with tunable noise and learning rate.
XY Model Topological Defects
Vortices and antivortices emerge and annihilate in the 2D XY model — explore Kosterlitz-Thouless physics and topological defect dynamics via Monte Carlo simulation.
Conway's Game of Life
Spawn gliders, spaceships, pulsars, and the Gosper gun in Conway's Turing-complete cellular automaton — draw cells and watch complexity emerge from four simple rules.
HP Lattice Protein Folding
Simulated annealing folds hydrophobic-polar sequences on a 2D lattice — NP-complete even in this simplified model, yet proteins solve it in milliseconds.
Cellular Automaton Rule 110
Watch Rule 110 — the simplest known Turing-complete system — evolve complex patterns from a single cell. Explore all 256 elementary rules including chaotic Rule 30 and fractal Rule 90.
Social Network Clustering
Compare Erdos-Renyi, Watts-Strogatz small-world, and Barabasi-Albert scale-free networks — click nodes to measure local clustering coefficients and path lengths.
Ising Mean-Field Phase Transition
Ferromagnet to paramagnet at T_c 2.269 — watch the 2D Ising model undergo Onsager's exact phase transition and compare simulation to mean-field theory predictions.
Lorenz Attractor (3D)
Drag to rotate the butterfly attractor in 3D — add a second trajectory to visualize exponential divergence and estimate the Lyapunov exponent live.
Diffusion-Limited Aggregation
Particles random-walk until sticking to a growing cluster — fractal dimension D ~1.71 emerges naturally, modeling lightning, snowflakes, and mineral dendrites.
BTW Sandpile (Self-Organized Criticality)
Drop grains one at a time and watch avalanches of all sizes follow a power law — the system self-tunes to criticality without any parameter adjustment.
Neural Network Backpropagation
Watch gradients flow backward through a live neural network learning XOR — visualize weights, activations, and the chain rule in real time.
2D Random Walk & Diffusion
Multiple walkers diffuse from the origin as MSD grows linearly with time — compare measured ⟨r²⟩ to the theoretical prediction 4Dt and watch Einstein's law emerge.
Hubble Expansion & Cosmological Redshift
Every galaxy recedes with velocity proportional to distance — explore the Friedmann equation, dark energy, and how redshift maps cosmic history.
Entropy & Ideal Gas Mixing
Remove the partition between two ideal gases and watch entropy rise irreversibly — Boltzmann's H-theorem and the statistical arrow of time made visible.
Self-Propelled Particle Flocking
The Vicsek model: local alignment rules drive a phase transition from disordered noise to collective motion — tune noise through the critical point and watch order emerge.
Fourier Series Wave Builder
Build square waves, sawtooth, and triangles from spinning circles — visualize Gibbs phenomenon at discontinuities and watch epicycles trace the reconstruction.
Tournament Selection & Evolutionary Dynamics
Populations climb fitness landscapes via tournament selection, mutation, and drift — explore multimodal Rastrigin, Ackley, and Rosenbrock functions.
Activator-Inhibitor Pattern Formation
Turing's 1952 morphogenesis model: two diffusing chemicals spontaneously form spots, stripes, and labyrinths — click to inject activator and perturb the patterns.
Power Law Fitting with MLE
Generate power-law data and compare the Clauset-Newman MLE estimator against naive log-log regression — visualize the likelihood surface and bootstrap confidence intervals.
Branching Process & Criticality
Galton-Watson trees at the critical point m=1 produce power-law avalanche size distributions — explore subcritical extinction, critical fluctuations, and supercritical explosion.
Julia Set Explorer
Escape-time coloring of Julia sets for f(z)=z²+c. Explore presets, drag to pan, and watch fractal boundaries shift as c changes.
Driven Harmonic Oscillator & Resonance
Mass-spring-damper driven by a periodic force. Watch amplitude explode at resonance and see the full frequency-response curve in real time.
Spring-Mass Lattice Wave Propagation
1D chain of masses on springs. Drive one end, observe standing waves, reflections, and the non-linear dispersion relation of a discrete lattice.
Schelling Segregation on a Torus
Two-group agents on a toroidal grid. Mild individual preferences produce dramatic macro-level segregation.
Metabolic Flux & Stoichiometry
Tune enzyme activities in a branched metabolic network and watch steady-state fluxes rebalance. Identifies bottleneck reactions automatically.
Boids Flocking — 3 Rules
Reynolds 1987 model: separation, alignment, cohesion. Zero out rules one at a time to isolate their contributions to collective motion.
Ant Colony Pheromone Trail Optimization
Foraging ants lay pheromone trails. Shorter paths accumulate more pheromone before evaporation — emergent shortest-path optimization.
Crystal Nucleation & Growth
Stochastic solidification with competing grains, anisotropic growth, and grain boundaries. Watch Avrami kinetics and microstructure form.
Spectral Clustering & Community Detection
Graph Laplacian eigenvectors reveal community structure. The Fiedler vector splits the graph; K-means in spectral space finds k communities.
Runge-Kutta ODE Solver Visualizer
Compare Euler, Midpoint, and RK4 on classic ODEs. Tune step size to see error accumulation and the power of higher-order methods.
SIR Epidemic: Network vs Mean-Field
Compare SIR epidemic spreading on a random network against mean-field ODE predictions. Tune β, γ, and R₀.
Heat Equation on a Rod
Finite-difference simulation of the 1D heat equation. Choose initial conditions, boundary temperatures, and watch diffusion.
Ecosystem Energy Flow & Trophic Levels
Visualize Lindeman's 10% rule: energy cascading through producer, herbivore, carnivore, and apex predator levels.
Elastic Waves & Phonon Dispersion
Animate a 1D diatomic lattice and explore acoustic vs optical phonon branches. See the gap open as mass ratio increases.
Swarmalators: Sync + Motion Coupling
Agents that synchronize their phase AND swarm in space simultaneously — a phase-coupled active matter model.
Polymer Chain & Entropic Spring
Sample random-walk conformations of a freely-jointed chain. See the Langevin entropic spring force vs Gaussian approximation.
Morse Code Oscillator
Type a message, hear it in Morse code, and watch the keyed waveform and its frequency spectrum in real time.
Greedy Circle Packing
Greedily pack non-overlapping circles into various container shapes. Approach Apollonian fractal density.
Rayleigh-Bénard Convection Rolls
Fluid heated from below forms convection rolls above Ra=1708. Adjust Ra and Pr to see onset and roll structure.
Bacterial Chemotaxis (Run-and-Tumble)
E. coli-style run-and-tumble navigation up chemical gradients. Tune tumble rate, sensitivity, and attractant sources.
Neutrino Oscillation Probability
Visualize how neutrinos oscillate between flavors as they travel, governed by mixing angles and mass-squared differences.
Morphogenesis (3-component Reaction-Diffusion)
Gray-Scott model: short-range activation and long-range inhibition spontaneously generate spots, stripes, and labyrinthine Turing patterns.
Opinion Leaders in Social Networks
Watch opinions spread through scale-free, random, and small-world networks where high-degree nodes hold outsized influence.
Colloidal Gel & Arrested Phase Separation
Short-range attractive colloids gel before reaching equilibrium — explore how volume fraction and interaction strength control microstructure.
Epidemic Vaccination Threshold (Herd Immunity)
SIR dynamics with vaccination: find the herd immunity threshold p_c = 1 − 1/R0 and see how coverage stops epidemic spread.
Carnot Cycle & Engine Efficiency
Animate the Carnot cycle on PV and TS diagrams. Explore how temperature reservoirs set the fundamental upper limit on heat engine efficiency.
Cellular Signaling & Phosphorylation Cascade
MAPK-like cascade: signal amplification through sequential kinase phosphorylation, ultrasensitivity via Hill cooperativity, and noise filtering.
Particle in a Box Energy Levels
Quantum wavefunctions and energy quantization in an infinite square well. Explore probability densities and superposition dynamics.
Arms Race Evolutionary Dynamics
Red Queen coevolution: prey defense and predator offense traits evolve in a perpetual arms race with no stable equilibrium.
Stokes Flow & Viscosity
Exact creeping-flow solution around a sphere. Animate tracers, streamlines, and velocity profiles; compute Stokes drag and Reynolds number.
Wilsonian EFT & Coarse-Graining
Integrate out high-momentum modes to reveal effective low-energy physics and renormalization group flow.
Hypergraph Higher-Order Contagion
Group interactions on hyperedges drive explosive spreading beyond what pairwise networks allow.
Hodgkin-Huxley Neuron CA
Action potential propagation along a neural cable — ion channel gating, refractory periods, traveling waves.
Hubbell's Neutral Theory of Ecology
All species ecologically equivalent — biodiversity emerges from drift, dispersal, and speciation alone.
XXZ Spin Chain
Exactly solvable quantum spin model via Bethe Ansatz — ferromagnet, Luttinger liquid, and antiferromagnet phases.
Adaptive Immunity & Clonal Selection
B-cell affinity maturation in shape space — somatic hypermutation guided by selection evolves antibodies.
Gravitational Lensing & Light Deflection
General relativity bends light around massive objects — Einstein rings, arcs, and magnification.
Stochastic Epidemics (Gillespie SSA)
Exact stochastic SIR simulation — discrete individuals, random events, and variance absent from mean-field ODEs.
Run-and-Tumble Active Particle
Bacterial motility — ballistic runs interrupted by random tumbles, chemotaxis, and motility-induced clustering.
Bell State Measurement & Quantum Correlations
Entanglement and CHSH inequality violation — quantum correlations exceed any classical hidden-variable bound.
Wavepacket Dispersion (Free Particle)
Quantum wavepacket spreading — Gaussian minimum-uncertainty states dispersing as different momentum components travel at different speeds.
Network Motifs & Feedforward Loops
Gene regulatory circuit dynamics — coherent and incoherent feedforward loops as sign-sensitive delay and pulse generators.
MHD Alfvén Wave Propagation
Transverse waves in magnetized plasma — magnetic tension drives non-dispersive Alfvén waves along field lines.
Forest Fire Cellular Automaton (SOC)
Drossel-Schwabl model self-organizing to criticality — power-law fire sizes emerge from slow growth and rare lightning.
Bayesian Inference with Conjugate Priors
Watch the posterior update in real time — Beta-Binomial, Gamma-Poisson, and Normal-Normal conjugate pairs.
Phase-Locked Loop Synchronization
VCO locking to a reference signal via phase error feedback — lock range, acquisition dynamics, and filter trade-offs.
Phonon Heat Transport & Umklapp Scattering
Crystal lattice vibrations carrying heat — Umklapp processes flip phonon momentum and govern thermal resistance at high T.
Public Goods Game & Social Dilemmas
Spatial evolutionary dynamics of cooperation, defection, and punishment — tragedy of the commons and its resolutions.
Rugged Fitness Landscape Evolution
NK model with tunable epistasis — smooth single-peak landscapes become maximally rugged as K increases, trapping populations.
Allopatric vs Sympatric Speciation
Geographic isolation vs ecological divergence — watch populations split into distinct species via drift, selection, and assortative mating.
Bootstrap Percolation
Threshold activation cascades and phase transitions on a lattice
Rényi Entropy
Generalized entropy with order α — from Hartley to min-entropy
Two-Temperature Model
Hot electrons, cool lattice — ultrafast laser nonequilibrium thermalization
Spectral Clustering
Graph Laplacian eigenvectors reveal non-convex cluster structure
Gillespie Algorithm
Exact stochastic simulation of chemical reactions via the master equation
Perona-Malik Diffusion
Edge-preserving image smoothing via nonlinear anisotropic PDE
Power Law & Pareto
Heavy-tailed distributions, Zipf's law, and scale-free phenomena
Spin Ice
Ice-rule frustration, magnetic monopoles, and Dirac strings
Turbulent Puff
Localized turbulent slugs propagating and splitting in pipe flow
Antiferromagnet & Néel Order
Alternating sublattice spins and the Néel phase transition
Game of Life
Emergent complexity from simple cellular automaton rules — gliders, guns, oscillators
Hodgkin-Huxley Neuron
Voltage-gated ion channels producing action potentials — animate Na⁺ and K⁺ gating
Cascade Failure
Interdependent network collapse — failure in one layer triggers cascades in the other
Boids Flocking
Emergent collective motion from alignment, cohesion, and separation rules
Heat Equation
Fourier diffusion of temperature in 1D and 2D — watch high modes decay fastest
Thermodynamic Cycles
Carnot, Otto, Stirling engines — entropy production on PV and TS diagrams
Magnetic Skyrmions
Topologically protected spin textures with integer winding number — skyrmion Hall effect
Fitness Network Model
Bianconi-Barabási model — nodes with varying fitness compete for links, condensation emerges
Network Aging
Edges decay with age and reform — watch degree distribution and clustering evolve
Logistic Map Bifurcation
Period-doubling route to chaos — Feigenbaum universality and the onset of chaos at r≈3.57
Firefly Synchronization
Pulse-coupled oscillators spontaneously synchronizing via phase response curves
Information Bottleneck
Compression vs. relevant information — IB curve and mutual information planes
Double Pendulum Chaos
Chaotic motion via Lagrangian mechanics — phase portrait and sensitivity to initial conditions
Nucleation & Crystal Growth
Classical nucleation theory — Gibbs free energy barrier and dendritic crystal growth
NaSch Traffic Model
Stochastic cellular automaton for highway traffic — jam formation and fundamental diagram
Worm Algorithm QMC
Quantum Monte Carlo worldline sampling — efficient ergodic updates via the worm
Rugged Energy Landscape
Spin glass landscape — basin hopping, simulated annealing, and escape dynamics
Lévy Flights
Lévy flights vs. Gaussian walks — superdiffusion, heavy tails, displacement distributions
Hard Sphere Jamming
Random packing approaching jamming — contact network, pressure divergence, φ_J
SEIR + Vaccination
Herd immunity threshold, vaccination campaigns, and outbreak dynamics on networks
Shear Band Formation
Localized plastic deformation in amorphous solids
Phonon Density of States
Debye model, real DOS, and heat capacity contributions
Protein Folding Energy Funnel
Funneled landscape guides chains to native structure
Hysteresis in First-Order Transitions
Bistability, metastability, and nucleation barriers
Tipping Points & Resilience
Early warning signals before catastrophic state shifts
Goldstone Modes & Zero Modes
Spontaneous symmetry breaking produces gapless modes
Elastic Wrinkling Instability
Thin film on soft substrate buckles into periodic wrinkles
Helfrich Membrane Mechanics
Lipid bilayer bending energy, fluctuations, and shape transitions
Bacterial Chemotaxis
Run-and-tumble motion biased up attractant gradient
Elastic Depinning Transition
Avalanches and critical force in driven elastic manifolds
RG Flow φ⁴ Theory
Wilson momentum-shell renormalization group, fixed points, and universality classes
Active Brownian Particles
Self-propelled particles with rotational diffusion and motility-induced phase separation
Stochastic Gene Toggle
Bistable genetic switch with noise-driven transitions and bimodal protein distributions
Topological Edge States
SSH chain bulk-boundary correspondence, winding number, and in-gap edge modes
Competitive Exclusion
Lotka-Volterra competition, niche overlap, and coexistence vs exclusion principle
Bingham Plastic Flow
Viscoplastic fluid with yield stress, plug flow, yield surface, and rheology curves
Superfluid Vortex Lattice
Quantized vortex nucleation in rotating BEC and Abrikosov lattice formation
Hegselmann-Krause Opinions
Bounded confidence opinion dynamics: fragmentation, consensus, and cluster formation
Turbulent Energy Budget
Kolmogorov K41 energy spectrum, cascade zones, and production-transfer-dissipation budget
Richardson Cascade
Richardson eddy cascade: big whirls to little whirls, animated energy flow to dissipation
Diffusion Maps
Nonlinear manifold learning via graph Laplacian diffusion; Swiss roll & S-curve embeddings
Chiral Symmetry Breaking
QCD quark condensate, Mexican hat potential, and pion as pseudo-Goldstone boson
Colloidal Crystal Assembly
Hard sphere colloids self-assembling into FCC/HCP crystals via entropy; phase diagram
Chimera States
Synchronized and desynchronized oscillators coexisting under nonlocal Kuramoto coupling
Phononic Crystal
Elastic wave band gaps in periodic media; dispersion relation and transmission spectrum
Network Motif Census
Feed-forward loops, bi-fans, and mutual dyads vs random null models; Z-score analysis
Reaction Network Stoichiometry
Stoichiometric matrix, null space, flux cone, and conservation laws in chemical networks
Fold Catastrophe
Saddle-node bifurcation normal form; hysteresis, catastrophe surface, and cusp extension
Quantum Zeno Effect
Frequent measurement freezes quantum evolution; survival probability vs measurement rate
Spectral Gap — Markov Chains
Second eigenvalue controls mixing time; convergence to stationarity on cycle, complete, barbell graphs
Polymer Brush
Grafted chains stretching with surface coverage — Alexander–de Gennes theory
Keystone Species
Remove a keystone predator, watch trophic cascade collapse food web
Neuronal Avalanches
Branching process at criticality — power-law avalanche size distributions
Pontryagin Control
Bang-bang optimal control for the double integrator — rocket car trajectory
Random Matrix Bulk & Edge
Wigner semicircle law and Tracy-Widom largest eigenvalue distribution
Edge Dislocation
Stress and strain field around a dislocation core — Burgers vector
Habitat Fragmentation
Metapopulation on patch network — extinction, colonization, rescue effect
Curvature-Driven Interface
Allen-Cahn equation — shrinking droplets and mean curvature flow
Amorphous Vibrational Modes
Boson peak, soft modes, and participation ratio in disordered solids
Boltzmann H-Theorem
Velocity distribution relaxing to Maxwell-Boltzmann — H(t) decreasing
Darcy Flow
Fluid flow through porous media, pressure fields, and permeability
Brownian Ratchet
Asymmetric potential drives directed transport from thermal noise
Herd Immunity Networks
Epidemic thresholds on heterogeneous networks: random vs scale-free
Nonequilibrium Steady State
Probability currents and detailed balance breaking in driven systems
Liquid Crystal Defects
Topological disclinations, director fields, and defect coarsening
Stochastic Bifurcation
Noise-induced transitions: bimodal PDF and Lyapunov sign changes
Sphere Geodesics
Great circles, parallel transport, and holonomy as Berry phase
Kleiber's Law
Metabolic scaling B ~ M^(3/4) across 27 orders of magnitude
Moran Process
Genetic drift and fixation probability in finite populations
Quantum Dot Confinement
Discrete energy levels, absorption spectra, and wavefunctions in quantum dots
Diffusion-Limited Aggregation
Brownian particles stick on contact, growing fractal clusters with Hausdorff dimension ~1.71 via diffusion field screening.
Nematic Liquid Crystal Order
Q-tensor dynamics of rod-like molecules under Frank elastic energy; ±1/2 topological defects form and annihilate.
Sandpile Toppling Avalanches
Abelian sandpile self-organized criticality: grain additions trigger power-law avalanches with exponent τ ≈ 1.2.
Directed Polymer in Random Media
Polymers minimize energy in a quenched random potential; free energy fluctuations scale as N^(1/3) — KPZ universality.
2D Vortex Topological Defects
XY model BKT transition: vortex-antivortex pairs bind below T_BKT and unbind above it, destroying quasi-long-range order.
Glassy Dynamics — Trap Model
Bouchaud's trap model: a particle hops between exponentially distributed energy traps, producing aging and non-exponential relaxation.
Entropy Production — Nonequilibrium
Driven diffusion between two heat baths visualizes entropy production rate; probability currents and the second law made visible.
Spin Glass — Parisi RSB
SK spin glass with random Gaussian couplings; Parisi's replica symmetry breaking gives a non-trivial overlap distribution P(q).
Kuramoto Mean-Field Synchronization
N oscillators with Lorentzian frequency spread lock above critical coupling K_c; order parameter R rises continuously from zero.
Hebbian Learning — Hopfield Network
Draw and store binary patterns; Hebb's rule encodes them as attractors and asynchronous updates recall them from noisy inputs.
Reynolds Boids Flocking
Emergent flocking from three local rules — separation, alignment, cohesion — as in Craig Reynolds' 1987 model.
2D Ising — Metropolis Critical Point
Spins flip via the Metropolis rule; at T_c ≈ 2.269 J/k the system undergoes a continuous phase transition with diverging correlation length.
Polymer Adsorption on Substrate
Self-avoiding walk chain near an attractive surface; below the critical temperature monomers adsorb and localize at the wall.
Granular Gas — Haff's Cooling Law
Inelastic hard-disk collisions dissipate kinetic energy as T(t) = T₀/(1 + t/τ)² — Haff's 1983 cooling law for granular gases.
Voter Model — Consensus Time
Voters copy random neighbors; opinion clusters coarsen and consensus is reached in time T ~ N ln N on 2D lattices.
Percolation Cluster Labeling
Hoshen-Kopelman algorithm labels connected clusters in site percolation; a spanning cluster appears at p_c ≈ 0.5927.
SIS Endemic Epidemic Model
Above R₀ = β/γ = 1, infection reaches endemic equilibrium I* = 1 − 1/R₀; below it the disease dies out.
Quantum Walk — Interference
Hadamard coin walk spreads ballistically (σ ∝ t) rather than diffusively (σ ∝ √t) due to quantum interference.
Jamming Transition — Hard Spheres
Hard disks compressed toward random close packing; at φ_J the system rigidifies and develops a yield stress.
Mean-Field Glass — RFOT Theory
p-spin spherical model shows two transitions: T_d where metastable states proliferate, and T_K where configurational entropy vanishes.
Epidemic on Heterogeneous Networks
SIR dynamics on scale-free networks. Hubs act as super-spreaders — the epidemic threshold vanishes as degree heterogeneity grows.
Ecological Patch Dynamics
Metapopulation model on a spatial grid: patches cycle through empty, colonized, and degraded states under Levins colonization-extinction dynamics.
Long-Range Ising Model (1D)
Power-law interactions J(r)~r^{-σ} give the 1D Ising model a real phase transition for σ<2. Tune temperature and decay to probe the boundary.
Shear Banding — Complex Fluids
Johnson-Segalman model: a non-monotonic flow curve drives separation into coexisting high- and low-shear-rate bands at a plateau stress.
Athermal Quasistatic Shear
Amorphous solids deform via discrete stress-drop avalanches. Eshelby stress redistribution at yielding sites produces scale-free burst statistics.
Fiber Bundle Model — Avalanches
N parallel fibers with random thresholds break under increasing load. Near the critical load, avalanche sizes follow D(s)~s^{-5/2}.
Aging & Renewal Theory
Heavy-tailed inter-arrival times break ergodicity and cause aging: the mean residual waiting time diverges and the inspection paradox becomes extreme.
Fracture & Crack Propagation
Lattice fracture with random bond thresholds. Stress concentrates at crack tips via the Griffith criterion — tune disorder to see brittle vs. quasi-brittle failure.
Turbulent Pair Dispersion
Richardson's t³ law: particle pairs in turbulence separate as ⟨r²⟩~εt³, not t². Scale-dependent diffusivity K(r)~r^{4/3} drives superdiffusion.
Lévy Superdiffusion & CTRW
Continuous-time random walks with power-law jumps (Lévy flights, index α) and waiting times (index β) produce anomalous diffusion ⟨x²⟩~t^{2β/α}.
Host-Pathogen Coevolution
Red Queen arms race: resistance and virulence alleles oscillate in perpetual frequency-dependent selection.
1D Toppling Sandpile
Abelian sandpile model: grains avalanche when a cell exceeds threshold, producing self-organized criticality.
Birth-Death Branching
Galton-Watson process with Poisson offspring — critical mean m=1 divides certain extinction from possible survival.
Hegselmann-Krause Opinions
Bounded-confidence model: agents average neighbors within ε. Below critical ε, consensus fractures into clusters.
Neural Field Turing Patterns
Mexican-hat connectivity drives spontaneous spatial patterns in a Wilson-Cowan neural field — a neural Turing mechanism.
Phase Ordering & Coarsening
Model B conserved dynamics: after a quench, domains coarsen as L∝t^(1/3) via Ostwald ripening.
Wetting & Contact Angle
Young's equation governs the droplet shape: tune surface energies to explore hydrophilic, hydrophobic, and complete wetting.
Ergodic Stadium Billiard
Bunimovich stadium: a chaotic billiard whose trajectories fill phase space uniformly — ergodic and mixing.
Hyperbolic Space Embedding
Poincare disk model of H²: trees embed with exponentially less distortion than in Euclidean space.
Motility-Induced Phase Separation
Active Brownian particles slow in dense regions, triggering MIPS — phase separation with no attractive forces.
Deformable Membrane
Helfrich bending energy governs lipid bilayer undulations. Watch thermal fluctuations drive membrane shapes under tunable rigidity and tension.
Reaction-Diffusion Activator-Inhibitor
Gierer-Meinhardt dynamics produce Turing patterns — spots, labyrinths, and stripes from short-range activation and long-range inhibition.
Epidemic Threshold & Herd Immunity
SIR model with vaccination. Watch R₀ and the herd immunity threshold p_c = 1 − 1/R₀ control whether the epidemic percolates.
Anomalous Diffusion & MSD Scaling
Lévy flights and subdiffusive CTRW. The MSD exponent α is measured live — normal (α=1), super (α>1), or subdiffusion (α<1).
Flocking — Vicsek Polar Order
Vicsek model: velocity alignment plus noise drives a kinetic phase transition from disorder to collective flocking. Order parameter φ tracked live.
Chaos & Lyapunov Spectrum
Lorenz attractor with ensemble divergence. Lyapunov exponent λ₁ and Kaplan-Yorke dimension converge as nearby trajectories separate exponentially.
KdV Solitons
Korteweg-de Vries equation: solitons propagate without dispersing and pass through each other unchanged — a hallmark of exact integrability.
Kink-Antikink Collisions φ⁴
Topological solitons in φ⁴ field theory collide, annihilate, or form oscillon bound states depending on impact velocity.
Minority Game
Adaptive agents learn strategies to be in the minority. Emergent coordination reduces volatility below the random-chance baseline.
Escape Rate — Kramers Theory
Overdamped Brownian particles escape a double-well potential. Empirical escape rates vs. Kramers formula: k ∝ exp(−ΔV/kT).
Block-Spin Renormalization
Ising spins coarse-grained by majority-rule RG. Watch coupling constants flow toward ordered, disordered, or critical fixed points.
Preferential Attachment
Barabási–Albert growing network: new nodes attach proportional to degree, generating scale-free hubs with P(k) ~ k⁻³.
Oslo Stochastic Sandpile
Rice-pile model with random thresholds exhibits self-organized criticality. Avalanche sizes follow a power law P(s) ~ s⁻¹·⁵⁵.
Rayleigh–Taylor Instability
Heavy fluid atop light fluid — small perturbations grow into mushroom plumes. Growth rate σ = √(Agk) sets the instability timescale.
Glassy Aging — Trap Model
Bouchaud's trap model: exponential energy landscape produces infinite mean escape times below Tg, breaking time-translational invariance.
Contact Angle & Wetting
Young's equation governs droplet shape on substrates. Tune surface energies to explore hydrophilic, hydrophobic, and complete-wetting regimes.
ASEP Exclusion Process
Asymmetric simple exclusion process: hard-core particles hop on a lattice. Exact current J = (p−q)ρ(1−ρ) with KPZ fluctuation universality.
SIR Epidemics on Trees
SIR spreading on branching tree networks. Epidemic threshold R₀ = β/γ·(⟨k²⟩−⟨k⟩)/⟨k⟩ separates extinction from outbreak.
Crystal Roughening Transition
Solid-on-solid model of a crystal surface. Below TR the surface is smooth; above TR height-height correlations diverge logarithmically.
Elastic Interface Depinning
Elastic line driven through disorder. Below Fc the interface is pinned; above it moves with velocity v ~ (F−Fc)^0.625 — a non-equilibrium critical point.
Fisher-KPP Population Waves
Reaction-diffusion invasion waves where a population spreads into empty territory at speed v = 2√(Dr). Adjust diffusion and growth rate to see wave dynamics.
Allometric Scaling — Kleiber's Law
Metabolic rate scales as B ∝ M^(3/4) across 27 orders of magnitude. West-Brown-Enquist fractal vascular network theory on a log-log plot with real animals.
Erdős-Rényi Random Networks
Phase transition in G(n,p) random graphs: a giant connected component suddenly emerges at p* = 1/n. Watch the percolation threshold in real time.
BTW Sandpile Avalanches
Self-organized criticality in the Bak-Tang-Wiesenfeld sandpile. Avalanche sizes follow a power law P(s) ∝ s^−1.5 without tuning any parameter.
Chimera States
Identically coupled oscillators spontaneously split into synchronized and incoherent groups — a striking symmetry-breaking phenomenon discovered in 2002.
Jamming Transition — Soft Spheres
Bidisperse soft spheres jam at φ_J ≈ 0.64. Near jamming, pressure and shear modulus diverge with distinct exponents — a new universality class.
Anomalous Heat Conduction
Fourier's law breaks down in 1D chains. Heat conductivity diverges with system size via Lévy-walk energy transport in a Fermi-Pasta-Ulam lattice.
Polydisperse Glass Dynamics
Size polydispersity frustrates crystallization. Watch the mean squared displacement cross from diffusive to caged as temperature drops — the glass transition.
Neuronal Avalanche Power Law
Cortical activity avalanches follow P(s) ∝ s^−3/2 at criticality (Beggs & Plenz 2003). Tune the branching ratio and watch the exponent shift away from criticality.
Invasion Percolation Frontier
Fluid invades porous media by always capturing the lowest-resistance boundary site. The fractal frontier has dimension d_f ≈ 1.89 — identical to critical percolation.
Trophic Cascade & Food Web
Three-level Lotka-Volterra ecosystem: apex predators, mesopredators, and plants. Remove the apex predator and watch mesopredator release collapse the vegetation.
Epidemics with Age Structure
SEIR model across four age cohorts with heterogeneous contact matrices. Age-assortative mixing amplifies outbreaks; targeted vaccination reshapes the epidemic curve.
Adaptive Network Rewiring
Susceptibles rewire links away from infected neighbors. The balance between infection spread and rewiring drives a phase transition between epidemic and fragmented network states.
Trap Model & Random Energy
Bouchaud's glassy dynamics: a particle hops between random energy traps with waiting times τ ∝ exp(E/T). Below T_g the mean waiting time diverges — aging appears.
Soliton Kinks — Sine-Gordon
Topological solitons of the sine-Gordon equation collide elastically and emerge unchanged. Explore kink–antikink annihilation and breather bound states in an integrable field theory.
Dynamic Percolation & Spreading
Fire spreads through a bond-percolation lattice with dynamic edge regeneration. Near the critical threshold p_c ≈ 0.5 spanning clusters form and spreading changes universality class.
Multifractal Turbulence
Turbulent dissipation concentrates on a fractal set with a spectrum of Hölder exponents. Explore the D(h) singularity spectrum, She-Leveque scaling exponents, and log-normal cascade intermittency.
Reaction Front & Combustion
Fisher-KPP and Arrhenius combustion fronts propagate at the minimum wave speed c* = 2√(Dk). Switch between bistable, KPP, and ignition-threshold kinetics to see how the front profile changes.
Elastic Chain & Phonons
A 1D mass-spring crystal with dispersion ω(k) = 2√(K/m)|sin(ka/2)|. Excite individual normal modes, pluck the chain, and watch wave packets reflect at zone boundaries.
Lipid Membrane Fluctuations
Thermal undulations of a lipid bilayer governed by the Helfrich Hamiltonian. Mode amplitudes follow ⟨|h_q|²⟩ = k_BT/(κq⁴ + σq²) — bending rigidity suppresses short-wavelength ripples.
Magnus Effect — Spinning Ball
How backspin and topspin curve a ball's trajectory through the air via pressure differences.
Penrose Rhombus Tiling
Aperiodic tiling with fat and thin rhombi — five-fold symmetry, golden ratio, never periodic.
Stellar Nucleosynthesis Chain
How stars forge elements from hydrogen to iron through successive fusion burning stages.
Phonon-Polariton Dispersion
Avoided crossing of photon and transverse optical phonon modes in polar crystals.
Epidemic Contact Network
SIR epidemic spreading on a Watts-Strogatz small-world network with tunable R₀.
Rayleigh-Plateau Instability
Surface tension breaks a liquid jet into droplets — the physics behind why faucets drip.
CMB Angular Power Spectrum
Acoustic oscillations in the early universe imprinted on the cosmic microwave background.
Molecular Motor — Brownian Ratchet
Directed transport from thermal noise via a flashing asymmetric periodic potential.
Edwards-Anderson Spin Glass
Frustrated magnetic order from quenched random interactions simulated by Metropolis MC.
Feynman Ratchet — Thermal Ratchet
Directed motion from two heat baths and why asymmetry alone cannot violate the second law.
Acoustic Phonon Dispersion 2D
Visualize vibration modes and dispersion curves of a 2D square lattice with nearest-neighbor spring coupling.
Rayleigh-Bénard Convection Rolls
Watch thermal convection rolls emerge above the critical Rayleigh number Ra = 1708 as heated fluid overturns.
Forest Fire Percolation Cluster
Ignite a random forest and watch fire spread through the spanning cluster near the site percolation threshold p_c ≈ 0.593.
Diffusion Equation Green's Function
Watch a delta-function initial condition spread into a Gaussian with variance σ² = 2Dt under the diffusion equation.
Young's Equation — Wetting
Explore how surface tensions γ_SV, γ_SL, γ_LV determine the contact angle and wetting regime via Young's equation.
Resonance Fluorescence — Mollow Triplet
Drive a two-level atom and observe the Mollow triplet spectrum with sidebands at ω_L ± Ω̃ emerging above saturation.
Firefly Phase Synchronization
N Kuramoto phase oscillators with distributed frequencies synchronize above the critical coupling K_c, like firefly swarms.
Saddle-Point / Laplace Approximation
See how Laplace's method approximates integrals by a Gaussian at the peak — the basis of Stirling's formula and path integrals.
Hertz Contact Mechanics
Compute contact radius, indentation, and the ellipsoidal pressure distribution when two elastic spheres are pressed together.
Capillary Length and Surface Tension
Explore Jurin's Law for capillary rise and the capillary length λ_c = √(γ/ρg) that governs when surface tension beats gravity.
Landau-Zener Transition
Watch a quantum two-level system swept through an avoided crossing — tune the coupling and sweep rate to see diabatic vs adiabatic transitions in real time.
2D Ising Model — Metropolis
Simulate spin-lattice ferromagnetism with the Metropolis algorithm and watch spontaneous magnetization emerge below Onsager's exact critical temperature Tc ≈ 2.269.
Vortex Filament Dynamics
Evolve vortex rings and trefoil knots under the Biot-Savart law and see leapfrogging, self-propulsion, and topological complexity unfold.
BTW Sandpile Avalanches
Drop grains onto the Bak-Tang-Wiesenfeld sandpile and observe self-organized criticality with power-law avalanche size distributions.
Double-Slit Quantum Interference
Watch individual particles build up the quantum interference pattern one by one, demonstrating wave-particle duality with tunable slit geometry.
Shockwave — Rankine-Hugoniot
Visualize the density, pressure, temperature, and velocity jumps across a normal shock and compute the Rankine-Hugoniot conservation conditions as you vary Mach number.
Curie-Weiss Mean-Field Ferromagnet
Explore the Curie-Weiss free energy landscape and order parameter: watch the pitchfork bifurcation at Tc and tune the external field to tip the double well.
Noise-Induced Phase Transition
See how multiplicative noise in a stochastic ensemble can induce symmetry breaking — disorder creating order in a counterintuitive phase transition.
Power-Law Spectral Density (1/f Noise)
Generate colored noise with arbitrary spectral exponent α via Fourier shaping and measure the resulting power spectral density on a log-log scale.
Breit-Wigner Nuclear Resonance
Plot the single-level Breit-Wigner cross-section for compound nucleus resonances and explore how resonance energy, width, and partial widths shape the sharp nuclear cross-section peak.
Little-Parks Effect
Oscillations of T_c in a superconducting ring threaded by magnetic flux quanta.
Density of States — 3D Band
The square-root DOS g(E) ∝ √E for a 3D parabolic band, with Fermi-Dirac filling.
Eikonal Ray Tracing
Rays bending through gradient-index media — |∇S|² = n²(r) — with animated wavefronts.
Josephson Junction RCSJ
The RCSJ model: a pendulum analogy for supercurrent, voltage oscillations, and Shapiro steps.
SSH Topological Edge States
Su-Schrieffer-Heeger chain: topological phase, winding number, and zero-energy edge modes.
Phonon Bandgap
Diatomic chain acoustic and optical branches — bandgap at the Brillouin zone boundary.
Electric Multipole Expansion
Monopole, dipole, quadrupole fields — V ~ 1/r^l — with field lines and equipotentials.
Born-Oppenheimer PES
Electronic energy as a function of nuclear coordinates — Morse potential, vibrational levels, and conical intersections.
Stokes Paradox
Creeping flow around a sphere and cylinder — why 2D Stokes drag diverges logarithmically.
Mode-Locked Laser Pulses
Coherent superposition of cavity modes forms ultrashort pulse trains — phase locking vs random phases.
Anharmonic Oscillator
Quantum corrections to the harmonic potential via perturbation theory — energy shifts, corrected wavefunctions, and λx⁴ anharmonicity.
Leidenfrost Effect
Droplets levitating on a vapor cushion above a superheated surface — film boiling, vapor gap, and the Leidenfrost point.
Kolmogorov Complexity
Measure the algorithmic randomness of any string via entropy, run-length encoding, and LZ78 phrase complexity.
Autocorrelation Decay
Memory in stochastic processes — exponential decay for OU and AR(1), power-law for fractional noise with long memory.
Optical Bloch Equations
Rabi oscillations and Bloch vector precession for a two-level atom driven by resonant light — detuning, decay, and pulse control.
Tidal Locking
Watch a moon's spin rate decay toward synchronous rotation as tidal dissipation locks it to its orbit.
Kondo Effect
Resistivity minimum in metals with magnetic impurities — logarithmic Kondo scattering competes with phonon T⁵ decrease.
Chip-Firing Automaton
Click to add chips to graph nodes; watch abelian avalanches propagate as overcritical vertices topple to neighbors.
Raman Scattering
Stokes and anti-Stokes spectra for graphene, silicon, diamond, and quartz — phonon peaks, Bose–Einstein population, laser color.
Wave Packet Spreading
Group velocity, phase velocity, and GVD across six dispersion relations — quantum, photonic, water waves, and phonons.
Dielectric Breakdown Fractal
DLA-like electrical discharge simulation: Laplacian growth where branching probability scales as |∇φ|^η, recreating lightning and Lichtenberg figures.
Quantum Eraser
Slide from full interference fringes to classical distribution as which-path information is marked — and erase it to restore quantum coherence.
Taylor-Couette Instability
Spin the inner cylinder past the Rayleigh criterion and watch laminar Couette flow break into stacked Taylor vortex pairs.
Information Cascade
Threshold model on random networks: seed a belief and watch it cascade — or fizzle — depending on connectivity and adoption threshold.
Fermi-Pasta-Ulam Recurrence
Nonlinear chain excited in one mode: energy spreads to neighbors then almost completely returns — the 1953 surprise that launched nonlinear dynamics.
Amphiphile Self-Assembly
Hydrophilic heads repel water while hydrophobic tails aggregate, driving spontaneous micelle formation above the critical micelle concentration.
Replica Trick & Spin Glass
Monte Carlo simulation of a ±J spin glass with visualized rugged free-energy landscape and the Edwards-Anderson order parameter.
Gravitational Wave Chirp
Binary black hole inspiral: watch the orbit shrink via Peters formula and the GW strain sweep up in frequency and amplitude to merger.
Hall Viscosity
Dissipationless Hall stress rotates the viscous response tensor 90°; explore how η_H and η_S shape velocity fields in topological fluids.
Nematic Liquid Crystal Director
Frank elastic relaxation of the director field with tunable splay/bend constants; click to inject ±½ topological defects and watch them annihilate.
Aharonov-Bohm Phase Shift
A charged particle encircling a magnetic flux tube acquires a quantum phase shift even though the field is zero along its path.
FRET Energy Transfer
Förster resonance energy transfer efficiency plummets with the sixth power of donor-acceptor distance — the molecular ruler of nanoscale biology.
Surface Plasmon Resonance
Evanescent light coupling in the Kretschmann geometry excites a plasmon wave at the gold-dielectric interface, producing a sharp reflectance dip.
Pecora-Carroll Chaos Sync
Two chaotic Lorenz systems synchronize through a unidirectional coupling signal — Lyapunov exponents of the response become negative.
Thermophoresis & Soret Effect
Colloidal particles drift along temperature gradients; the Soret coefficient sets direction and magnitude, with applications from biosensors to the origin of life.
Quantum Reflection
An attractive evanescent potential can reflect particles at ultra-low energies — not by a barrier but by rapid variation in the de Broglie wavelength.
Euler Buckling & Bone
Long bones are hollow tubes optimized against Euler buckling; vary load, length, and cortical geometry to find the critical compressive force.
Turbulent Boundary Layer
The universal log-law of the wall u⁺ = (1/κ)ln(y⁺) + B collapses turbulent velocity profiles from pipes to the atmosphere.
Photonic Crystal Bandgap
A periodic dielectric stack creates stop bands where Bragg reflections forbid light propagation — the origin of structural color and laser mirrors.
Dipole Radiation Pattern
The sin²θ donut of an oscillating electric dipole governs antenna directivity, Rayleigh scattering, and atomic spontaneous emission.
Dipole-Dipole Magnetic Interaction
Drag two magnetic dipoles and watch orientation-dependent attraction and repulsion emerge from the dipole field equations.
Josephson Junction Supercurrent
Cooper pairs tunnel through an insulating barrier, generating a phase-dependent supercurrent and AC voltage oscillations.
Rayleigh-Taylor Instability
Heavy fluid atop light fluid grows exponentially unstable — fingering downward as plumes rise, seen from supernovae to ocean mixing.
Explosive Percolation Transition
The Achlioptas process suppresses then dramatically sharpens the percolation transition by selecting edges that minimize component merging.
Kuramoto Order Parameter Onset
N coupled oscillators with random frequencies spontaneously synchronize above a critical coupling K_c, with order parameter r jumping from zero.
Polymer Chain End-to-End Distance
A freely jointed polymer chain performs a random walk; tune stiffness and excluded volume to shift the scaling exponent from ν=1/2 toward ν≈0.588.
Anderson Localization
Any disorder localizes all quantum states in 1D — watch wavefunctions decay exponentially with localization length set by disorder strength W.
Spin Glass & Replica Symmetry Breaking
Random ±J couplings on a 2D lattice create frustrated states that freeze into a disordered spin glass below the Edwards-Anderson transition.
Extreme Value Statistics & Gumbel
Block maxima of iid samples converge to Gumbel, Fréchet, or Weibull distributions depending on the tail of the parent — the three universality classes of extremes.
Renormalization Group Flow
Click to trace RG trajectories in coupling-constant space; fixed points reveal universality classes across 2D Ising, φ⁴ theory, and Kosterlitz-Thouless systems.
Ostwald Ripening
Small droplets dissolve as large ones grow — Gibbs-Thomson driven coarsening with ⟨R⟩ ~ t^(1/3) (LSW theory).
Stick-Slip Friction
A spring-pulled block alternates between sticking and slipping — the mechanism behind earthquakes, squeaky brakes, and bow strings.
Majorana Zero Modes
Topological superconductor Kitaev chain hosts edge-localized Majorana fermions — particles that are their own antiparticles.
Vesicle Thermal Fluctuations
Lipid bilayer vesicles flicker with thermal energy kT — Helfrich bending elasticity governs the mode spectrum.
Epidemic Contact Tracing
SIR spreading on a contact network with digital tracing — quarantining contacts breaks transmission chains.
Entropy Rate
A Markov chain generates symbols — watch how the entropy rate H = −Σπᵢ Σⱼ Pᵢⱼ log₂ Pᵢⱼ tracks order vs randomness.
Gap Junction Neural Sync
Electrical coupling via gap junctions drives neural synchrony — FitzHugh-Nagumo neurons linked in a small-world network.
Mexican Hat Symmetry Breaking
A ball rolls in the Higgs potential V = −½μ²|φ|² + ¼λ|φ|⁴ — spontaneous U(1) symmetry breaking visualized in 3D.
Anharmonic Duffing Oscillator
The Duffing oscillator with cubic nonlinearity exhibits period doubling, chaos, and strange attractors via Poincaré sections.
Jamming Transition
Granular packing jams above φ_J ≈ 0.64 — force chains emerge at the isostatic point where Z → 2d contacts per particle.
Quantum Walk on a Lattice
Coined quantum walks on a 1D lattice spread ballistically via interference, unlike the diffusive spread of classical random walks.
Population Genetics: Drift & Selection
Wright-Fisher model showing how genetic drift and natural selection jointly drive allele fixation or extinction across multiple lineages.
Ecological Niche & Competitive Exclusion
MacArthur-Gause resource competition: niche overlap drives Lotka-Volterra dynamics, showing when species coexist or competitively exclude.
Traffic Flow: LWR Model
Lighthill-Whitham-Richards conservation law for traffic density — shock waves form and propagate backward as jams. Click to add congestion.
Bootstrap Percolation & Nucleation
r-neighbor bootstrap percolation: infection spreads only if a cell has ≥ r infected neighbors, exhibiting a sharp threshold and nucleation.
Dissipative Solitons (Gray-Scott)
Gray-Scott reaction-diffusion system produces self-sustaining dissipative solitons — spots, stripes, and labyrinthine patterns driven by feed and kill rates.
Phase Locking & Arnold Tongues
Circle map dynamics: forced oscillators lock to rational frequency ratios forming Arnold tongues in parameter space — the devil's staircase.
Sand Pile Toppling Waves
Bak-Tang-Wiesenfeld sandpile model: cascading toppling avalanches with power-law size distributions — a canonical model of self-organized criticality.
Fiber Bundle Model of Fracture
Equal-load-sharing fiber bundles under increasing stress: random fiber failures trigger cascading avalanches until catastrophic rupture.
KPZ Surface Growth & Roughening
Kardar-Parisi-Zhang equation for stochastic interface growth — the nonlinear term drives roughening with universal exponents α=1/2, β=1/3.
Worm Algorithm Monte Carlo
XY model sampled by a worm defect that grows and shrinks on the lattice — signatures of the BKT transition.
Protein Folding Funnel
Simulated annealing on an HP lattice protein explores the energy funnel toward the native folded state.
Mode-Coupling Viscosity Divergence
Mode-coupling theory predicts viscosity diverges as η ~ (T−T_c)^{−γ} near the glass transition.
Navier-Stokes Vorticity-Streamfunction
2D incompressible flow solved via vorticity-streamfunction; click to inject interacting vortex pairs.
Mean First Passage Time
Stochastic particles escape a double-well potential; Kramers' law predicts exponential barrier dependence.
Wigner-Weisskopf Spontaneous Emission
Two-level atom decays exponentially into the vacuum, emitting photons with a Lorentzian spectrum.
Topological Charge & Skyrmions
Spin textures with integer winding number Q that cannot be smoothly deformed away — topological protection visualized.
Random Walk Return Probability
Pólya's theorem: walks are recurrent in d=1,2 but escape forever in d≥3 — watch the dimension crossover live.
Quantum Tunneling: Double Well
Wave packet oscillates between wells via quantum tunneling; tunnel splitting Δ is exponentially small in barrier height.
Hawking Radiation & Black Hole Evaporation
Virtual pairs at the event horizon cause black holes to radiate thermally and slowly evaporate as T_H ~ 1/M.
Nonlinear Schrödinger — Modulational Instability
Split-step Fourier simulation of the focusing NLS equation showing Benjamin–Feir instability breaking plane waves into solitons.
Birth-Death Master Equation
Probability distribution evolving under birth-death dynamics, converging to the Poisson stationary state.
Spin-Torque Oscillator
LLG-Slonczewski equation drives sustained magnetization precession via spin-polarized current — the physics behind nanoscale oscillators.
Fokker-Planck Stationary Distribution
Probability density flowing through a potential landscape toward the Boltzmann equilibrium P*(x) ∝ exp(−U/D).
Lattice QCD — Confinement String
Monte Carlo sampling of SU(2) gauge links on a lattice; the Cornell potential σr−α/r shows the quark confinement string.
Ising Model — Finite-Size Scaling
Metropolis simulation near Tc≈2.269 illustrating critical exponents β=1/8, γ=7/4, ν=1 of the exact 2D Ising solution.
Epidemic Pair Approximation
Beyond mean-field SIS: pair approximation tracks [SI],[SS],[II] correlations on a network, correcting the epidemic threshold.
Immune Epitope — Affinity Maturation
Darwinian evolution of B-cell clones navigating a rugged affinity landscape through somatic hypermutation and selection.
Conformal Field Theory — Operator Algebra
Visualize conformal maps of the complex plane and how primary operators transform under z², exp(z), Möbius, and Joukowski maps.
Voronoi Power Diagram — Weighted
Interactive power diagram where each site has a weight; the power distance πᵢ(x)=|x−pᵢ|²−wᵢ generalizes ordinary Voronoi cells.
Game of Life: Methuselah Patterns
Tiny initial configurations that evolve for thousands of generations before stabilizing — seeds of enormous complexity.
Quantum Reflection at a Barrier
A wave packet encounters a potential barrier, simultaneously reflecting and transmitting — quantum mechanics defies classical intuition.
Epidemic with Heterogeneous Mixing
Social network hubs act as super-spreaders, dramatically lowering epidemic thresholds compared to well-mixed populations.
Heat Equation: Diffusion Spreading
Paint heat sources onto a 2D grid and watch temperature diffuse — the fundamental PDE governing thermal conduction.
Brownian Bridge: Conditioned Paths
A Wiener process pinned to start and end at fixed values — random walk constrained to return, with a characteristic variance envelope.
Zero-Temperature Glauber Dynamics
Ising spins that instantly align with local majority produce coarsening, stripes, and metastable frozen states — order from disorder.
Optimal Transport: Earth Mover's Distance
The minimum work to reshape one distribution into another — Wasserstein geometry made visual with interactive histograms.
Stochastic Resonance
Adding noise to a weak subthreshold signal can actually improve detection — a counterintuitive phenomenon found throughout nature.
Classical Chaos: Double Pendulum
The simplest deterministic system that is chaotic — nearby trajectories diverge exponentially, quantified by the Lyapunov exponent.
Hysteresis Loop: Magnetic Domains
Ferromagnets remember their history — watch domains flip as the applied field cycles, tracing the characteristic M–H hysteresis loop.
Network Epidemic Threshold
Spectral theory of SIS epidemics: the epidemic threshold β/γ = 1/λ₁ where λ₁ is the largest eigenvalue of the adjacency matrix.
Multifractal Spectrum f(α)
The singularity spectrum f(α) computed via Legendre transform of τ(q), revealing the fractal dimension of sets with Hölder exponent α.
Dislocation Glide in Crystals
Edge dislocations glide along slip planes under shear stress — the atomic-scale mechanism of plastic deformation in metals.
Penrose Tiling via Substitution
Aperiodic Penrose tilings built by subdividing fat and thin rhombi via golden ratio inflation rules, with 5-fold symmetry.
Rabi Oscillations
A two-level atom driven near resonance oscillates on the Bloch sphere at the generalized Rabi frequency Ω_R = √(Ω₀² + Δ²).
Granular Gas Inelastic Cooling
Inelastic particle collisions drive Haff's cooling law T ∝ (1 + t/t₀)⁻² and spontaneous clustering instability at late times.
Critical Droplet Nucleation
Classical nucleation theory: a free energy barrier ΔG* separates dissolving embryos from supercritical droplets that grow spontaneously.
Autocorrelation Time — Monte Carlo
Critical slowing down near the Ising transition: autocorrelation time τ diverges with z≈2 for Metropolis, but z≈0 for Wolff cluster moves.
Gillespie Stochastic Reactions
Exact stochastic simulation of chemical kinetics: Gillespie's algorithm fires reactions at the correct random times, revealing molecular noise.
Cavity Polariton Condensate
Exciton-polaritons form a non-equilibrium BEC in driven microcavities, governed by the dissipative Gross-Pitaevskii equation.
Magnetic Skyrmion Hall Effect
Topological spin textures deflect perpendicular to applied current via the Magnus force from their topological charge Q = −1.
Omori Aftershock Law
Aftershock rate decays as R(t) = K/(c+t)^p — Omori's law — with stochastic seismicity and epicenter map.
Polymer Translocation Through a Nanopore
Watch a polymer thread through a nanopore under voltage bias; translocation time scales as τ ~ N^(1+ν) with anomalous diffusion.
Noise-Induced Transitions in Bistable Systems
A particle in a double-well potential escapes via thermally-activated Kramers transitions with rate r ~ exp(−ΔU/D).
Flocking & Polar Active Matter
Vicsek model of self-propelled particles shows a disorder-to-order phase transition to collective motion — long-range order in 2D active matter.
Cellular Senescence & Aging
Cells accumulate DNA damage, enter permanent growth arrest, and spread inflammation via SASP — the engine of biological aging.
Quantum Dot Shell Filling
Artificial atom with discrete shells filled by Hund's rules; Coulomb blockade addition spectrum reveals magic numbers 2, 6, 12.
Turbulent Mixing & Scalar Decay
Passive scalar advected by turbulence forms Batchelor-scale lamellae as variance decays — Kraichnan k^−1 scalar spectrum.
Branching Brownian Motion
Diffusing particles that split at rate β generate a KPP traveling wave front advancing at velocity v = √(2Dβ).
Synchronization of Pulse-Coupled Oscillators
Integrate-and-fire neurons coupled by excitatory pulses self-organize into global synchrony — Mirollo-Strogatz theorem.
Geodesic Flow & Curvature
Straight lines on curved surfaces reveal geometry — geodesics focus on spheres, diverge on saddles, guided by Gauss-Bonnet.
Avalanche Size Distribution
The Bak-Tang-Wiesenfeld sandpile self-organizes to a critical state where avalanche sizes follow a power law P(s) ~ s⁻¹·².
Ecological Network — Trophic Cascades
Food web dynamics across trophic levels: remove the apex predator and watch prey explode, cascading through the network.
Defect-Mediated Turbulence
Phase vortices proliferate in the complex Ginzburg-Landau equation past the Benjamin-Feir instability, annihilating in pairs.
Random Field Ising Model
Quenched disorder fragments ferromagnetic order into a mosaic of domains — the Imry-Ma argument made visible.
Debye Screening in a Plasma
Mobile charges assemble a screening cloud around any test charge, replacing 1/r Coulomb with the Yukawa potential e⁻ʳ/λD/r.
Lenia — Continuous Life
A smooth generalization of Conway's Game of Life with kernel-based rules that supports stable gliders and lifelike organisms.
Cross-Diffusion Patterns
Off-diagonal diffusion terms let species respond to each other's gradients, generating patterns impossible in standard Turing systems.
Ricci Flow — Geometry Evolving
Hamilton's geometric heat equation smooths curvature on surfaces — the tool Perelman used to prove the Poincaré conjecture.
Dislocation Dynamics — Crystal Plasticity
Edge dislocations glide through crystal lattices under shear stress, annihilate in pairs, and produce permanent plastic deformation.
Smoluchowski Cluster Growth
Particles merge and fragment: watch cluster size distributions evolve toward gelation under different coagulation kernels.
Hamiltonian Chaos: Poincaré Section
Explore the Hénon-Heiles system — click to launch trajectories and watch KAM tori dissolve into chaos as energy rises.
Active Gel & Cytoskeleton
Actin filaments driven by molecular motors spontaneously organize into flows, asters, and vortices — life's active matter.
Osmotic Pressure: Van't Hoff
Solute particles drive water across a semipermeable membrane; watch pressure build and water levels shift as you tune concentration.
Spectral Radius of Random Graphs
Generate Erdős-Rényi graphs, compute the adjacency spectrum, and see the Wigner semicircle law emerge from randomness.
Wetting Transition & Contact Angle
Young's equation in action: tune three surface energies and watch a droplet morph from super-hydrophilic to super-hydrophobic.
Fiber Bundle Fracture & Avalanches
Statistical failure cascades in a fiber bundle: load fibers until catastrophic breakdown, with power-law avalanche size distributions.
Disordered Hyperuniformity
Point patterns that look random yet suppress large-scale density fluctuations — the hidden order found in retinas and amorphous silicon.
SSH Model: Topological Edge Modes
The simplest topological insulator: tune hopping amplitudes to cross the phase transition and watch protected edge states appear.
NLS Bright Solitons
Self-reinforcing wave packets that never spread: dispersion and nonlinearity cancel perfectly, enabling elastic soliton collisions.
Random Hard Sphere Packing
Sequential addition of non-overlapping disks racing toward the jamming limit — where packing fraction freezes and the contact network rigidifies.
Rule 110: Edge of Chaos
The simplest known Turing-complete system — a 1D cellular automaton proved capable of universal computation through glider collisions.
Discrete-Time Quantum Walk
A quantum coin and a shift operator spread a particle ballistically rather than diffusively — the quadratic speedup at the heart of quantum search.
Spin Glass Aging
Frustrated random bonds trap the system in exponentially many metastable states — energy creeps downward forever, never reaching equilibrium.
Chemical Garden
Click to plant metal salt seeds in silicate solution — osmotic pressure ruptures membranes and grows hollow mineral tubes upward.
Frustrated Kuramoto Network
Phase oscillators in competing communities battle between within-group synchrony and between-group frustration — cluster sync wars on a network.
Minimal Surfaces
Soap films span wire frames by minimizing area — zero mean curvature everywhere, from the classical catenoid to Costa's 1982 surprise.
Worm-Like Chain Polymers
Thermal fluctuations bend semiflexible polymers with a characteristic persistence length — from rigid DNA to floppy random coils.
Lévy Flights
Heavy-tailed step distributions produce superdiffusion — the anomalous spreading strategy used by foraging sharks, albatrosses, and human mobility.
Cellular Potts Model
Cell sorting emerges from differential adhesion — mix two cell types and watch Steinberg's 1963 hypothesis play out in Monte Carlo steps.
Perron-Frobenius Theorem
Every positive matrix has a unique dominant real eigenvalue — watch power iteration converge to the Perron eigenvector that underlies PageRank and population dynamics.
Topological Pump
Thouless's 1983 adiabatic pump transports exactly one quantum of charge per cycle — a topologically protected process immune to small perturbations.
Jamming Transition
Inflate disks until they jam — pressure diverges at random close packing as the system transitions from fluid to rigid at the jamming point φ_J.
Autocatalytic Networks
Kauffman's RAF sets show how chemistry bootstraps itself — above a critical connectivity, a self-sustaining reaction network spontaneously appears.
Flocking Boids
Reynolds' three rules — separation, alignment, cohesion — produce lifelike flocking; the Vicsek order parameter tracks the noise-driven phase transition.
Müller-Lyer Illusion
Equal lines appear different lengths depending on arrowhead direction — measure your own perceptual bias and explore depth-cue and Bayesian explanations.
Forest Fire Percolation
Ignite a forest at varying tree densities to see the sharp percolation threshold at p_c ≈ 0.593 where fire first spans the entire grid.
Slime Mold Aggregation
Dictyostelium amoebae aggregate via cAMP chemotaxis waves into a multicellular slug — a live demonstration of the Keller-Segel model.
Vortex Merger
Point vortices evolve under Biot-Savart — co-rotating vortices orbit and merge while counter-rotating pairs translate, conserving Kelvin circulation.
Barchan Dune Migration
Crescent-shaped dunes march downwind at speed ∝ 1/height — smaller dunes overtake larger ones, collide, and merge in Bagnold's aeolian dynamics.
Hyperboloid Geodesics
Trace shortest paths on the one-sheeted hyperboloid — a doubly-ruled surface where geodesics wind around the waist, governed by Clairaut's integral.
Ecosystem Tipping Points
A fold bifurcation drives sudden ecosystem collapse: drag the stress slider past the critical point to watch the healthy state vanish and hysteresis trap the system below.
Langmuir Adsorption
Watch gas molecules adsorb onto a surface in real time — the Langmuir isotherm θ = KP/(1+KP) emerges from kinetic balance between adsorption and desorption rates.
Nonreciprocal Waves
Break time-reversal symmetry with a gyroscopic metamaterial term and watch waves propagate one-way — an acoustic analogue of the quantum Hall edge state.
Optical Tweezers
A focused laser beam creates a harmonic trap that captures Brownian particles — click to move the focal point and watch thermal fluctuations compete with the gradient force.
Cosserat Micropolar Elasticity
Each grain carries an independent microrotation that couples to strain via a new length scale — the size effect that makes thin beams stiffer than thick ones.
Bethe Lattice Percolation
The Bethe lattice is exactly solvable: p_c = 1/(z−1), with mean-field exponents β=1 and γ=1 — watch the giant cluster emerge as bonds cross the critical threshold.
Morphogen Gradients
Diffusion and degradation shape a French-flag concentration profile that gives each cell its positional identity — Wolpert's 1969 model brought to life.
Active Nematics
Self-propelled rods align head-to-tail and spontaneously form ±½ topological defects that drive large-scale active turbulence — as seen in microtubule-kinesin gels.
Penrose Quasicrystal
Inflate thick and thin rhombi by the golden ratio φ to generate an aperiodic tiling with 10-fold symmetry — the mathematical skeleton of Shechtman's 1984 Nobel discovery.
Elastic Turbulence
Chaos in polymer solutions at low Reynolds number driven by stretched chain elasticity.
Evolutionary Prisoner's Dilemma
Spatial strategy competition where cooperators survive by clustering against defectors.
Richards Equation
Unsaturated water flow through layered soils — the wetting front and perched water tables.
Burridge-Knopoff Earthquake Model
Spring-block stick-slip dynamics that reproduce the Gutenberg-Richter law of seismicity.
Ant Colony Optimization
Emergent shortest-path finding through pheromone stigmergy — collective intelligence without a leader.
Second Harmonic Generation
How nonlinear crystals convert infrared light to visible via χ² interactions and phase matching.
Budyko Climate Model
Ice-albedo feedback, bifurcations, and the snowball Earth instability in a 1D energy balance model.
Amphiphile Self-Assembly
Spontaneous micelle formation as hydrophobic tails cluster to escape water — the driving force of life.
Complex Contagion Cascades
Threshold-based social adoption on networks — when seeds trigger global cascades and when they do not.
Open Quantum Systems
Lindblad master equation — watch decoherence shrink the Bloch sphere as a qubit couples to its environment.
7000 Simulations: A Journey Through Science
A milestone celebration — morphing particle systems, live mini-sims, and a meditation on emergence across all scientific domains.
Kinematic Dynamo
Watch a conducting fluid flow amplify a magnetic seed field via stretching and folding, driving a self-sustaining dynamo.
Spin-Torque Oscillator
Spin-polarized current drives sustained magnetization precession on the Bloch sphere via the LLGS equation.
Osmotic Shock & Cell Volume
Apply osmotic shocks and watch a cell swell or shrink as water flows across its membrane down the osmotic gradient.
Epidemics on Hypergraphs
Higher-order group interactions create explosive epidemic transitions absent in ordinary pairwise network models.
Polar Active Fluid
Self-propelled particles align via Vicsek rules, revealing the non-equilibrium flocking transition predicted by Toner-Tu theory.
Baroclinic Instability
Temperature gradients and planetary rotation conspire to generate atmospheric vortices and extratropical cyclones.
Fracture Mechanics: Crack Tip
Visualize the singular stress fields around a crack tip in LEFM, with Mode I/II stress intensity factors and plastic zone.
Social Learning & Cultural Evolution
Watch traits spread through populations via imitation, innovation, and selection — exploring Rogers's paradox on dynamic networks.
Berry Phase & Holonomy
A spin-1/2 adiabatically transported around the Bloch sphere acquires a geometric phase equal to half the enclosed solid angle.
Magnetic Reconnection
Plasma topology change and energy release
Shear Banding
Localized flow in viscoplastic materials
Elasticity Tensor
Stress, strain, and continuum mechanics
Euler Vorticity
Inviscid fluid flow and vortex dynamics
Kramers Escape
Thermal activation over energy barriers
Network Motifs
Recurring subgraph patterns in networks
Soap Film
Minimal surfaces and surface tension
Hysteresis Loop
Magnetization and magnetic memory
Fiber Bundle Fracture
Statistical model of material failure
Charge Density Waves
Periodic electron density modulations
Granular Hopper Flow
Intermittent flow and jamming in granular systems
Frustrated Magnets
Geometric frustration on triangular lattice
Acoustic Resonance
Standing waves and cavity modes
Brownian Ratchet
Directed transport from thermal fluctuations
Fractal Diffusion
Anomalous diffusion on Sierpiński gasket
Vibration Modes
Normal mode shapes of vibrating structures
Flocking & Toner-Tu
Collective motion and polar active matter
Landau-Ginzburg
Order parameter field theory near phase transitions
Anderson Localization
Wave localization in disordered media
Hopfield Network
Associative memory and attractor dynamics
Thin Film Interference
Structural color from wave interference in thin layers
Nucleation & Growth
Classical nucleation theory and droplet dynamics
Josephson Junction
Superconducting plasma oscillations and phase dynamics
Quantum Ising Model
Transverse field quantum phase transition
River Delta Networks
Bifurcation and self-organization in river deltas
Circle Swimmers
Active Brownian particles with circular trajectories
Kicked Rotor Chaos
Quantum chaos and dynamical localization
Active Nematics
Self-propelled rods and spontaneous flow
Burgers Equation
Shock waves and viscous dissipation in 1D flow
Fourier & Sampling
DFT, aliasing, and the Nyquist theorem
String Harmonics
Standing waves and harmonic series on vibrating strings
Electron-Phonon Coupling
Cooper pairing and lattice vibration interactions
Langmuir Waves
Electrostatic plasma oscillations and wave dispersion
JKR Contact Mechanics
Adhesion forces and elastic contact at nanoscale
Couette Flow
Viscous flow between rotating cylinders
Prey Switching
Frequency-dependent predation and population cycles
Supercritical Fluid
Critical opalescence and density fluctuations
Elastic Stress Waves
Wave propagation and reflection in elastic rods
Needleman-Wunsch
Dynamic programming for sequence alignment
Lyapunov Escape-Time
Mandelbrot set colored by Lyapunov exponents
Waveguide Dispersion
TE/TM modes and cutoff frequencies in rectangular waveguides
Laplace Pressure
Membrane tension, curvature, and pressure balance
Coupled Pendulums
Normal modes and chaotic dynamics in coupled pendulums
Ghost Imaging
Quantum correlations and coincidence detection
Rayleigh Quotient
Variational eigenvalue bounds and power iteration
Protein Contact Maps
Residue contacts and folding energy landscapes
Helfrich Fluctuations
Thermal undulations of lipid bilayer membranes
Binary Alloy Decomposition
Spinodal decomposition and phase separation in alloys
Adiabatic Invariants
Action variables and slow parameter change in Hamiltonian systems
Heat Bath MCMC
Gibbs sampling and heat bath algorithm for spin systems
Random Walk Polymer Chain
A freely-jointed chain of N monomers performs a random walk; end-to-end distance grows as R ~ N^0.5 with Gaussian statistics.
Max-Flow Min-Cut Theorem
Ford-Fulkerson algorithm finds maximum flow from source to sink; by the theorem this equals the minimum cut capacity separating them.
Polymer Reptation vs Rouse
Two polymer dynamics regimes: Rouse diffusion D~N⁻¹ for unentangled chains, reptation D~N⁻² for entangled melts in a tube.
Critical Slowing Down
Near a phase transition, relaxation time τ diverges as |T−Tc|^(−νz) — systems become sluggish and fluctuations grow at criticality.
Stochastic Resonance
Adding the right amount of noise to a subthreshold signal enhances detection — SNR peaks at an optimal noise level, not at zero noise.
Landau-Zener Avoided Crossing
Two quantum levels repel and avoid crossing; slow sweeps follow adiabatically while fast sweeps cause diabatic transitions with P=exp(−2πΓ).
Frustrated Kuramoto Oscillators
Phase oscillators with random frequencies and frustration phase α compete between coupling and disorder, producing partial synchronization and chimera hints.
Lévy Flights & Heavy Tails
Power-law step distributions produce superdiffusive Lévy flights where rare long jumps dominate and MSD scales as t^(2/α) instead of t.
Adiabatic Invariance & Action
When a harmonic oscillator's frequency changes slowly, the action J=E/ω is conserved; rapid changes destroy the invariant.
Müller Matrix Polarimetry
The 4×4 Müller matrix acts on Stokes vectors to transform polarization; explore wave plates, polarizers, and rotators on the Poincaré sphere.
Scale-Free Network Fitness
Bianconi-Barabási fitness model: nodes compete for links via η·k attachment probability, producing Bose-Einstein condensation where the fittest node dominates.
Cyclic Competition Automaton
Rock-paper-scissors cellular automaton with three-state cyclic dominance on a grid, generating self-organizing spiral waves and coexistence dynamics.
Quantum Zeno Effect
Frequent measurement slows quantum decay: survival probability [cos²(ΩT/2N)]^N approaches 1 as measurement frequency N → ∞.
Vicsek Model: Active Matter
N polar self-propelled particles align with neighbors within radius r; a noise-driven phase transition from disorder to collective flocking motion emerges.
Hopf Bifurcation Normal Form
The canonical Hopf bifurcation ṙ=r(μ−r²) births a limit cycle at r*=√μ when μ crosses zero, with interactive phase plane and potential visualization.
Poincaré Section of Lorenz Chaos
A stroboscopic slice through the Lorenz attractor reveals fractal Cantor-set structure; chaotic trajectories fill an infinitely fine strange set.
Basin of Attraction — Newton Fractals
Newton's method for complex polynomials partitions the plane into fractal basins of attraction, with boundaries of Hausdorff dimension 2.
Tracer Diffusion in Crowded Media
Obstacle-obstructed diffusion produces anomalous subdiffusion MSD~t^α with α<1; explore the percolation threshold where diffusion nearly halts.
Host-Pathogen Coevolution
Red Queen dynamics: resistance and virulence alleles chase each other in perpetual arms-race oscillations driven by frequency-dependent selection.
Van der Pol Relaxation Oscillator
Nonlinear damping μ(1−x²)ẋ drives a stable limit cycle transitioning from sinusoidal to sawtooth relaxation oscillations as μ increases.
Cooperative Binding & Hill Function
The sigmoidal Hill equation θ = [L]ⁿ/(Kⁿ+[L]ⁿ) models allosteric protein binding; the Hill coefficient n controls cooperativity from Michaelis-Menten (n=1) to switch-like (n→∞).
Fiber Bundle Failure Model
A bundle of N fibers fails progressively as load redistributes after each break; avalanche statistics distinguish brittle catastrophic fracture from quasi-ductile failure.
Chiral Symmetry Breaking
The Mexican hat potential V = λ(σ²+π²−v²)² governs spontaneous chiral symmetry breaking; Nambu-Goldstone pions emerge when the quark condensate ⟨ψ̄ψ⟩ ≠ 0 at low temperature.
Josephson Junction Phase Dynamics
The RCSJ model φ̈ + γφ̇ + sin(φ) = I/Ic describes a superconducting junction as a phase particle in a tilted washboard potential, producing IV curves and Shapiro steps under RF drive.
Ising Mean-Field Theory
The self-consistency equation m = tanh(βJzm + βh) is solved graphically; spontaneous magnetization emerges below Tc = Jz/k with mean-field critical exponent β = 1/2.
Turbulent Pipe Flow Transition
Hagen-Poiseuille parabolic flow transitions to turbulence above Re ≈ 2300; explore velocity profiles, the Moody friction factor diagram, and pressure-drop scaling across flow regimes.
Quasiperiodic Fibonacci Chain
A 1D quasicrystal built by the substitution A→AB, B→A has tile ratio converging to the golden ratio τ, a Cantor-set energy spectrum, and sharp quasiperiodic diffraction peaks.
Lattice Boltzmann Fluid Simulation
D2Q9 streaming and BGK collision steps evolve fluid distributions on a lattice; vortex shedding past an obstacle emerges from purely local rules — click to draw new obstacles.
Zero-Temperature Quantum Criticality
The transverse-field Ising model has a quantum critical point at Γ/J = 1 where quantum fluctuations close the spectral gap and drive a T=0 phase transition with Ising universality class.
DNA Replication Fork Stochastic
Origins fire stochastically during S-phase and bidirectional forks merge neighboring replicons; the distribution of replication completion times depends on origin density and firing kinetics.
Housekeeping Entropy Production
Nonequilibrium steady states decompose total entropy production into housekeeping (maintained by nonconservative forces) and excess parts via the Hatano-Sasa relation.
Nematic Defect Annihilation
Nematic liquid crystal ±½ topological defect pairs attract via Frank elastic forces and annihilate, releasing stored orientational energy.
Delay-Coupled Oscillators
A time delay τ in coupling creates phase frustration, stability boundaries in (τ,κ) space, and Hopf bifurcations that destroy synchronized states.
Evolutionary Branching & Adaptive Dynamics
Adaptive dynamics tracks trait evolution under invasion fitness; disruptive selection at singular strategies causes populations to branch into distinct morphs.
Triadic Closure & Network Evolution
Triadic closure drives clustering coefficient growth as open triangles close; Granovetter weak ties bridge clusters in evolving social networks.
Population Vector Neural Coding
N directionally tuned neurons decode stimulus angle via weighted population vector; Fisher information sets the Cramér-Rao precision bound.
Autocatalytic Sets & RAF Theory
Reflexively autocatalytic food-generated sets model Kauffman's origin-of-life chemistry, emerging spontaneously above a catalytic complexity threshold.
Bead-Spring Polymer Viscoelasticity
The Rouse model connects bead-spring chain dynamics to G'(ω) and G''(ω) viscoelastic moduli via normal mode relaxation times.
Transition State Theory & Saddle Points
An index-1 saddle defines the transition state; Arrhenius kinetics k = A·exp(−Eₐ/kT) and Kramers theory connect barrier height to reaction rate.
Mixed Strategy Nash Equilibria
Players randomize to satisfy indifference conditions; replicator dynamics converge to mixed NE or cycle in rock-paper-scissors.
Worm-Like Chain: DNA Stretching
Entropic elasticity of semiflexible polymers — the WLC force-extension relation and 50 nm persistence length of dsDNA.
Spectral Radius & Matrix Stability
May-Wigner threshold σ√(SC)=1: when random ecological networks destabilize as complexity grows.
Noisy Majority-Vote Model
Spin flips against local majority with noise q — continuous order-disorder transition at qc≈0.075 in the 2D Ising universality class.
Heteroclinic Cycles: Rock-Paper-Scissors
Cyclic species competition creates saddle connections on the simplex — intermittent slow-fast dynamics near each vertex.
Telegraph Process: Two-State Switching
Two-state Markov switching with exponential dwell times — Lorentzian power spectrum, gene expression bursting, and the diffusion limit.
SIR on Heterogeneous Networks
Degree-based mean-field epidemic: threshold 1/⟨k²/k⟩ vanishes on scale-free networks, revealing the superspreader hub effect.
Polymer Brush: Alexander-de Gennes
Tethered polymer chains stretch to height H~N(a/d)^(2/3) — osmotic pressure versus entropic elasticity at grafted surfaces.
Shear Transformation Zones
Flow defects in amorphous solids: STZ activation, stress-strain curves, yield stress, and shear band nucleation in metallic glasses.
Quantum Dot: Exciton-Biexciton Cascade
Two-photon cascade through the biexciton-exciton ladder, photon antibunching g²(0)=0, and Purcell-enhanced single-photon emission.
Correlated Equilibrium
A mediator's joint recommendation distribution that no player wants to deviate from — the CE polytope contains all Nash equilibria and more.
Neutral Theory of Biodiversity
Hubbell's unified neutral model: speciation-drift-dispersal balance generates log-series species abundance distributions from a single parameter θ.
Laser Threshold & Population Inversion
Four-level rate equations for gain vs loss: output power switches on sharply above threshold pump rate as N₂ crosses the critical inversion density.
Wetting & Contact Angle
Young's equation balances three surface energies; spreading coefficient S determines whether a liquid spreads completely or beads into a partial-wetting droplet.
Critical Percolation Hull Walk
Site percolation at p_c ≈ 0.5928 on the square lattice; cluster external boundaries trace fractal paths with dimension d_f = 7/4 converging to SLE₆.
Flashing Ratchet
An asymmetric sawtooth potential switching on and off drives overdamped Brownian particles in one direction — directed transport from noise without a net force.
Flory-Huggins Polymer Blends
Free energy of mixing with entropic chain-length penalties and the χ interaction parameter; spinodal and binodal boundaries drawn live on the phase diagram.
Stress Drop Avalanches
Amorphous plasticity under shear: local yielding events cascade into power-law distributed stress drops with mean-field exponent τ ≈ 1.5.
Sznajd Opinion Dynamics
Aligned neighbor pairs persuade their surrounding agents; depending on noise rate, the system either reaches full consensus or freezes in fragmented coexistence.
Kramers Escape Rate
Mean first-passage time over a double-well barrier follows the Arrhenius law k ∝ e^(−ΔU/kT); interactive barrier height and temperature reveal the exponential sensitivity.
Lempel-Ziv Complexity
LZ76 parses a binary sequence into maximally novel subwords; the count normalized by n/log(n) distinguishes random, chaotic, periodic, and structured signals.
Topological Quantum Computation
Non-Abelian anyons braid world-lines in 2D to implement fault-tolerant quantum gates — topological winding number encodes information immune to local errors.
Stochastic Resonance
Adding noise to a subthreshold signal paradoxically improves detection in a bistable system — the signal-to-noise ratio peaks at an optimal noise level.
Polymer Coil-Globule Transition
A polymer chain swells into a fractal coil in good solvent (ν≈0.59) and collapses into a compact globule in poor solvent (ν=1/3) as solvent quality changes.
Invasion Percolation & Trapping
A fluid invades porous media by always entering the lowest-resistance neighboring cell, self-organizing to the critical threshold and trapping isolated pockets.
Spectral Clustering
Eigenvectors of the normalized graph Laplacian embed data so non-convex clusters become linearly separable — solving the minimum normalized-cut problem.
Abelian Sandpile Model
Sand grains topple when cells overflow, triggering power-law avalanches — the set of stable configurations forms an abelian group with a striking fractal identity element.
Rényi Entropy
A one-parameter family of entropies H_α interpolates from Hartley (α=0) through Shannon (α=1) to min-entropy (α=∞), each sensitive to a different moment of the distribution.
Geodesic Flow on Hyperbolic Surface
Circular arcs in the Poincaré disk are geodesics of hyperbolic geometry; nearby geodesics diverge exponentially — a prototype of Anosov chaos and the Selberg trace formula.
Zero-Temperature Quantum Phase Transition
The transverse-field Ising model transitions from ordered ferromagnet to quantum paramagnet at Γ/J=1 — a quantum critical point driven by fluctuations, not heat.
Arrhenius Kinetics
Reaction rates follow k = A·exp(−Ea/RT) — molecular simulation shows only high-energy collisions surmount the activation barrier, with exponential temperature sensitivity.
Quantum Spin Liquid
Geometric frustration on the triangular lattice prevents magnetic ordering even at T=0 — spins remain in a quantum superposition of competing configurations, a exotic disordered ground state.
Turing Pattern Formation
The Gray-Scott reaction-diffusion system spontaneously generates spots, stripes, and mazes from homogeneous noise when an activator diffuses slower than its inhibitor.
Granular Jamming Transition
As packing fraction φ exceeds φ_c ≈ 0.64, granular particles abruptly transition from a flowing state to a rigid amorphous solid supported by percolating force chains.
Bose-Hubbard Mott Insulator
Competing hopping J and on-site repulsion U drive a quantum phase transition from a coherent superfluid to a Mott insulator with exactly one boson per site.
Anomalous Hall Effect
Berry curvature in momentum space acts as a synthetic magnetic field, deflecting electrons transversely without any external B-field — its integral gives the quantized Chern number.
Matrix Product States & DMRG
The DMRG algorithm variationally optimizes a chain of tensors to capture 1D quantum ground states, exploiting the area law that limits entanglement in gapped systems.
Random Close Packing
Random sphere packings jam at φ_RCP ≈ 0.64, far below the crystal optimum — coordination number colored discs reveal defects and the boundary between amorphous and ordered packing.
Continuous-Time Quantum Walk
Quantum walkers evolve under H = −γA and spread ballistically (∝ t) rather than diffusively — the basis for quantum search speedup, with Anderson localization killing transport at strong disorder.
Bifurcation Normal Forms
Saddle-node, transcritical, pitchfork, Hopf, and cusp bifurcations are universal templates for how dynamical systems change qualitative behavior — classified by codimension.
Spin Glass & Replica Symmetry Breaking
Random frustrated couplings create a rugged free-energy landscape with exponentially many minima — Parisi's RSB solution (Nobel 2021) encodes their ultrametric hierarchy in the overlap distribution P(q).
Topological Defects & Winding Number
Place +1 and −1 defects in an ordered medium and watch the director field wind around them — winding number counts rotations along a closed loop and determines topological stability.
Chaotic Map & Symbolic Dynamics
Encode logistic-map orbits as binary sequences (L/R) and see how symbolic dynamics captures the full topology of chaos — cobweb diagrams reveal sensitivity to initial conditions.
Self-Organized Criticality
BTW sandpile model self-tunes to a critical threshold — avalanches span all scales and their size distribution follows a power law without any parameter tuning.
Critical Exponents & Data Collapse
Adjust β and ν until five finite-size magnetization curves collapse onto a single universal function — the precision of this collapse reveals the Ising universality class exponents.
Predator-Prey Spatial Patterns
Spatial Lotka-Volterra dynamics on a 2D lattice produce spiral waves and refugia that stabilize the population cycle far beyond what the mean-field ODE predicts.
Free Energy Landscape & Metastability
Langevin particles navigate a double-well potential — Kramers' escape rate k ∝ exp(−ΔF/kT) makes barrier crossing exponentially rare, and thermal noise drives spontaneous symmetry breaking.
Dynamical Mean-Field Theory
Tune the Hubbard U/W ratio and watch the spectral function transform from a coherent quasiparticle peak to split Hubbard bands as the Mott metal-insulator transition is crossed.
Quantum Tunneling & WKB
WKB approximation gives tunneling transmission T ≈ exp(−2κa) through a classically forbidden barrier — the wavefunction decays exponentially inside yet emerges coherently on the far side.
Perron-Frobenius Theorem
Power iteration on a random nonneg matrix converges to the unique positive Perron eigenvector — node size encodes the stationary distribution underlying PageRank and Markov chains.
Synchrony: Coupled Phase Oscillators
Kuramoto model shows a sharp synchronization transition at K_c = 2σ/π — watch oscillators with spread natural frequencies suddenly lock into a coherent rotating cluster as coupling increases.
Winding Number & Topological Charge
Count how many times a vector field winds around a defect — integer topological charges that cannot be removed by smooth deformations, classified by π₁(S¹) = ℤ.
CFT Two-Point Correlators
Conformal invariance fixes the form of all two-point functions exactly: ⟨O(r₁)O(r₂)⟩ = C/|r₁−r₂|^{2Δ} — drag operators and watch correlations decay with a single scaling dimension Δ.
Fitness Landscape & Epistasis
High epistasis creates rugged multi-peaked landscapes where evolving populations get trapped at local optima — explore the NK model from smooth single-peak to maximally rugged.
Stochastic Gene Expression
Transcription happens in random bursts, producing wide protein-level variation even in genetically identical cells — watch noise η² = 1/⟨n⟩ + b/k_on set the cell-to-cell spread.
Loschmidt Echo & Quantum Fidelity
Evolve forward then backwards under a slightly perturbed Hamiltonian — chaotic systems lose quantum fidelity exponentially at the Lyapunov rate, independent of perturbation size.
Anomalous Diffusion & Subdiffusion
Beyond Brownian motion: ⟨x²⟩ ~ t^α where α<1 traps particles in crowded media (CTRW) and α>1 drives superdiffusive Lévy flights — watch the MSD exponent shift live.
Directed Network Bow-Tie Structure
The Web's giant SCC core, IN and OUT wings, and disconnected tendrils revealed by Kosaraju's algorithm — Broder et al. 2000 found ~25% in each major compartment.
Quantum Decoherence & Pointer States
Environment interaction kills quantum coherences exponentially — watch off-diagonal density matrix elements vanish as the Bloch vector collapses toward a classical pointer state.
Colloidal Gel & Percolation Network
Attractive colloidal particles cluster into a spanning gel network at the percolation threshold — simulate aggregation and watch the system-spanning cluster appear as a phase transition.
Entropy of Mixing & Gibbs Paradox
Remove the partition between two gases: distinguishable species gain mixing entropy −R Σ xᵢ ln xᵢ, but identical gases gain nothing — the N! correction that foreshadowed quantum indistinguishability.
Microcanonical Ensemble & Phase Space
Isolated particles explore the energy shell in phase space — watch the Maxwell-Boltzmann distribution emerge from equal a priori probabilities as entropy S = k ln Ω.
Climate Tipping Points & Bifurcations
Fold bifurcations create irreversible climate tipping — hysteresis means the path back differs from the path forward, and critical slowing down signals the approaching catastrophe.
Gillespie Algorithm & Chemical Kinetics
Exact stochastic simulation of molecular reactions — draw waiting times from exponential distributions to capture intrinsic noise that deterministic ODEs miss at low copy numbers.
Spiral Waves in Excitable Media
Broken wavefronts in FitzHugh-Nagumo excitable media curl into stable rotating spirals — the same mechanism behind cardiac arrhythmia and the Belousov-Zhabotinsky reaction.
Mutual Information & Channel Capacity
Shannon's channel capacity C = max I(X;Y) — explore BSC, BEC, and AWGN channels, visualize the information Venn diagram, and see how noise erodes the bits you can reliably send.
Variational Autoencoder Latent Space
The ELBO = reconstruction − β·KL regularizes the latent space toward a Gaussian prior — watch classes cluster, disentangle under high β, and interpolate smoothly between concepts.
Active Brownian Particles & MIPS
Self-propelled particles with rotational diffusion undergo Motility-Induced Phase Separation — a purely non-equilibrium gas-liquid transition with no attractive forces required.
Mean First Passage & Barrier Crossing
Kramers (1940) theory of thermally activated escape — MFPT ∝ exp(ΔV/kT) creates exponential sensitivity to barrier height, the foundation of chemical kinetics and protein folding.
Replica Method & Spin Glasses
Parisi's replica symmetry breaking reveals infinitely many pure states arranged ultrametrically — the exact solution (Guerra-Talagrand 2003) of the SK model and its P(q) overlap distribution.
Adiabatic Quantum Computing & Gap
Slowly evolve from a trivial Hamiltonian to a hard optimization problem — the adiabatic theorem guarantees the ground state if T ≫ 1/Δ²_min, but the spectral gap can be exponentially small.
Topological Soliton — Sine-Gordon
Kink solitons in the sine-Gordon equation carry conserved topological charge ±1, pass through each other unchanged, and annihilate with antikinks — exact solutions from complete integrability.
Flory-Huggins Polymer Mixing
The entropy of mixing for polymers is drastically reduced by chain length N — explore how the χ parameter and spinodal/binodal boundaries control whether polymer blends phase-separate.
Belousov-Zhabotinsky Reaction
Chemical spiral waves and target patterns in the BZ reaction — a dissipative structure sustained far from equilibrium, modeled by the Oregonator reaction-diffusion system.
Quantum Annealing & Tunneling
Quantum tunneling through energy barriers versus thermal hopping — watch how a transverse field Γ lets particles escape local minima that trap classical simulated annealing.
Eden Growth Model — Surface Roughening
Random perimeter growth produces KPZ universality class surfaces with roughness exponent β≈1/3, connecting bacterial colony growth to random matrix theory via Tracy-Widom statistics.
Transfer Entropy & Information Flow
Measure directed causal influence between time series — transfer entropy TE(X→Y) quantifies how much X's past reduces uncertainty about Y's future beyond Y's own history.
2D Ising Universality Class
Onsager's exact solution (1944) at Tc=2/ln(1+√2) with exact critical exponents β=1/8, γ=7/4 — the paradigmatic example of universality and the renormalization group.
Smoluchowski Coagulation & Aggregation
Cluster aggregation kinetics with constant, sum, product, and Brownian kernels — the product kernel K~ij undergoes gelation, forming an infinite cluster at finite time.
Optical Bistability — Kerr Effect
S-shaped input-output hysteresis in a nonlinear optical cavity — the Kerr effect shifts the resonance with intensity, enabling all-optical switching between bright and dim states.
Lanczos Algorithm — Eigenvalues
Krylov subspace iteration that finds extreme eigenvalues of large sparse matrices in k steps — the workhorse of quantum chemistry, lattice QCD, and spectral graph analysis.
Jacobi Iteration Convergence
Iterative solver for linear systems Ax=b — each component updated from its neighbors until convergence. Guaranteed for strictly diagonally dominant matrices; visualizes log-residual decay.
Diffusion-Limited Aggregation
Fractal growth by random-walking particles that freeze on contact with a growing cluster. Produces stunning dendritic patterns with Hausdorff dimension ≈ 1.71 — universal in snowflakes and mineral dendrites.
Magnetic Monopole — Hedgehog Field
The 't Hooft–Polyakov monopole as a hedgehog configuration: Higgs field points radially outward, a topological soliton classified by π₂(S²) = ℤ. Rotate the 3D field line structure.
Quantum Zeno Effect
Frequent measurement suppresses quantum decay — survival probability follows P ~ 1−(t/τ)² between projections, so rapid measurement "freezes" the system in its initial state.
Voigt Profile — Spectral Line Shape
Convolution of Lorentzian (natural broadening) and Gaussian (Doppler broadening) line shapes. The physically correct spectroscopic profile, interpolating between exponential and power-law wings.
Entropy Production & Irreversibility
Irreversible thermodynamics: σ = Σ JᵢXᵢ ≥ 0 always. Heat conduction between reservoirs drives entropy production that is always non-negative — a live demonstration of the Second Law.
Double-Slit Wave Interference
Young's experiment (1801): two coherent slits produce fringes at d sinθ = mλ. Adjust wavelength and slit separation to see intensity I ∝ cos²(πd sinθ/λ)·sinc²(πa sinθ/λ).
Spin-Transfer Torque — LLG Dynamics
Landau–Lifshitz–Gilbert equation with Slonczewski spin-transfer torque. Above a critical current, spin-polarized electrons overcome damping and switch or oscillate the free-layer magnetization.
Polymer 3D Self-Avoiding Walk
A polymer chain as a self-avoiding lattice walk. End-to-end distance scales as ⟨R⟩ ~ N^ν with Flory exponent ν ≈ 0.588 in 3D — excluded volume swells the coil beyond a simple random walk.
Hopf Fibration S³ → S²
Every point on S² lifts to a great circle on S³, and any two fibers are linked. Hopf's map (1931) generates π₃(S²) = ℤ — visualized via stereographic projection of the fiber bundle structure.
Toda Lattice Solitons
Integrable nonlinear particle chain with exact soliton solutions. Solitons pass through each other elastically with only a phase shift — visualized as a scrolling space-time diagram with conserved energy and momentum.
Lid-Driven Cavity — Navier-Stokes
2D viscous flow in a square cavity driven by a sliding lid. Finite-difference vorticity-streamfunction solver reveals a central vortex and corner eddies as Reynolds number increases — a canonical CFD benchmark.
Wigner-Ville Distribution
Joint time-frequency representation of non-stationary signals with maximal resolution. Reveals instantaneous frequency of chirps and pulses — but nonlinear cross-terms appear when multiple components coexist.
Riemann Zeta — Critical Strip
The Riemann zeta function plotted in the complex plane. Non-trivial zeros appear as dark rings on the critical line Re(s)=½. The Riemann Hypothesis — all zeros lie here — remains unproven and is a Millennium Prize Problem.
Bernoulli Percolation Clusters
Bond percolation on a square lattice near the critical threshold p_c=½. At criticality, clusters appear at all scales with fractal dimension 91/48. Drag p across the threshold to see the spanning cluster snap into existence.
Phonon Thermal Conductivity
Molecular dynamics simulation of a 1D anharmonic lattice. The Green-Kubo formula relates thermal conductivity to heat flux autocorrelation. Pure harmonic chains have infinite κ — anharmonicity (FPUT model) introduces scattering and finite conductivity.
Nonlinear Wave Breaking — Burgers
The inviscid Burgers equation ut + u·ux = 0 steepens any downslope into a shock discontinuity. Watch characteristics converge in the (x,t) plane and the wave overturn at the exact breaking time t* = −1/min(du₀/dx).
Quantum Harmonic Chain
Normal modes, zero-point energy, and phonon occupancy in a 1D lattice. Select any mode to see its displacement pattern; the dispersion curve shows the characteristic sine shape with acoustic behavior near k=0.
Hindmarsh-Rose Neuron
Three-variable model capturing quiescence, tonic spiking, square-wave bursting, and chaos. Tune external current and slow adaptation parameters; watch the membrane voltage and phase portrait (x vs z slow manifold).
Scale-Free Network Growth
Preferential attachment grows a network where new nodes connect proportional to k^α. α=1 gives the Barabási-Albert power law P(k)~k⁻³; log-log degree histogram reveals fitted exponent γ in real time.
Worm-Like Chain Polymer
Sample ensembles of semiflexible polymer chains. Tangent-tangent correlations decay with persistence length Lp. Track the rod-to-coil crossover in ⟨R²⟩/L² vs Lp, matching DNA, actin, and microtubule stiffnesses.
MSD and Diffusion Coefficient
Ensemble of 2D random walkers with tunable anomalous exponent α. The log-log MSD plot fits slope and diffusion coefficient D in real time — sub-diffusion, normal Brownian motion, and super-diffusion regimes.
Hénon-Heiles Chaos Transition
Poincaré sections reveal integrable-to-chaotic transition in a galactic potential as energy increases.
Quantum Kicked Rotor
Dynamical localization suppresses classical diffusion — quantum interference freezes the momentum distribution.
Arctic Circle Domino Tiling
Random Aztec diamond tilings spontaneously freeze outside an inscribed circle — the arctic circle theorem.
Braid Group Word Problem
Build braid words from Artin generators, visualize over/under crossings, and close braids into links.
SOC Rice Pile Model
Oslo rice pile self-organizes to criticality — avalanche sizes follow a power law with no fine-tuning.
Crystallization Nucleation Theory
Classical nucleation: critical nucleus size and nucleation rate vs supercooling from the free energy barrier.
Block-Spin RG — 1D Ising
Exact block-spin renormalization shows coupling flows to zero — no phase transition in 1D confirmed by RG.
Quorum Sensing Switch
Bacterial autoinducer bistability: population density triggers a collective switch via positive feedback.
Chua's Circuit — Double-Scroll
Piecewise-linear Chua diode drives a chaotic attractor spiraling between two unstable foci — simplest electronic chaos circuit.
Persistent Homology Filtration
Grow ε-balls on point clouds and track when components merge and loops form. Persistence diagram reveals robust topology.
Sandpile Group Identity
The abelian sandpile group's identity element produces an exact self-similar fractal — pure arithmetic generating fractal geometry.
Boolean Network Attractor Basins
Kauffman NK random Boolean networks: state-space basins, attractor cycles, and the K=2 "edge of chaos" scaling law.
Saddle-Node Bifurcation
Fold catastrophe: normal form ẋ=r+x², phase portrait, ghost slowing near the bifurcation point.
Pipe Flow Transition to Turbulence
Directed percolation universality: puff dynamics, splitting vs decay, Re threshold at Rec≈2040.
Quantum Graph: Star Spectrum
Star metric graph: secular equation Σcot(kLⱼ)=0, eigenvalues, wave functions on edges.
Rijke Tube Instability
Thermoacoustic feedback: Rayleigh criterion, heater position, mode growth and saturation.
Polymer Brush Scaling
Alexander–de Gennes: brush height H~σ^(1/3) vs grafting density, mushroom-to-brush crossover.
Writhe + Twist = Linking Number
Calugareanu–White–Fuller theorem: Lk=Tw+Wr, DNA supercoiling, topological ribbon invariant.
Stochastic SIS Endemic State
Quasi-stationary distribution, endemic persistence, extinction times scaling as exp(N·f(R₀)).
Shilnikov Chaos
Homoclinic orbit to saddle-focus: ρ/λ<1 condition, spiral attractor, logarithmic return map.
Kitaev Chain: Majorana Modes
Topological superconductor: MZMs at chain ends, Z₂ phase diagram, BdG spectrum gap closing.
Chirikov Standard Map
Resonance overlap criterion: KAM tori, island chains, and chaos onset in phase space.
Eigenvalue Repulsion
GUE/GOE level spacing statistics: Wigner surmise vs Poisson — random matrix level repulsion.
Moody Diagram
Darcy-Weisbach friction factor vs Reynolds number across laminar, transitional, and turbulent regimes.
D2Q9 Lattice Boltzmann
BGK collision-streaming fluid simulation: channel flow and cylinder wake from microscopic kinetics.
Landau Mean-Field Theory
Free energy double-well landscape, order parameter vs temperature, and spontaneous symmetry breaking.
Driven Brownian Ratchet
Asymmetric sawtooth potential rectifies thermal noise into directed transport — no net force needed.
Magnetic Frustration
Antiferromagnetic Ising spins on a triangular lattice: geometric frustration and spin liquid degeneracy.
Wright-Fisher Genetic Drift
Allele frequency trajectories, fixation probability, and the power of population size on neutral evolution.
Vesicle Shape: Helfrich Energy
Bending energy shapes vesicles: sphere, prolate, oblate, stomatocyte vs reduced volume.
Phonon-Polariton Dispersion
Avoided crossing between photon and TO phonon: upper/lower polariton branches and Reststrahlen gap.
Stochastic Block Model
Planted partition network: spectral community detection and Kesten-Stigum threshold.
Quantum Repetition Code
Bit-flip error correction: syndrome measurement, logical vs physical error rates, code distance threshold.
Hamiltonian Monte Carlo
Leapfrog dynamics on potential landscapes: HMC vs random walk for efficient posterior sampling.
Auxetic Metamaterial
Re-entrant honeycomb with negative Poisson ratio — expands laterally when stretched.
Holstein Polaron
Electron-phonon coupling, self-trapping transition, and polaron mass enhancement in the Holstein model.
Shilnikov Chaos
Homoclinic orbit to a saddle-focus generates spiral chaos in 3D systems — Smale horseshoes from a single orbit.
Kitaev Honeycomb Anyons
Exactly solvable spin liquid on a honeycomb lattice with Z₂ gauge field, Majorana fermions, and non-Abelian anyons.
de Bruijn Sequence
Every binary string of length n appears exactly once in this cyclic sequence — generated by an Eulerian circuit in the de Bruijn graph.
Crumpling Transition
Elastic sheet phase transition from flat to crumpled — energy concentrates in 1D ridges and conical vertices as FvK number grows.
SSH Model Bulk-Edge
Su-Schrieffer-Heeger chain: tune hopping amplitudes to cross topological phase transition and watch protected edge states appear at zero energy.
Contact Process
Absorbing-state phase transition at λ_c ≈ 3.30 in directed percolation universality class — active epidemic vs. extinct absorbing state.
Surface Hopping
Tully's fewest-switches surface hopping: quantum-classical trajectories hop between adiabatic surfaces at avoided crossings via Landau-Zener transitions.
Voter Model Coarsening
Agents copy random neighbors — domains coarsen as L(t) ~ √t and interface density decays toward consensus.
Fisher-KPP Wavefront
Traveling invasion wave in reaction-diffusion: pulled waves select speed c* = 2√(Dr), pushed waves exceed it via nonlinear growth.
Topological Magnons
DMI breaks time-reversal symmetry in honeycomb ferromagnets, giving magnon bands nonzero Chern number and chiral edge currents — the magnon Hall effect.
Secretary Problem
Optimal stopping: the 1/e strategy — skip ⌊N/e⌋ candidates then hire the next best — gives 37% probability of finding the absolute best applicant.
Riemann Surfaces
Branch points and monodromy for √z and log(z): domain coloring reveals how multi-valued functions become single-valued on stacked Riemann sheets.
Landau Damping
Collisionless plasma wave absorption via phase-space resonance: particles near the wave phase velocity exchange energy without any collisions.
Anderson Localization
Disorder-driven metal-insulator transition: random on-site potentials cause quantum wavefunctions to localize exponentially — all states localize in 1D.
Mean Field Games
Nash equilibrium for infinite rational populations: coupled HJB-Fokker-Planck equations yield optimal crowd motion balancing congestion and target attraction.
Fokker-Planck Equation
Kolmogorov forward equation evolves probability densities under drift and diffusion — stationary distribution p∞ ∝ exp(−V/D) is the Boltzmann distribution.
Calogero-Moser System
Exactly integrable N-body system with 1/r² repulsion: Lax pair encodes N conserved charges, particles never cross, and eigenvalues of L are constants of motion.
Feigenbaum Universality
Period-doubling cascade with universal ratio δ = 4.6692… — the same constant for every unimodal map, explained by renormalization group fixed points in function space.
Random Sequential Adsorption
Objects placed randomly jam at Rényi's parking constant M ≈ 0.7477 in 1D and ~54.7% in 2D — far below close-packing, with no reversibility allowed.
Turing Spots
Activator-inhibitor reaction-diffusion instability spontaneously forms spots, stripes, and labyrinths — Turing's 1952 mechanism for biological pattern selection.
2D CFT Correlators
Primary field correlators in 2D conformal field theory: exact two-point and four-point functions constrained by Virasoro symmetry and the operator product expansion.
First Passage Time
Random walk on the half-line: the time to reach the absorbing boundary follows an inverse Gaussian distribution, with drift shifting the peak and tail algebraically.
Ultrametricity & Tree Distance
Hierarchical trees generate ultrametrics where all triangles are isoceles — a structure appearing in p-adic numbers, spin glass landscapes, and single-linkage clustering.
Piecewise Linear Map Chaos
Tent maps and their variants are exactly solvable: Lyapunov exponent λ = log r, uniform invariant measure, and symbolic dynamics are all analytically computable.
Gosper Glider Gun
The first pattern in Conway's Life proven to grow without bound, emitting a new glider every 30 generations — proving unlimited growth and enabling Turing completeness.
Lévy Stable Distributions
Generalized central limit theorem attractors with heavy tails: for stability index α < 2, variance diverges; for α ≤ 1, even the mean fails to exist.
Magnetic Reconnection
Field line topology change at current sheets converts magnetic energy to plasma heat and jets — Sweet-Parker predicts slow rates while Petschek allows fast reconnection via standing shocks.
Ising Energy Landscape
Metastable states, domain walls, and nucleation barriers in the 2D Ising model: a reversed field traps the system until a critical droplet of size r* ~ J/h nucleates the transition.
Adaptive Dynamics & ESS
Trait evolution via invasion fitness and successive trait substitutions: the pairwise invasibility plot reveals evolutionarily stable strategies where the fitness gradient vanishes.
Heterogeneous R₀
Network structure shifts the epidemic threshold: scale-free graphs with diverging ⟨k²⟩ make it vanish entirely, while targeted vaccination of hubs restores herd immunity efficiently.
Quantum Walk: Coin on Lattice
A quantum walker with an internal coin qubit spreads ballistically (σ~t) rather than diffusively (σ~√t), producing a characteristic twin-peaked interference pattern — a purely quantum signature absent in all classical random walks.
Vicsek Flocking Transition
N self-propelled particles align with neighbors under noise η: below a critical threshold the system spontaneously orders into collective flocking, a first-order nonequilibrium phase transition characterized by the order parameter ⟨v⟩/v₀.
KdV Two-Soliton Collision
Two KdV solitons collide elastically and emerge unchanged except for a phase shift — computed exactly via Hirota's tau-function. The exact 2-soliton formula reveals that solitons are indestructible particle-like solutions of the nonlinear wave equation.
Chip-Firing: Toppling Waves
Chip-firing on graphs fires vertices with chips ≥ degree, cascading chips along edges until a fixed point is reached. The Abelian sandpile property guarantees the final stable state is order-independent — the identity element of the sandpile group.
Benjamin–Feir Modulational Instability
The focusing NLS equation makes uniform plane waves unstable to sidebands with |q| < 2A: perturbations grow as cosh(Ωt) with rate Ω(q) = |q|√(2A²−q²/2). This Benjamin–Feir instability is the seed of rogue wave formation and wave turbulence.
Finite-Time Carnot Efficiency
The ideal Carnot engine achieves maximum efficiency η_C = 1−T_c/T_h but zero power; real engines trade off efficiency for speed, producing entropy irreversibly. The Curzon–Ahlborn result η_CA = 1−√(T_c/T_h) gives the efficiency at maximum power output.
Grassberger–Procaccia Correlation Dimension
The GP algorithm estimates the fractal dimension D₂ of a strange attractor by measuring how the correlation integral C(r) ~ r^D₂ scales with radius. The log-log slope reveals the attractor's intrinsic dimensionality — Lorenz ~2.06, Hénon ~1.26.
Chromatic Polynomial & Graph Coloring
The chromatic polynomial P(G,k) counts proper k-colorings of a graph, computed exactly by deletion-contraction recursion P(G,k) = P(G−e,k) − P(G/e,k). The smallest positive integer root gives the chromatic number χ(G).
Weisfeiler–Leman Isomorphism Test
The 1-WL algorithm iteratively recolors vertices by their neighborhood color multisets until stable; two graphs are distinguished if histograms differ. WL is complete for trees but fails on non-isomorphic regular graphs — and modern GNNs are exactly as powerful as 1-WL.
Abel–Ruffini: Why Quintics Have No Formula
Galois proved the quintic x⁵+px+q has no radical solution because its Galois group is S₅, which is not solvable: A₅ is the smallest non-abelian simple group and cannot appear as an abelian quotient in any subnormal tower. Roots are computed numerically via the Aberth–Ehrlich method.
Dissipative Soliton — Complex Ginzburg-Landau
The cubic-quintic CGLE ∂tA=(μ+…)A governs driven-dissipative pattern formation. Dissipative solitons are stable localized structures sustained by gain-loss balance — unlike conservative solitons, they have fixed amplitude and width set by the nonlinear dissipation.
Network Robustness — Targeted vs Random Attack
Scale-free networks are resilient to random failures but collapse rapidly under targeted removal of high-degree hubs. This simulation tracks the giant component fraction S(f) as nodes are removed, revealing the catastrophic vulnerability of preferential-attachment networks to degree-based attacks.
Asymptotic Freedom — QCD Running Coupling αs
The strong coupling αs(Q) decreases logarithmically at high energies (asymptotic freedom) due to the one-loop beta function b₀=(11Nc−2Nf)/2π being positive for QCD. This lab plots the one- and two-loop running coupling with experimental data points from τ decays to LHC.
Hyperbolic Discounting — Temporal Preference Reversal
Humans discount future rewards hyperbolically D(t)=1/(1+kt) rather than exponentially, causing preference reversals: the curves for two delayed rewards can cross as time passes, flipping which one is preferred. This explains why we procrastinate even when we plan not to.
Tikhonov Regularization — Inverse Problems
Solving ill-conditioned inverse problems without regularization amplifies noise catastrophically. Tikhonov regularization xλ=(AᵀA+λLᵀL)⁻¹Aᵀb trades data fidelity for smoothness, with the optimal λ found at the corner of the L-curve where both residual and solution norm are small.
Wigner Crystal — Electron Lattice Crystallization
At low density (large rs=a/aB), Coulomb repulsion dominates kinetic energy and electrons spontaneously crystallize into a triangular lattice. Molecular dynamics shows the transition from correlated liquid to Wigner crystal as rs exceeds ~37 in 2D, with Madelung energy E≈−1.106/rs.
Geometric Frustration — Triangular Antiferromagnet
On a triangular lattice, antiferromagnetic spins cannot all satisfy their pairwise constraints — geometric frustration prevents long-range order even at T=0. Monte Carlo simulation shows the highly degenerate ground state manifold of the triangular AFM compared to the ordered Néel state on a square lattice.
Energy Landscape — Basin-Hopping Optimization
The energy landscape paradigm describes complex systems via their potential surface with many local minima (inherent structures). Basin-hopping replaces the landscape with a staircase of local minima — step, minimize, accept/reject — finding the global minimum far more efficiently than simple Monte Carlo.
SGD Noise — Flat Minima and Generalization
SGD noise (scale η∝lr/batch_size) drives optimization toward flat minima with wide basins that generalize well, while full-batch GD converges to sharp minima that generalize poorly. This 2D loss landscape visualization shows how noise scale controls exploration and the final basin width.
Signal Detection Theory — Morse Code in Noise
SDT separates perceptual sensitivity d′=z(HR)−z(FAR) from response bias using overlapping Gaussian distributions for signal+noise vs noise alone. The ROC curve traces all operating points as the decision criterion shifts, with Morse code as the signal model and tunable SNR and threshold.
Sznajd Model — Pair-Driven Social Influence
Two neighbours sharing an opinion convince their neighbours to adopt it. This "united we stand" rule drives rapid consensus on lattices and networks, exhibiting sharp transitions between ordered and disordered phases depending on initial opinion density.
Fermi Pressure — Pauli Exclusion Principle
At T=0, electrons fill energy levels up to the Fermi energy — Pauli's exclusion principle prevents two fermions from sharing a state. This degeneracy pressure, visualised through Fermi-Dirac statistics and level occupation, supports white dwarf stars against gravitational collapse.
Rotating Frame — Coriolis & Centrifugal Forces
In a rotating reference frame, freely-moving objects appear to curve due to fictitious Coriolis (−2Ω×v) and centrifugal (Ω²r) forces. Compare inertial vs rotating trajectories side-by-side — the same straight path becomes a spiral, explaining Foucault pendulums and cyclone rotation.
Mechanical Synchronization — Huygens' Clocks
Christiaan Huygens (1665) noticed pendulum clocks on a common beam synchronize through tiny vibrations. Simulate coupled metronomes on a free platform: watch the Kuramoto order parameter rise from disorder to phase-locked consensus as coupling and initial spread are varied.
Grand Canonical Ensemble — Chemical Potential
The grand canonical ensemble allows particle number to fluctuate; the chemical potential μ controls ⟨N⟩ through the fugacity z=e^(μ/kT). Energy level populations follow Boltzmann weights and the particle-number distribution is Poissonian with variance equal to the mean.
Membrane Fluctuations — Helfrich Bending Energy
The Helfrich Hamiltonian H=∫(κ/2)(2H)²dA governs lipid bilayer elasticity. Thermal modes are excited with amplitude ⟨|h_q|²⟩=kT/(κq⁴+σq²) — soft membranes undulate wildly at long wavelengths. Adjust bending rigidity κ and surface tension σ to explore the fluctuation spectrum.
Euler Characteristic — V−E+F=2
For any convex polyhedron, vertices minus edges plus faces always equals 2 — a topological invariant χ=2−2g where g is the genus. Spin through the Platonic solids and a genus-1 torus, verifying χ=2 for spherical topology and χ=0 for the torus.
Aharonov-Bohm Effect — Gauge Invariance
An electron traveling two paths around a solenoid acquires a phase difference Δφ=eΦ/ℏ even though B=0 along both paths. The flux-tunable interference pattern demonstrates that the vector potential A is physically real — a purely quantum phenomenon with no classical analogue.
Two-Temperature Model — Ultrafast Laser Ablation
Femtosecond laser pulses heat electrons to kilokelvin temperatures while the lattice stays cold. The TTM couples them through electron-phonon coupling G: C_e dT_e/dt = −G(T_e−T_l)+S(t). Tune pulse energy, width, and coupling to watch the non-equilibrium dynamics relax.
Quantum Zeno Effect — Measurement Freezes Decay
Frequent measurement of an unstable quantum state slows its decay: P(τ)≈1−(Ωτ)² for short intervals, so N measurements in time T leave survival probability cos²(ΩT/N)^N→1 as N→∞. The anti-Zeno effect shows how measurement rate can also accelerate decay at resonance.
Arnold Tongues
Frequency-locking regions in parameter space via the circle map — Devil's staircase, rational winding numbers, and the boundary between quasiperiodic and locked motion.
Phase Locking
A driven nonlinear oscillator snaps into synchrony when detuning falls within the lock-in range. Phase portrait and Adler equation reveal the pull into resonance.
Topological Defect Coarsening
Vortex and anti-vortex pairs annihilate in the 2D XY model after a quench. Domain coarsening follows the universal power law L(t) ~ t^(1/2).
Kibble-Zurek Mechanism
Rapid phase transitions leave behind topological defects because the correlation length cannot diverge fast enough. Defect density scales as a power of the quench rate.
Nematic-Isotropic Transition
Landau free energy with a cubic term drives a discontinuous jump in liquid crystal order parameter S at the nematic-isotropic transition — a prototypical first-order transition.
Duffing Oscillator
Cubic nonlinearity tilts the resonance curve, creating bistability and a jump phenomenon: amplitude leaps discontinuously as drive frequency sweeps through the bent peak.
Smoluchowski Coagulation
Cluster-size distributions evolve under coagulation kernels. The multiplicative kernel K=ij triggers a gelation transition at t=1 where an infinite cluster forms suddenly.
Random Energy Model
Derrida's REM: 2^N random energies, exact free energy via the replica trick. Below Tg the system freezes into exponentially few states — entropy crisis and glass transition.
Microswimmer Squirmer
The squirmer model distinguishes pushers (bacteria) from pullers (algae) by the sign of β = B₂/B₁. Flow field arrows reveal the force-dipole structure of each swimmer type.
Geometric Brownian Motion
Log-normal price paths under GBM, Black-Scholes call pricing, and the terminal distribution. Adjust drift, volatility, and strike to see how option value and delta respond.
Worm-Like Chain Model
Semiflexible polymer physics: tune persistence length and contour length to explore the crossover from rigid rod to flexible coil. End-to-end distance follows the exact WLC formula with thermal fluctuations.
Renormalization Group Flow
RG flow diagrams for φ⁴ theory, 2D Ising, and Kosterlitz-Thouless: fixed points, relevant vs irrelevant operators, universality basins, and streamlines in coupling space.
Neutrino Oscillations
PMNS mixing and two-flavor oscillation probability P(νₑ→νₑ) vs. distance and energy. Adjust Δm², mixing angle θ, and detector position to match reactor and atmospheric neutrino experiments.
Jaynes-Cummings Model
Two-level atom in a cavity: vacuum Rabi oscillations, dressed-state splitting, and the effect of detuning and decay. The energy level diagram updates live as you change coupling and photon number.
Purcell Effect
Cavity-enhanced spontaneous emission: tune Q-factor and mode volume to control the Purcell factor. Compare enhanced vs free-space decay curves and observe the transition from weak to strong coupling.
Quantum Discord
Quantum correlations beyond entanglement: discord D = I(A:B) − J(A|B) is nonzero even for separable mixed states. Explore how mutual information, classical correlations, and discord partition quantum information.
Tropical Geometry
Min-plus algebra replaces classical arithmetic: tropical polynomials are piecewise-linear, their zeros are kinks, and tropical curves in 2D are polyhedral fans. Adjust coefficients to see the structure.
ZND Detonation Model
Zel'dovich-von Neumann-Döring detonation structure: von Neumann spike, reaction zone, and Chapman-Jouguet condition. Switch between spatial profiles, Hugoniot diagrams, and D_CJ vs. heat release.
Cosmic String Formation
Kibble mechanism in 2D: a complex scalar field acquires random phase domains after a phase transition. Topological defects form where winding number is ±1; watch them annihilate under relaxation.
Wilsonian EFT
Wilson's renormalization group: running couplings, operator dimension hierarchy, and EFT flow diagrams. Relevant operators grow at low energy; irrelevant operators are suppressed by powers of E/Λ.
Minority Game
Agents repeatedly choose between two scarce resources; the minority group wins. Strategy tables and memory length drive emergent coordination — watch volatility and efficiency as you tune the ratio α = N/2^m.
Quantal Response Equilibrium
Boundedly rational agents choose actions proportional to exp(λ·payoff). At λ→0 choices are random; at λ→∞ QRE converges to Nash. Explore the logistic best-response curves and the QRE locus for a coordination game.
Bianconi-Barabási Fitness Model
New nodes attach preferentially to high-fitness, high-degree nodes. Above a critical fitness heterogeneity, Bose-Einstein condensation occurs — one node captures a finite fraction of all links and dominates the network.
Diffusion Fundamental Solution
The heat equation's Green's function is a spreading Gaussian G(x,t) = (4πDt)^{−½} exp(−x²/4Dt). Width grows as √t, peak falls as t^{−½}, and total probability is conserved — visualize superposition of multiple point sources.
Quantum Harmonic Chain — Phonons
A 1D chain of quantum oscillators supports phonon modes with dispersion ω(k) = 2ω₀|sin(ka/2)|. Each mode carries zero-point energy ½ℏω, causing vacuum fluctuations visible even at T=0. Add temperature to populate Bose-Einstein modes.
Galton-Watson Branching Process
Each individual produces a random number of offspring; extinction probability satisfies q = G(q) where G is the PGF. The critical point μ=1 separates certain extinction from possible survival — watch the phase transition in real simulations.
Exceptional Points — Non-Hermitian
In a gain-loss PT-symmetric dimer, eigenvalues coalesce at an exceptional point where eigenvectors merge. The splitting scales as √|ε−ε_EP|, giving extreme parameter sensitivity exploited in EP-enhanced sensors.
Non-Hermitian Skin Effect
The Hatano-Nelson model has asymmetric hopping t±g. Under open boundaries ALL eigenstates pile up at one edge — the skin effect. The OBC spectrum collapses to a real segment while PBC gives a complex ellipse.
Saddle-Point Approximation
Integrals ∫e^{N·f(z)}dz are dominated by the saddle f'(z*)=0; expanding to Gaussian order yields the steepest-descent approximation. Applied to n!, it gives Stirling's formula with relative error ~1/N.
Polymer Translocation — Pore Ratchet
A polymer threads through a nanopore driven by an electric field and thermal noise on the entropic free-energy landscape F(s). The ratchet effect gives directed transport; translocation time scales as τ ~ N^{1+ν} with Flory exponent ν≈0.588.
KPZ Interface Growth
Kardar-Parisi-Zhang stochastic interface growth: ∂h/∂t = ν∇²h + (λ/2)(∇h)² + η. The nonlinear λ term drives the KPZ universality class with roughness exponent β≈1/3. Watch the interface roughen and measure the scaling in real time.
Debye Screening in Electrolyte
Mobile ions screen a central charge: φ(r) = (q/4πε)·e^(−r/λ_D)/r. The Debye length λ_D shrinks with ionic concentration and grows with temperature. Tune parameters to see how the screening cloud tightens or expands.
Casimir Effect: Vacuum Fluctuations
Two parallel conducting plates suppress vacuum electromagnetic modes between them, creating an attractive force F/A = −π²ℏc/(240d⁴). Drag the plate separation to see the d⁻⁴ force law and mode counting visualization.
Cusp Catastrophe — Normal Form
The cusp catastrophe V(x)=x⁴/4+ax²/2+bx has equilibria satisfying x³+ax+b=0. Click the (a,b) control plane to move through the cusp bifurcation set and watch the potential well suddenly jump between stable states.
Quantum Zeno Effect
A watched quantum never decays: frequent measurements freeze a two-level system. Compare unmonitored P(t)=cos²(Ωt/2) decay against the Zeno-slowed survival under repeated projections at interval τ.
Poincaré Return Map
Stroboscopic snapshots of a driven damped pendulum θ''+γθ'+sin(θ)=A cos(ωt) reveal period-1 orbits, period-doubling cascades, and chaotic strange attractors as driving amplitude A increases.
Jaynes-Cummings Cavity QED
A two-level atom in a cavity undergoes vacuum Rabi oscillations at Ω_n=g√(n+1). Tune coupling g and photon number n; for a coherent state input, quantum revivals emerge from the discrete photon spectrum.
Quantum Zeno Subspace
Repeated projections onto a quantum subspace confine a qubit's evolution. Watch the Bloch sphere trajectory freeze toward the projection axis as measurement interval τ shrinks — the generalized Zeno effect.
Ginzburg-Landau Vortex
A single vortex in a type-II superconductor: order parameter ψ(r)≈tanh(r/ξ)e^(iθ) vanishes at the core, phase winds by 2π. Tune GL parameter κ=λ/ξ to distinguish type-I and type-II regimes.
Riemann ζ Zeros & Random Matrices
Montgomery's conjecture: the spacing statistics of Riemann ζ non-trivial zeros match GUE random matrix eigenvalue statistics. Plot the nearest-neighbor spacing histogram for 50 known zeros and compare to the Wigner surmise.
Abelian Sandpile SOC
Grains topple when height ≥ 4, redistributing to neighbors. The system self-organizes to criticality with power-law avalanche size distributions—click the grid to trigger cascades.
Mollow Triplet
Resonance fluorescence of a strongly driven two-level atom: the emission spectrum splits into three peaks separated by the generalized Rabi frequency Ω_gen = √(Ω_R² + Δ²).
Bogoliubov-de Gennes Quasiparticles
BCS superconductor quasiparticle dispersion E_k = √(ξ_k² + Δ²) with coherence factors u_k², v_k²—tune the gap Δ and chemical potential μ across the BEC-BCS crossover.
SSH Winding Number
Su-Schrieffer-Heeger model: when intercell hopping t₂ > t₁, the winding number jumps to W=1 and protected zero-energy edge states appear in the bulk gap.
Kibble-Zurek Defects
Quench a 2D XY model through its critical temperature—causally disconnected domains freeze in random phases, generating vortex-antivortex defects whose density obeys a power law in quench rate.
Lissajous & Parametric Resonance
Lissajous figures from rational frequency ratios x=A·sin(ω₁t+φ), y=B·sin(ω₂t), alongside the Mathieu equation stability diagram showing parametric resonance tongues.
May's Ecological Chaos
The logistic map x_{n+1} = r·x_n(1−x_n) as a population model: click the bifurcation diagram to watch period-doubling cascade into chaos with Feigenbaum constant δ ≈ 4.669.
Majorana Zero Modes
Kitaev chain model for topological superconductivity: zero-energy Majorana modes appear at chain ends when |μ| < 2t, exponentially localized and protected by topology.
Sznajd Opinion Dynamics
Agreeing adjacent agents convince their neighbors—"united we stand, divided we fall." The 1D ring and 2D grid always reach full consensus; compare to voter model dynamics.
RSA Jamming
Irreversible random deposition of circles approaches the Rényi parking constant θ* ≈ 0.547—far below close-packing. Watch coverage saturate and compare circles, dimers, and squares.
Dissipative Soliton — Complex Ginzburg-Landau
Stable localized structures in the complex Ginzburg-Landau equation—tune dispersion and nonlinear phase to explore soliton breathing, stability, and Benjamin-Feir turbulence.
Fluctuation Theorem & Jarzynski Equality
Simulate forward and reverse work distributions for a dragged particle and verify P_F(W)/P_R(−W) = exp(W/kT)—the detailed fluctuation theorem with Jarzynski free energy recovery.
Mode-Coupling Theory — Glass Transition
Watch the intermediate scattering function F(q,t) develop two-step relaxation as temperature approaches the MCT critical point T_c, with diverging α-relaxation time τ ~ (T−T_c)^{−γ}.
Ising Critical Clusters — Wolff Algorithm
Fractal Ising clusters at T_c = 2.269 simulated with the Wolff algorithm; cluster size distribution follows P(s) ~ s^{−τ} with τ = 187/91 ≈ 2.055 from conformal field theory.
Photon Blockade — Kerr Cavity
Anharmonic energy ladder E_n = nω_c + n(n−1)U/2 blocks two-photon absorption; tune the Kerr nonlinearity U/κ to control g²(0) from coherent to strongly sub-Poissonian light.
Frustrated Magnetism — Kagome Lattice
Corner-sharing triangles make every antiferromagnetic triangle unsatisfiable—Monte Carlo reveals the disordered spin-liquid ground state with no long-range order down to T → 0.
Quantum Hall Edge States
Landau levels bend at the sample boundary, crossing the Fermi level to create ν chiral edge channels immune to backscattering—each filled level contributes e²/h to σ_xy.
Adaptive Dynamics — Evolutionary Branching
Traits evolve up the fitness gradient until reaching a branching point where a single population splits into two coexisting species—controlled by the competition-to-resource width ratio σ_c/σ_r.
Floquet-Bloch Bands — Driven Lattice
Periodic driving folds static bands into the Floquet-Brillouin zone, renormalizing hopping via J_eff = t₀J₀(V/ω)—at Bessel zeros the band flattens and topological transitions appear.
Brazil Nut Effect — Granular Segregation
Large heavy particles rise to the top under vertical vibration—simulate bidisperse granular media with convection-driven segregation and tune the dimensionless shaking strength Γ = Aω²/g.
Dynamical Decoupling — Pulse Sequences
CPMG π-pulse sequences extend qubit coherence T₂ by filtering 1/f dephasing noise — watch Bloch vector decay slow as pulse count N rises.
Quantum Discord — Beyond Entanglement
Discord D = I(A:B) − J(A:B) captures quantum correlations in separable states — nonzero even when concurrence vanishes at p < 1/3.
Spin-Transfer Torque — Magnetic Switching
The Landau–Lifshitz–Gilbert–Slonczewski equation drives magnetization reversal when current exceeds the critical threshold I_c — observe switching on the Bloch sphere.
Lévy Stable Distributions & Flights
Stable distributions with heavy tails (P(x) ~ x^{−α}) generate superdiffusive Lévy flights with ⟨r²⟩ ~ t^{2/α} — tune stability index from Gaussian (α=2) to Cauchy (α=1).
Langton's Self-Replicating Loop
An 8-state cellular automaton that copies itself — Langton's 1984 loop detects its own boundary and constructs offspring, the first artificial self-replicating machine.
Magnetic Skyrmion — Topological Soliton
DMI-stabilized spin texture with topological charge Q = ±1 — watch the skyrmion drift at the Hall angle ~40° under applied current, protected against collapse by topology.
Chimera State — Partial Synchrony
In symmetric coupled Kuramoto populations one group locks while the other remains incoherent — the chimera paradox discovered by Kuramoto & Battogtokh in 2002.
Non-Hermitian Skin Effect
Asymmetric hopping t_R ≠ t_L collapses all bulk eigenstates to one edge under OBC, while the PBC spectrum traces a loop in the complex plane — a topological origin.
Graphene Phonon Dispersion
Six phonon branches from graphene's 2-atom unit cell — the unique quadratic ZA flexural branch and Dirac-like linear acoustic cones at the K point.
Photonic Crystal Band Gap
Transfer matrix method for 1D dielectric stacks reveals photonic band gaps — quarter-wave condition n₁d₁ = n₂d₂ = λ/4 maximizes the gap width.
Thermodynamic Uncertainty Relation
TUR bound Q=Var(J)·⟨Σ⟩/⟨J⟩²≥2 for a driven 3-state Markov cycle — tune rates and watch current fluctuations saturate the bound.
Klein-Gordon Soliton Kink
φ⁴ field theory kink solitons and kink-antikink collisions — watch breather formation and energy density waves in the nonlinear wave equation.
Hall Viscosity — Topological Fluid
Odd (Hall) viscosity rotates stress 90° from strain rate — animate the parity-breaking flow patterns of fractional quantum Hall fluids.
Kramers Escape Rate
Brownian particle escaping a double-well: Kramers formula k=(ω_aω_b/2πγ)e^{-ΔU/kT} with Langevin simulation showing the underdamped/overdamped turnover.
Growing Network Fitness Model
Bianconi-Barabási fitness model: broad fitness distributions cause Bose-Einstein condensation where one node captures a finite fraction of all links.
Chern-Simons Theory & Anyons
2+1D anyons braid and accumulate topological phase e^{iθ} — visualize worldline winding and compare bosons, fermions, and intermediate anyonic statistics.
Topological Band Theory — Berry Curvature
Berry curvature heatmap over the Brillouin zone for a 2-band model; tune the mass parameter m through topological transitions and watch the Chern number jump.
Crystal Surface Roughening Transition
SOS model Monte Carlo: below T_R the surface stays smooth and faceted; above it, interface width W diverges logarithmically — a BKT-type transition.
Aging & Glassy Dynamics — Trap Model
Bouchaud's trap model: for T<T_g the mean waiting time diverges, ⟨E⟩(t) never equilibrates, and the system ages forever with power-law waiting times.
Quantum Gravity — Spin Foam
Spin foam model of LQG: a tetrahedron with face spins j giving discrete area eigenvalues A=8πγℓ_P²√(j(j+1)) and Ponzano-Regge amplitude from {6j}-symbols.
Feigenbaum Universality
Period-doubling cascade and self-similar bifurcation structure revealing the universal constant δ≈4.669.
Kitaev Chain Topology
1D topological superconductor with Majorana zero modes emerging at chain ends in the topological phase.
Soft Mode Transition
Phonon soft mode whose frequency vanishes at the critical temperature, driving structural phase transition.
Stochastic Resonance
Optimal noise enhances signal detection — SNR peaks at the sweet spot between too little and too much noise.
Flory Polymer Scaling
Self-avoiding random walks as polymer models, with Flory exponent ν≈0.75 governing chain expansion.
Snap-Through Instability
Bistable elastic arch snapping between stable states — the mechanics of buckling and elastic instability.
Basset Memory Force
History-dependent hydrodynamic force giving algebraic t⁻¹/² velocity decay, contrasting Stokes drag.
Entropy Production
Non-equilibrium dissipation rate as a colloidal particle is driven far from thermal equilibrium.
Modulation Instability
Benjamin-Feir instability breaks uniform wave trains into envelope solitons via NLS dynamics.
Quantum Zero-Point Motion
Ground-state probability density spills beyond classical turning points — quantum fluctuations at absolute zero.
Kink-Antikink Collision
Topological solitons in φ⁴ field theory colliding and annihilating with localized energy bursts.
Maxwell's Demon
An information-powered demon sorts particles, extracting work at the cost of kT ln2 per bit of memory.
Critical Opalescence
Near-critical density fluctuations span all length scales, scattering light and turning fluids milky.
Umklapp Scattering
Phonon momentum folded back by a reciprocal lattice vector — the key to thermal resistance in crystals.
Absorbing State Transition
Directed percolation phase transition between active spreading and an absorbing dead state.
Rayleigh-Plateau Instability
Surface tension breaks a fluid jet into droplets — fastest growth at wavelength 9x the jet radius.
Ostwald Ripening
Large droplets grow at the expense of small ones via surface diffusion — coarsening follows R^3 ~ t.
Lichtenberg Discharge
Fractal dielectric breakdown patterns grown via DLA on a Laplace potential field — nature's lightning art.
Frustrated Magnetism
Antiferromagnetic spins on a triangular lattice — geometric frustration prevents any fully ordered ground state.
Parametric Squeezing
Time-varying spring constant squeezes quantum noise below vacuum level in one quadrature while amplifying the other.
Heteroclinic Orbits
Trajectories connecting saddle points — slow intermittent switching and Milnor attractor dynamics.
Quantum Walk
Ballistic spreading and interference fringes in a discrete quantum walk — far faster than classical diffusion.
Van Hove Singularities
Logarithmic and square-root divergences in the phonon density of states mark critical points in the dispersion.
DLA Fractal Growth
Random-walking particles aggregate into branching fractal clusters with dimension D_f ≈ 1.71.
Adiabatic Elimination
Fast variables slaved to slow ones — when timescale separation justifies dimensional reduction.
SSH Topological Edge States
Bulk-boundary correspondence in the SSH chain — zero-energy edge modes protected by chiral symmetry.
Network Epidemic Threshold
Heterogeneous mean-field theory shows scale-free networks have vanishing epidemic threshold — any pathogen spreads.
Z₂ Gauge Confinement
Confinement-deconfinement transition in Z₂ lattice gauge theory — area law vs perimeter law Wilson loops.
Manna Sandpile
Stochastic self-organized criticality with power-law avalanche distributions in the Manna universality class.
Quorum Sensing Switch
Autoinducer positive feedback creates a bistable switch — bacteria collectively flip between silent and active states.
Ginzburg-Landau Spirals
Rotating spiral waves and phase turbulence in the complex Ginzburg-Landau equation.
KPZ Interface Growth
The KPZ universality class governs rough interface growth — roughness scales as t^{1/3}, not t^{1/4}.
HP Protein Folding
Lattice model protein folding — hydrophobic collapse drives compact native structures via H-H contact minimization.
Jaynes-Cummings Model
Vacuum Rabi splitting and collapse-revival of Rabi oscillations in cavity QED with a coherent photon field.
Continuum Percolation
Random disk packings forming a spanning cluster — the Boolean model percolation threshold in continuous space.
Kelvin-Helmholtz Instability
Shear flow vortex sheet rollup into cat-eye patterns — the instability behind clouds, ocean waves, and Jupiter's bands.
Lévy Flight Diffusion
Power-law jump distributions give superdiffusive MSD ~ t^{2/α} — anomalous transport in disordered media.
Non-Abelian Berry Phase
Degenerate adiabatic loops produce non-commuting holonomy matrices — the geometry behind topological quantum gates.
BEC Solitons (GPE)
Dark and bright matter-wave solitons in a Bose-Einstein condensate governed by the Gross-Pitaevskii equation.
Hidden Markov Entropy Rate
The entropy rate of a hidden Markov model is strictly less than the observed output entropy — information hidden in state.
Barkhausen Avalanches
Domain wall avalanches in a driven random-field Ising magnet produce power-law Barkhausen noise statistics.
Rayleigh-Bénard Convection
Convection rolls emerge when the Rayleigh number Ra exceeds ~1708, solved via stream-function vorticity dynamics with stream function Poisson equation.
Spectral Zeta Function
The spectral zeta ζ_A(s) = Σλ_n^{-s} encodes eigenvalue geometry; analytic continuation to s=-½ yields vacuum zero-point energy across different domain shapes.
Casimir Effect: Parallel Plates
Quantum vacuum fluctuations produce an attractive force P = -ℏcπ²/(240a⁴) between uncharged conducting plates, visualized with quantized interior modes.
Orr-Sommerfeld Stability
The neutral stability curve for plane Poiseuille flow shows instability above Re_c ≈ 5772 at wavenumber α ≈ 1.02, with Tollmien-Schlichting wave eigenmodes.
Smoluchowski Coagulation
Cluster-cluster aggregation via the Smoluchowski equation with constant, additive, and product kernels — the product kernel exhibits finite-time gelation.
Chaotic Scattering
The deflection function Θ(b) of particles scattering off multiple hard disks develops a fractal Cantor-set singularity structure — a hallmark of chaotic scattering.
Epidemic Metapopulation
SIR dynamics on a network of coupled patches with travel coupling — hub networks sustain epidemics below the local threshold via network-level amplification.
Bethe Lattice Phonons
The exact phonon density of states on a Bethe lattice (Cayley tree) computed via the cavity method, with animated vibrational modes and Van Hove singularities.
Driven Lattice Gas (KLS)
The Katz-Lebowitz-Spohn model: an external drive breaks detailed balance in an Ising lattice gas, producing anisotropic nonequilibrium ordered phases.
TASEP Exclusion Process
The totally asymmetric simple exclusion process has an exact Bethe ansatz solution with three phases — LD, HD, MC — meeting at α=β=½ with J_max=¼.
KPZ Interface Growth
Stochastic surface growth via the Kardar-Parisi-Zhang equation; roughness scales as W~t^(1/3) with tunable noise, diffusion, and nonlinearity.
Duffing Oscillator & Chaos
Double-well anharmonic oscillator with periodic forcing; explore period-doubling cascades and chaotic attractors via phase portrait and Poincaré sections.
Rule 184 Traffic Flow
Wolfram's Rule 184 as an exact cellular automaton model of traffic; a phase transition at density 0.5 separates free-flow from congested regimes with shockwaves.
Free Energy Landscape
Brownian particles in a double-well potential; Kramers escape rate governs metastability as temperature and well asymmetry are tuned.
Frustrated Triangular Ising
Antiferromagnetic Ising model on a triangular lattice; geometric frustration yields extensive ground-state entropy and no long-range order at any T>0.
Heisenberg XXZ Spin Chain
Anisotropic spin-1/2 chain interpolating between XY, isotropic Heisenberg, and Ising limits; visualize spin correlations and phase regimes.
Fiber Bundle Fracture
Equal-load-sharing fiber bundle with Weibull disorder; catastrophic failure at critical load with power-law avalanche statistics near the critical point.
Topological Magnon Bands
Honeycomb ferromagnet magnon band structure; DM interaction opens a topological gap with nonzero Chern number, producing chiral edge magnon modes.
Superfluid-Mott Transition
Quantum rotor model on a 2D lattice; competition between hopping J and on-site repulsion U drives the superfluid-to-Mott insulator quantum phase transition.
Colloidal Crystal Phonons
2D hexagonal colloidal crystal with Yukawa interactions; acoustic and optical phonon modes in real space with live dispersion relation.
Wave Turbulence & Weak Kinetics
Weakly nonlinear waves redistribute energy via resonant four-wave interactions, forming the Kolmogorov-Zakharov spectrum E(k) ~ k⁻⁷/³ as energy cascades to high wavenumbers.
Magnetic Reconnection
Oppositely directed magnetic field lines meet at a Harris current sheet, reconnect, and release stored energy as plasma jets — visualized via Sweet-Parker MHD.
Stochastic Block Model
Nodes belong to latent blocks with intra-block probability p_in and inter-block p_out; the Kesten-Stigum threshold separates detectable from undetectable community structure.
Gray-Scott Spots & Stripes
Two reacting chemical species with autocatalysis (U+2V→3V) and decay (V→P) spontaneously self-organize into spots, stripes, or labyrinthine patterns depending on feed and kill rates.
Deconfined Quantum Criticality
At the Néel-to-VBS transition in 2D quantum magnets, fractionalized spinons deconfine and monopoles in the emergent U(1) gauge field drive the phase change.
Explosive Percolation (Achlioptas)
The product rule offers two random edges and selects the one minimizing merged component product, suppressing giant-cluster growth until an abrupt explosive transition.
Shear Band Formation
Strain-softening materials under shear develop localized deformation bands; Johnson-Cook thermal softening drives adiabatic runaway as heat reduces local flow stress.
Fick Transport & Non-Equilibrium
Fick's second law ∂c/∂t = D∇²c governs concentration spreading; maintained gradients produce non-equilibrium steady states with entropy production σ = D(∇c)²/c.
Lévy Ratchet — Asymmetric Drift
Particles performing Lévy flights (α-stable heavy-tailed jumps) in a sawtooth potential exhibit net directed drift — the asymmetry rectifies long jumps without external bias.
Quantum Brownian Motion (Caldeira-Leggett)
The Wigner function W(x,p,t) of a quantum oscillator coupled to a bosonic bath evolves under dissipation and quantum diffusion; cat-state interference fringes decohere rapidly.
SOC Forest Fire
Drossel–Schwabl forest-fire model self-organizes to criticality with power-law fire avalanches — no tuning required.
Point Vortex Gas
Exact Hamiltonian dynamics of 2D Euler point vortices; conserved energy, momentum, and angular momentum with Onsager negative-temperature clustering.
Nonlinear Schrödinger Solitons
Split-step Fourier simulation of the NLS equation — bright solitons, modulational instability, and rogue wave formation.
Wigner–Dyson Level Statistics
Random matrix theory predicts universal GOE/GUE level-spacing distributions; compare empirical histograms to Wigner surmise p(s) = (π/2)s exp(−πs²/4).
Classical Nucleation Theory
Visualize the free energy barrier ΔG(r) = 4πr²γ − (4/3)πr³|Δμ|/v_m and stochastic droplet dynamics near the critical radius r*.
Motility-Induced Phase Separation
Self-propelled particles spontaneously phase-separate at high Péclet number — purely kinetic MIPS with no attractive interactions.
Vicsek Flocking (Topological)
Metric-free Vicsek model uses k nearest neighbors; watch the flocking order parameter φ collapse as noise exceeds the critical threshold.
Hydrogen Orbitals (DFT)
Exact hydrogen wavefunctions ψ_nlm via associated Laguerre and Legendre polynomials — visualize electron density cross-sections for any (n,l,m).
Kosterlitz–Thouless Transition
2D XY model Monte Carlo shows vortex-antivortex pairs binding below T_KT and unbinding above — the Nobel Prize topological phase transition.
Quantum Chaos — Kicked Rotor
Chirikov standard map Poincaré sections reveal KAM tori dissolving at K_c ≈ 0.9716; quantum mode shows dynamical localization in momentum space.
Random Field Ising Model
Quenched disorder hᵢ~N(0,Δ²) competes with ferromagnetic coupling, fragmenting domains. Below the critical disorder, long-range order survives in d≥3 — above it, the ordered phase is destroyed.
RFIM Avalanche Dynamics
T=0 RFIM driven by a slow external field exhibits power-law avalanches P(s)~s⁻¹·⁶⁶ near critical disorder Δc≈2.21 — self-organized criticality in a magnetic system.
Glass Transition: Adam-Gibbs Theory
Relaxation time τ = τ₀ exp(C/T·Sc) diverges as configurational entropy Sc→0 at the Kauzmann temperature — the entropy crisis underlying vitrification.
Angell Fragility Plot
Angell's log η vs Tg/T diagram separates "strong" glass formers (Arrhenius, SiO₂ m≈20) from "fragile" ones (super-Arrhenius, o-terphenyl m≈81) via the steepness index m.
Stochastic Lotka-Volterra
Multiplicative demographic noise added to predator-prey ODEs can drive extinction that the deterministic system forbids — noise breaks the conserved quantity of the classic equations.
Quantum Walk with Decoherence
A Hadamard-coin quantum walker spreads ballistically (σ~t); introducing coin decoherence probability p smoothly crossovers to classical diffusion (σ~√t).
Colloidal Gel: Arrested Phase Separation
Short-range attractive colloids undergo spinodal decomposition that freezes when the dense phase vitrifies, creating a space-spanning gel network — a non-equilibrium solid.
Spin-Torque Switching (Stochastic LLG)
The stochastic Landau-Lifshitz-Gilbert equation with spin-transfer torque and thermal fluctuations governs probabilistic magnetization reversal in STT-MRAM free layers.
Leidenfrost Drop Levitation
Above ~200°C a vapor film forms under a water droplet, levitating it frictionlessly. Lifetime scales as R₀^(8/5); tilt propels the drop as a vapor-cushioned droplet rocket.
Swift-Hohenberg Pattern Selection
∂ₜu = [ε−(1+∇²)²]u − u³ selects patterns at wavenumber q_c=1. Above threshold ε>0, stripes and labyrinthine morphologies emerge from the Turing-like instability.
Rayleigh-Bénard Convection Rolls
Fluid heated from below forms organized convection rolls above Ra~1708. Particle tracers and temperature field reveal roll patterns that transition to chaos at high Rayleigh numbers.
Stochastic Resonance
Adding optimal noise to a subthreshold signal enhances its detection. Watch noise-assisted threshold crossings synchronize with the hidden signal — a paradox exploited by neurons and sensory systems.
Abelian Sandpile Fractals
Drop sand grains on a grid; cells with ≥4 grains topple to neighbors. The boundary of a million-grain pile reveals breathtaking fractal geometry — a hallmark of self-organized criticality.
Kibble-Zurek Mechanism
Quenching an XY-model through its phase transition creates topological vortex defects. Causally disconnected domains freeze into mismatched phases — a mechanism from cosmology to superfluids.
Anomalous Diffusion & fBm
Fractional Brownian motion with Hurst exponent H controls MSD ~ t²ᴴ. Tune H to switch between subdiffusion (crowded cells) and superdiffusion (active matter) and watch the MSD log-log slope shift.
Ecological Food Web
Trophic network with Lotka-Volterra population dynamics. Explore how connectance and species count govern stability via the May-Wigner threshold SC½ — and watch cascades when apex predators vanish.
Nonlinear Wave & Shock Formation
Burgers equation simulation: watch smooth initial conditions steepen into shock discontinuities. Explore how viscosity controls shock thickness and see characteristics converge at the shock front.
Magnetic Domain Wall Motion
Landau–Lifshitz–Gilbert dynamics of a Bloch wall under applied field. See the Walker breakdown: above a critical field the wall precesses and time-averaged velocity actually decreases.
Information Cascade in Networks
Granovetter threshold model on a Watts–Strogatz network. Set adoption thresholds and watch how small seeds trigger global cascades — or die out — depending on network density and structure.
Marangoni Flow
Surface tension gradients drive convective flow — temperature or concentration differences create Marangoni circulation.
Wegner Flow / SRG
Similarity Renormalization Group: flow equation drives Hamiltonians toward band-diagonal form, decoupling energy scales.
Bloch Oscillations
Electron in periodic lattice + DC field undergoes Bloch oscillations; Wannier-Stark ladder of discrete energy levels.
Saffman-Taylor Finger
Surface tension selects a unique finger width λ=1/2 when air invades oil in a Hele-Shaw cell.
Toda Lattice Solitons
Exact soliton solutions of the integrable Toda lattice — two solitons collide and pass through with only a phase shift.
Kosterlitz-Thouless Transition
Topological phase transition: vortex-antivortex pairs bind below T_KT and unbind above it, destroying quasi-long-range order.
Falicov-Kimball Model
Heavy localized and light itinerant electrons interact — charge-density wave ordering as a function of U and temperature.
Poisson-Voronoi Statistics
Random Poisson points generate Voronoi cells with universal statistics: mean 6 sides, Gamma-distributed areas.
Smoluchowski Coagulation
Cluster growth kinetics and gelation: product kernel K=ij causes a giant cluster to form in finite time.
Zhabotinsky Scroll Wave
3D BZ reaction scroll waves rotating around a filament — cross-section simulation with Barkley excitable medium model.
Calogero-Moser Particles
N particles on a line with 1/r² repulsion — an exactly integrable Hamiltonian system with conserved Lax eigenvalues.
Kibble-Zurek Defect Scaling
Topological defects formed during a symmetry-breaking quench: density n ~ τ_Q^(−ν/(1+νz)) — the KZ power law.
Halperin-Nelson Hexatic Melting
2D melting via two KT transitions — solid to hexatic to liquid — visualized through bond-orientational order ψ₆.
Karhunen-Loève Expansion
Optimal eigenmode decomposition of a 2D random field — KL modes capture maximum variance in minimum dimensions.
Benjamini-Schramm Networks
Unimodular random graphs with local convergence — explore the mass-transport principle on random planar graphs.
Gross-Pitaevskii Dark Soliton
Dark soliton — density dip and phase kink — propagating and colliding in a Bose-Einstein condensate.
Eden Growth — KPZ Interface
Eden cluster growth produces a rough interface in the KPZ universality class: width w ~ t^β with β ≈ 1/3.
Zaslavsky Stochastic Web
Hamiltonian chaos map creates an infinite stochastic web in phase space with q-fold symmetry — KAM theory breakdown.
Bogoliubov Squeezing
Bogoliubov transformation mixes creation and annihilation operators — visualize squeezed state Wigner functions.
KPZ Height Fluctuations
KPZ equation ∂h/∂t = ν∇²h + λ(∇h)²/2 + η — Tracy-Widom statistics and β=1/3 roughness exponent.
Frenkel-Kontorova Model
Commensurate-incommensurate transition: atoms on a periodic substrate, Aubry transition, and kink solitons.
SYK Quantum Scrambling
Sachdev-Ye-Kitaev model: all-to-all random couplings, OTOC decay, and Lyapunov exponent λ_L = 2π/β.
Altland-Zirnbauer Tenfold Way
Periodic table of topological phases: ten symmetry classes (T, C, S) and their topological invariants in 1D/2D/3D.
Baxter Hard Hexagon Model
Exact solution via Yang-Baxter equation; Rogers-Ramanujan density curve; Monte Carlo valid hexagon configurations.
Peierls Instability & CDW
1D metal at half-filling: lattice dimerization opens a gap, Kohn anomaly, and charge density wave formation.
Lindblad Decoherence
Open qubit coupled to environment: Bloch sphere trajectory with T₁/T₂ decay, amplitude damping vs pure dephasing.
Regge Calculus: Discrete Gravity
Triangulated 2-sphere: deficit angles concentrate curvature at vertices; Regge action converges to Einstein-Hilbert.
Taylor-Green Vortex
2D Navier-Stokes with exact TG initial condition; energy spectrum developing −5/3 Kolmogorov slope.
Poincaré Section: Hénon-Heiles
KAM tori, resonance islands, and chaotic sea in the Hénon-Heiles Hamiltonian — interactive energy slider.
KAM Tori: Standard Map
Chirikov standard map: KAM tori survive for K < K_c ≈ 0.9716; last golden-mean torus breaks at criticality.
Atiyah–Singer Index: Heat Kernel
Heat kernel proof of the index theorem — spectral zeta function, small-t asymptotics, and Index = dim ker D − dim coker D on a 2D surface.
McKay Correspondence & ADE
Finite subgroups of SU(2) ↔ ADE Dynkin diagrams ↔ affine Lie algebras — animated McKay quiver and resolution of ℂ²/Γ singularity.
Crooks Theorem & Jarzynski Equality
P_F(W)/P_R(−W) = e^{β(W−ΔF)} — forward/reverse work distributions, crossing-point free energy, and Jarzynski exponential average.
Ginzburg–Landau Abrikosov Lattice
Type-II superconductor: GL order parameter |ψ|² and Abrikosov vortex lattice forming above H_c1 — triangular vs square lattice.
Adler–Bardeen Chiral Anomaly
Triangle (AVV) diagram: ∂_μj^μ_5 = e²/16π² · F∧F̃ — routing ambiguity, Chern-Pontryagin density, and exact one-loop non-renormalization.
Efimov Three-Body Universal Scaling
Infinite tower of three-body bound states with geometric ratio E_{n+1}/E_n ≈ 1/515 — log-periodic spectrum and universal hyperspherical wave function.
BKT Transition: Vortex Unbinding
2D superfluid: vortex-antivortex pairs unbind at T_KT; universal jump in superfluid density ρ_s — Nelson-Kosterlitz 1977.
Migdal–Kadanoff Block Spin RG
Real-space renormalization of the Ising model — bond-moving decimation, recursion K' = f(K), fixed points, and interactive RG flow diagram.
Zamolodchikov C-Theorem RG Flow
2D RG: c-function decreasing UV→IR — coupling-space landscape with Ising (c=½), free boson (c=1), tricritical (c=7/10) fixed points and flow arrows.
Araki–Woods Type III Factors
Von Neumann algebras in QFT — Tomita-Takesaki modular flow, KMS condition, Rindler wedge, and the Unruh thermal spectrum.
Shilnikov Chaos & Homoclinic Orbits
3D saddle-focus chaos: when γ/ρ > 1 the homoclinic orbit spawns Smale horseshoes — infinitely many periodic orbits coexisting with chaos.
Sachdev-Ye-Kitaev Model
N Majorana fermions with random all-to-all interactions — maximal quantum chaos, conformal Green functions, and holographic duality to JT gravity.
Topological Order & Ground State Degeneracy
Kitaev toric code on a torus: GSD=4, anyonic excitations, Wilson loops, and topologically protected quantum memory.
Kondo Effect — Spin Screening
Magnetic impurity screened by conduction electrons below T_K — logarithmic resistivity rise and the Kondo cloud visualized in real and energy space.
Mott Insulator — Hubbard Transition
Correlation-driven metal-insulator transition: quasiparticle weight Z→0, Hubbard band splitting, and the Brinkman-Rice diverging effective mass.
Dirac Fermions in Graphene
Massless Dirac cones at K and K' — Berry phase π, anomalous half-integer QHE, Landau levels with √B spacing, and honeycomb lattice structure.
Flux Quantization in Superconductors
Φ = nΦ₀ = nh/2e — Abrikosov vortex lattice in type-II SCs with interactive vortex dynamics, phase winding, and BTK phase diagram.
Andreev Reflection & Proximity Effect
Electron-to-hole retroreflection at N-S interface — BTK conductance doubling below the gap, Cooper pair injection, and proximity-induced pairing.
Weyl Semimetal & Chiral Anomaly
Berry monopoles in k-space — Fermi arcs connecting Weyl nodes, chiral charge pumping via E·B, and negative longitudinal magnetoresistance.
Rashba-Dresselhaus Spin-Orbit Coupling
Spin splitting from broken inversion symmetry — spin textures, band structure, and the persistent spin helix when α=β quenches Dyakonov-Perel relaxation.
Topological Insulator Surface States
Dirac cone band structure and spin-momentum locked Fermi surface of a Z₂ topological insulator — helical edge states immune to backscattering.
Abrikosov Vortex Lattice
Quantized flux vortices (Φ₀=h/2e) forming a triangular Abrikosov lattice in a type-II superconductor — tunable field, penetration depth λ, and coherence length ξ.
Quantum Hall Chiral Edge States
Landau levels and chiral edge channels in the integer quantum Hall effect — σ_xy = νe²/h, skipping orbits, and topological protection from backscattering.
Bogoliubov Quasiparticle Spectrum
BCS superconductor quasiparticles as electron-hole superpositions — E(k)=√(ξ²+Δ²) band structure, coherence factors u(k)/v(k), and the divergent BCS density of states.
Majorana Bound States in a Nanowire
Topological phase diagram of a Kitaev nanowire — zero-bias conductance peak as Majorana signature, Zeeman field tuning, and non-Abelian braiding potential.
Ising Anyons & Topological Computation
Non-Abelian anyon braiding in the Ising model — worldline diagrams, fusion rules σ⊗σ=1⊕ψ, and topologically protected quantum gate matrices.
Quantum Critical Point — Hertz-Millis
Zero-temperature phase transition driven by quantum fluctuations — quantum critical fan, non-Fermi liquid ρ∼T behavior, and dynamic exponent z in the phase diagram.
Heavy Fermion Kondo Lattice
Kondo screening of f-electron moments by conduction electrons — hybridized heavy bands, Abrikosov-Suhl resonance in the DOS, and m*~1000mₑ quasiparticles.
Nodal Line Semimetal — Drumhead States
Topological semimetal with a nodal ring in the Brillouin zone — flat drumhead surface states bounded by the projected ring, and spin-orbit coupling gap opening.
Magnon Dispersion & Spin Waves
Quantized collective spin excitations in ferro- and antiferromagnets — ω(k) dispersion, thermal magnon population, and Bloch T^{3/2} magnetization decay law.
Mott Insulator & Hubbard Gap
Watch the metal-insulator transition as electron repulsion U overwhelms kinetic energy, splitting bands and localizing charges on a Hubbard lattice.
Kondo Effect — Resistance Minimum
Explore the anomalous resistance minimum from magnetic impurity scattering and the Kondo logarithm competing with phonon transport.
Quantum Spin Liquid
A frustrated kagome lattice evades magnetic order, instead birthing fractionalized spinon excitations that roam free through quantum entanglement.
Skyrmion Dynamics
Topologically protected magnetic whirlpools with winding number Q=±1 drift under spin-Hall currents and cannot be unwound without energy cost.
Dirac Semimetal
Massless Dirac fermions at a linear band crossing carry Berry phase π, producing exotic transport and a vanishingly small Fermi surface.
Rashba Spin-Orbit Coupling
Structural inversion breaking locks spin perpendicular to momentum in a 2DEG, splitting the Fermi circle into two helical sub-bands.
Cooperative Jahn-Teller
Electronic degeneracy drives spontaneous lattice distortion; when sites couple cooperatively, orbital order sweeps the crystal in a collective phase transition.
Spin Ice — Kagome Frustration
Ice-rule constraints on a kagome lattice create a macroscopically degenerate ground state with residual entropy and emergent magnetic monopoles.
Berezinskii-KT Transition
Vortex-antivortex pairs bind below T_BKT giving finite superfluid stiffness; above it they unbind in a universal topological phase transition.
Anderson Localization
Random disorder causes quantum interference to exponentially localize electron wavefunctions — even an infinitesimal impurity localizes all states in 1D and 2D.
Fractional Quantum Hall & Composite Fermions
Explore Laughlin ν=1/3 wavefunctions and flux attachment that transforms fractional QHE into integer-like physics.
Andreev Reflection & Josephson Effect
Watch electrons retroreflect as holes at a normal-superconductor interface, driving Josephson supercurrents via Andreev bound states.
Magneto-Optical Kerr Effect
Polarized light rotates on reflection from a magnetic surface — trace hysteresis loops and magnetic domain switching in real time.
Van Hove Singularities & Density of States
Critical k-points where group velocity vanishes create logarithmic DOS divergences that drive instabilities in 2D metals.
Charge Density Wave & Fermi Nesting
Parallel Fermi surface sheets nested by Q=(π,π) trigger the Peierls instability, opening a CDW gap and dimerizing the lattice.
Excitonic Insulator — Semimetal Transition
Spontaneous exciton condensation bridges BCS and BEC physics as Coulomb attraction overcomes the band overlap.
Shubnikov–de Haas Oscillations
Landau levels sweep through the Fermi energy as B rises, generating periodic resistance oscillations that map the Fermi surface.
Topological Crystalline Insulator
Mirror symmetry protects surface Dirac cones in SnTe with mirror Chern number C_M = ±2, gapped only by symmetry breaking.
Kagome Flat Band & Localized States
Destructive interference on the Kagome lattice produces a dispersionless band where eigenstates are trapped on isolated hexagons.
Orbital Magnetization — Modern Theory
Berry curvature and orbital moment together define the bulk magnetization of Bloch electrons, linking to anomalous Hall conductivity.
Quantum Metric & Fubini-Study
The quantum geometric tensor decomposes into the Fubini-Study metric (quantum distance) and Berry curvature, visualized across the Brillouin zone.
Nematic Superconductor
C₄→C₂ symmetry breaking in a superconductor produces anisotropic gap and a rich Ginzburg-Landau free energy landscape with nematic order.
Fractionalization & Slave Boson
Electrons fractionate into charged holons and neutral spinons via slave-boson decomposition, driving the Mott transition at half-filling.
Quantum Spin Hall Edge States
Helical edge states in a Z₂ topological insulator carry spin-up right and spin-down left, protected by time-reversal symmetry at Kramers points.
Topological Magnon Thermal Hall
DM-interaction gives magnons Berry curvature, producing a transverse thermal Hall current — topology in a bosonic quasiparticle system.
Polaron Band Narrowing
Holstein model polaron: electron-phonon coupling dresses electrons with a phonon cloud, narrowing bandwidth as Z = exp(−g²/ω₀²).
Spectral Function & Self-Energy
A(k,ω) = −ImG/π reveals quasiparticle weight Z, lifetime, and kink structure arising from the complex self-energy Σ(ω).
d-Wave Gap Anisotropy
The cuprate d-wave gap Δ(k) = Δ₀(cos kx − cos ky) vanishes at four nodal points, producing a V-shaped density of states.
Tomonaga-Luttinger Liquid
1D interacting electrons fractionalize into spinons and holons with different velocities, seen as two branches in the spectral function.
Quantum Criticality (Hertz-Millis)
At a quantum critical point, diverging fluctuations destroy the Fermi liquid and yield non-Fermi-liquid resistivity ρ~T^n in the QC fan.
DMFT Impurity & Mott Transition
Self-consistent Kondo resonance narrows and collapses as Hubbard U drives the spectral function through a metal-insulator transition.
RVB State & Rokhsar-Kivelson Point
At the exactly solvable RK point the quantum dimer model ground state is an equal-weight superposition of all coverings, a deconfined critical liquid.
Quantum Hall Plateau Transition
Plateau-to-plateau criticality with exponent ν≈2.3 collapses conductance curves from all temperatures onto a single universal scaling function.
Vortex Defects in Superfluid XY Model
Monte Carlo reveals BKT physics: vortex-antivortex pairs bind below T_BKT and proliferate above, destroying quasi-long-range superfluid order.
Weyl Semimetal Chiral Landau Level
Only the n=0 chiral Landau level disperses linearly through zero energy, connecting opposite-chirality Weyl nodes and driving the chiral anomaly.
Magnon Dispersion in Ferromagnets
Holstein-Primakoff spin waves give ω∝k² ferromagnetic magnons — sharply contrasting with linear antiferromagnetic Goldstone modes.
Triplet Superconductor & d-vector
p-wave Cooper pairs carry a d-vector in spin space; the chiral p+ip state fully gaps the Fermi surface and hosts a topological Chern number.
Kitaev Honeycomb Anyons
Exactly solvable via Majorana fermions, the Kitaev honeycomb hosts vison flux excitations whose worldline braiding accumulates a topological π phase.
Kubo Formula & Optical Conductivity
Current-current correlations yield the Drude peak and interband absorption in σ(ω), all constrained by the exact f-sum rule.
Wilson RG Momentum Shell
Integrating out a thin shell of high-k modes generates flow equations that steer couplings toward the Wilson-Fisher fixed point governing all second-order transitions.
Bose-Einstein Condensate
GPE simulation of a macroscopic quantum state: solitons, vortices, and condensate density evolving in a harmonic trap.
Supersolid Stripe Phase
A quantum state that simultaneously breaks U(1) gauge symmetry and translational symmetry — superfluid and crystal at once.
Many-Body Localization
Disorder-driven ergodicity breaking: watch entanglement entropy switch from volume-law (ETH) to area-law (MBL) as disorder grows.
Sachdev-Ye-Kitaev Model
Maximally chaotic non-Fermi liquid saturating the MSB bound λ_L = 2πT/ℏ, with power-law spectral function and holographic dual.
Toric Code
Kitaev's topological quantum error-correcting code on a torus: watch anyonic e and m excitations emerge from errors and annihilate.
Non-Hermitian Skin Effect
Asymmetric hopping collapses all bulk eigenstates to the boundary; the winding number of the PBC spectrum loop counts skin modes.
Floquet Prethermalization
Periodically driven quantum systems thermalize to an effective Hamiltonian for an exponentially long prethermal plateau before heating.
Measurement Phase Transition
Random circuits with projective measurements undergo a phase transition at critical rate p_c between volume-law and area-law entanglement.
Boson Sampling
Indistinguishable photons through a linear optical network sample a distribution whose probabilities require computing #P-hard matrix permanents.
CFT Operator Spectrum
Explore primary operators, Virasoro descendants, OPE coefficients, and conformal bootstrap for Ising, free boson, and tricritical CFTs.
Anyon Fusion Rules
Interactive fusion trees for Fibonacci, Ising, and toric code anyons — visualize non-Abelian quantum dimensions.
String-Net Condensation
Levin-Wen model on a honeycomb lattice: local string rules and Monte Carlo dynamics reveal emergent topological order.
Quantum Double D(Z₂)
Place and fuse toric code anyons {1, e, m, ε} on a grid — drag to observe braiding phases and fusion rules.
Entanglement Spectrum
Li-Haldane conjecture: the ES of a fractional quantum Hall state mirrors its chiral edge spectrum, revealing topological order.
MPS Bond Dimension
Matrix Product States: tune bond dimension χ and watch how entanglement entropy, truncation error, and area law behave.
Variational Monte Carlo
Jastrow wavefunction VMC for correlated electrons — optimize variational parameters and watch pair correlations emerge.
Exact Diagonalization
Full spectrum of the spin-½ Heisenberg chain: level statistics, structure factor S(k), and ground state correlations.
DMRG Sweeping
Watch the density matrix renormalization group converge sweep by sweep, tracking entanglement entropy and truncation error.
QMC Sign Problem
The fermionic sign problem in Hubbard QMC: average sign decays exponentially — explore sign-free cases at half-filling.
Neural Quantum State
RBM wavefunction trained via variational Monte Carlo — watch hidden unit activations and energy converge to the ground state.
Ricci Flow
Ricci flow on 2D surfaces — watch bumpy profiles smooth to constant curvature, the key tool behind Perelman's proof of the Poincaré Conjecture.
Mirror Symmetry
Mirror symmetry on Calabi-Yau threefolds — Hodge diamond visualization with h¹¹↔h²¹ exchange and rotating CY cross-sections.
Kagome Spin Liquid
Frustrated kagome antiferromagnet Monte Carlo — geometric frustration suppresses ordering, producing a quantum spin liquid ground state.
AdS/CFT Dictionary
Holographic correspondence: bulk-to-boundary propagators and Ryu-Takayanagi geodesics in Poincaré AdS — entanglement entropy as area.
MPS Entanglement
Matrix product state tensor network — area law vs. volume law entanglement entropy, Schmidt spectrum, and bond dimension χ.
Chern-Simons Invariants
Chern-Simons theory computes topological link invariants — Jones polynomial from Wilson loops for Hopf link, trefoil, Borromean rings.
Causal Dynamical Triangulations
CDT path integral over Lorentzian random geometries — Monte Carlo on triangulated spacetime strips with Regge action.
Luttinger Liquid
1D bosonization maps interacting fermions to a free boson — power-law correlators with Luttinger parameter K governing all exponents.
Polymer Scaling & Flory Exponent
Self-avoiding walk polymer chains on a lattice — end-to-end distance R ~ N^ν with Flory exponent ν≈0.588 in 3D, log-log scaling plot.
KPZ Interface Roughening
Kardar-Parisi-Zhang stochastic growth equation — interface width W ~ t^β with β=1/3 in 1+1D, real-time log-log scaling measurement.
Jamming Transition — Soft Spheres
Soft-sphere packings approach rigidity at φ_J ≈ 0.64 — contact number jumps to Z=2d, pressure rises as P ~ (φ−φ_J)^α.
Vicsek Model — Flocking
Self-propelled particles align with neighbors — polar order parameter Φ transitions from disorder to collective flocking as noise decreases.
Stochastic Resonance
Noise enhances weak signal detection in a bistable double-well — SNR peaks at optimal noise level σ* predicted by Kramers escape theory.
BTW Sandpile — Self-Organized Criticality
Bak-Tang-Wiesenfeld sandpile self-tunes to criticality — avalanche size power law P(s) ~ s^−τ with τ≈1.29 emerges without fine-tuning.
Gray-Scott Reaction-Diffusion
Two-chemical autocatalytic system generates spots, stripes, labyrinths, and worms — explore the rich Turing pattern phase diagram by tuning F and k.
Hénon Map Strange Attractor
Hénon map iteration reveals a fractal strange attractor with Hausdorff dimension ≈1.26 — explore density coloring and fractal basin boundaries.
Gravitational Wave Chirp Signal
Binary black hole inspiral generates a chirp — watch frequency and amplitude sweep from 10 Hz to ISCO as the waveform sweeps to merger, with optional audio playback.
Big Bang Nucleosynthesis Network
Primordial reaction network predicts D/H, ⁴He, ³He, ⁷Li as functions of baryon density — tune η, neutron lifetime, and Neff against observational bounds.
Galaxy Rotation Curve & Dark Matter
Flat rotation curves demand an NFW dark matter halo — decompose stellar disk, gas disk, and dark halo contributions for Milky Way, NGC 3198, and custom galaxies.
Lane-Emden Stellar Structure
Polytropic stellar interiors from the Lane-Emden equation — explore density, pressure, temperature, and mass profiles for red giants, Sun-like stars, and white dwarfs.
Rayleigh-Bénard Convection
Thermal convection rolls between heated and cooled plates — watch patterns form above the critical Ra≈1708, with streamlines and temperature field evolving in real time.
XY Model Topological Defects
Vortex and antivortex dynamics in the 2D XY model — watch BKT physics: pairs bind below T_BKT≈0.89 J/k_B and proliferate above it, destroying quasi-long-range order.
Phase-Field Crystal Solidification
Dendritic crystal growth via the PFC model — atomic-scale density waves nucleate from a seed and grow fractal arms driven by undercooling parameter ε.
Microcanonical Entropy S = k ln Ω
Boltzmann's entropy for N harmonic oscillators at fixed energy — explore the density of states Ω(E,N), temperature emergence, heat capacity, and equipartition.
Kuramoto Synchronization & Order Parameter
N coupled phase oscillators synchronize above critical coupling K_c = 2σ/π — watch the complex order parameter r·e^(iψ) grow from incoherence to collective locking.
Faraday Waves — Parametric Resonance
Standing waves on a vertically vibrated fluid layer — Mathieu instability tongues open at half-integer drive frequencies, generating subharmonic standing patterns.
Rule 110 — Universal Computation
Wolfram's Rule 110 cellular automaton is Turing-complete (Cook 2004) — explore all 256 elementary CA rules, click to edit initial conditions, compare Wolfram classes.
Double Pendulum — Poincaré Section
Hamiltonian chaos visualized: trajectory animation and Poincaré map at θ₂=0 — KAM tori appear as closed curves, chaotic trajectories fill 2D regions of phase space.
Quantum Harmonic Oscillator — Ladder Operators
Eigenstates ψₙ via Hermite polynomials, probability densities, classical turning points, and coherent state wavepacket dynamics for the prototypical quantum system.
Logistic Map — Feigenbaum Universality
Period-doubling bifurcation diagram with Lyapunov exponent coloring — the universal constant δ≈4.6692 governs the route to chaos across all unimodal maps.
Quantum Walk — Discrete Interference
Discrete-time quantum walk on a line: coin operator creates bimodal interference pattern with ballistic σ~t spread, contrasted against classical Gaussian diffusion.
Wigner-Ville Distribution
Time-frequency phase space representation: unique bilinear distribution satisfying both marginals — cross-term interference fringes reveal quantum-like structure in classical signals.
SSH Model — Topological Edge States
Su-Schrieffer-Heeger chain: winding number ν=0/1 controls topological phase, zero-energy edge states localized at chain ends, visualized with finite-chain spectrum and Bloch loop.
Dissipative Soliton — Complex Ginzburg-Landau
CGLE amplitude evolution: gain-loss balance sustains self-organized pulse attractors; tune cubic gain ε and quintic loss μ to watch solitons form, breathe, or dissolve.
Lloyd's Algorithm — Centroidal Voronoi
Iterative relaxation moves each seed to its Voronoi centroid, minimizing quantization energy E=Σ∫‖x−cᵢ‖²dx — converges to the optimal uniform tessellation.
Lévy Flight vs Brownian Motion
Power-law step sizes with index α∈(0,2) produce superdiffusion ⟨r²⟩~t^{2/α}: rare giant leaps dominate over many small steps, shown with MSD log-log scaling and step-size histogram.
Entropy Production — Irreversible Thermodynamics
Onsager reciprocal relations and entropy production σ=ΣJᵢXᵢ≥0 in coupled heat-mass transport; Soret/Dufour cross-coupling drives coupled T and c fields on a 1D bar.
2D Lid-Driven Cavity — Navier-Stokes
Vorticity-streamfunction Navier-Stokes in a square cavity: primary vortex shifts toward center and Moffatt corner eddies emerge as Reynolds number increases.
Geodesic Flow — Negative Curvature
Geodesics on the hyperbolic plane (Poincaré disk) diverge exponentially with Lyapunov exponent λ=√|K| — the geometric origin of Anosov chaos, mixing, and ergodicity.
Fibonacci Quasicrystal Spectrum
Tight-binding spectrum of a Fibonacci quasiperiodic chain: Cantor-set eigenvalues, devil's staircase IDS, and self-similar gap structure as a function of hopping ratio tA/tB.
Symplectic Integrator — Kepler Orbit
Störmer-Verlet (symplectic) vs forward Euler integration of Kepler orbits: energy oscillates but never drifts for the symplectic method, while Euler spirals outward.
PCA — Geometry of Covariance
PCA as eigendecomposition of the covariance matrix: eigenvectors align with axes of maximum variance, illustrated with rotating confidence ellipses and variance-explained bar charts.
Osmotic Pressure — van't Hoff
Semipermeable membrane simulation: Π = iMRT drives net water flux from pure solvent to solution, building a hydrostatic pressure head visualized by rising water columns.
Chemical Master Equation — Gillespie SSA
Exact stochastic simulation of the Schlögl bistable reaction network via the Gillespie algorithm, revealing bimodal stationary distributions and spontaneous switching between states.
GARCH(1,1) — Volatility Clustering
GARCH(1,1) generates realistic financial return series with volatility clustering and fat tails: calm and turbulent periods emerge naturally from autoregressive variance dynamics.
Real-Space RG — 1D Ising Decimation
Exact real-space RG decimation K' = ½ ln(cosh 2K) for the 1D Ising model: cobweb flow diagram shows all couplings flowing to K*=0, explaining the absence of a finite-T phase transition.
Wavefunction Collapse & Born Rule
Quantum measurement postulate: superposition, collapse, Born rule probabilities across energy eigenstates
Feynman Checkerboard Path Integral
Relativistic particle on a spacetime lattice: sum over paths, kink amplitudes, Dirac propagator emergence
Nullclines & Phase Portraits
2D dynamical systems: nullclines, vector field, limit cycles, fixed point stability, Van der Pol & more
Fourier Series & Gibbs Phenomenon
Gibbs ~8.9% overshoot at discontinuities persists for all N: square wave, sawtooth, triangle approximations
Prime Number Theorem & Zeta Zeros
π(x) vs Li(x), explicit formula, Riemann zeros on critical strip, and prime counting statistics
GOE Random Matrices & Wigner Semicircle
Wigner semicircle law and level repulsion statistics from Gaussian Orthogonal Ensemble matrices
Hyperbolic Tessellation — Poincaré Disk
{p,q} tilings of the hyperbolic plane via Möbius reflections: {4,5}, {3,7}, {6,4} and more
Bifurcation Diagrams: Saddle-Node to Cusp
Interactive bifurcation zoo: saddle-node, pitchfork, Hopf, cusp catastrophe, logistic period-doubling
Pair Approximation on Networks
SIS/SIR beyond mean-field: pair correlations, clustering effects, network epidemic thresholds
Avoided Level Crossing & Mode Hybridization
Coupled oscillator anticrossing: normal mode splitting, Hopfield weights, Rabi oscillations, transmission spectrum
Schwarzschild Geodesics & Photon Sphere
Null and timelike geodesics around a black hole: photon sphere at r=3M, ISCO, effective potential, orbital capture vs deflection
Quantum Decoherence & Pointer Basis
Environment-induced superselection: density matrix dephasing, Bloch sphere shrinkage, einselection of classical pointer states
Shannon Channel Capacity
C = B·log₂(1+SNR): mutual information, bandwidth vs power tradeoffs, AWGN channel capacity curves
Preferential Attachment Network Growth
Barabási-Albert model: rich-get-richer dynamics, power-law degree distribution P(k)~k⁻³, hub formation
Stochastic Block Model & Detectability
Planted partition communities, Kesten-Stigum threshold, adjacency matrix block structure, detectability SNR
Ecological Tipping Points & Fold Bifurcation
Saddle-node bifurcation with hysteresis: bistable vegetation model, catastrophic collapse, early warning signals
Plasma Oscillations & Langmuir Waves
Bohm-Gross dispersion ω²=ωₚ²+3k²v_th², plasma frequency, Debye screening, animated density waves
Lattice Gauge Theory & Wilson Loops
U(1) lattice gauge theory, Wilson loop order parameter, confinement vs deconfinement, Monte Carlo Metropolis
BEC Ground State & Gross-Pitaevskii
Imaginary-time GPE solution, Thomas-Fermi parabolic profile, chemical potential, Gaussian-to-TF crossover
Magnetic Hysteresis Loop
Animated B-H hysteresis loop showing coercivity, remanence, and saturation as ferromagnetic domains respond to a cycling applied field.
Age-Structured Predator–Prey
Leslie matrix population dynamics with three prey age classes and size-selective predation, revealing how age structure shapes ecological cycles.
Bekenstein-Hawking Black Hole Entropy
Interactive black hole explorer computing S=A/4 horizon entropy, Hawking temperature, and simulating mass loss by thermal radiation evaporation.
Fiber Bundle & Holonomy
Parallel transport of a vector around closed loops on circle, sphere, and torus fiber bundles, visualizing the geometric phase holonomy angle.
Ising RG: Position-Space Decimation
Block-spin renormalization group on the 1D Ising chain, with cobweb RG flow diagram showing fixed points and the exact K'=½ ln cosh(2K) map.
Gauss-Bonnet Theorem
Rotating 3D surfaces demonstrating how integrated Gaussian curvature plus boundary terms always equals 2πχ, a topological invariant.
Kitaev Toric Code & Anyons
Interactive toric code lattice showing vertex and plaquette stabilizers, anyon creation from qubit flips, and topological error correction.
Power Iteration Convergence
Watch a vector converge to the dominant eigenvector of a 2×2 matrix with live Rayleigh quotient tracking and convergence rate visualization.
Born-Oppenheimer Potential Surface
Morse potential energy surface for a diatomic molecule with tunable well depth, equilibrium distance, and quantized vibrational energy levels.
Floquet-Bloch Band Topology
Quasi-energy bands of a periodically driven lattice in the Floquet-Brillouin zone, with topological gap openings controlled by drive amplitude and frequency.
Akhmediev Breather & Rogue Waves
Exact NLS breather solution showing modulation instability growing on a uniform background and peaking as a rogue wave at amplitude 3× the background.
Lieb-Liniger 1D Bose Gas
Bethe ansatz ground-state energy of 1D bosons with delta-function interactions, interpolating from BEC weak-coupling to Tonks-Girardeau fermionization.
Mean-Field Ising Susceptibility
χ = ∂M/∂H diverges at Tc=1 in mean-field theory; tune field H and watch magnetization and susceptibility curves evolve.
Finite-Size Scaling Collapse
Rescale susceptibility curves for different system sizes L onto a single universal function by tuning critical exponents ν and γ/ν.
SSH Model — Topological Edge States
Tune intracell/intercell hopping in the Su-Schrieffer-Heeger chain and watch zero-energy edge states appear in the topological phase.
Andreev Reflection — BTK Model
Differential conductance at a normal-metal/superconductor interface shows doubled conductance below the gap Δ and a dip for high barrier Z.
Saddle-Node Bifurcation
Two fixed points collide and vanish at r=0 in ẋ=r+x²; watch the phase portrait and bifurcation diagram transform in real time.
Poincaré Map — Driven Pendulum
Stroboscopic sampling of the driven nonlinear pendulum reveals period-doubling cascades and strange attractors as drive amplitude increases.
Jaynes-Cummings Spectrum
Vacuum Rabi splitting and anharmonic dressed-state ladder of the Jaynes-Cummings model; tune coupling g, detuning δ, and linewidth κ.
Josephson Junction Washboard
Phase particle rolls in the tilted washboard potential; below I_c the junction is trapped (supercurrent) and above it runs (AC Josephson).
Wigner-Eckart Selection Rules
Clebsch-Gordan coefficient table and angular momentum selection rules for operators of rank k acting between states j and j'.
Gray-Scott Reaction-Diffusion
Tune feed and kill rates to explore Pearson's pattern atlas: spots, stripes, labyrinths, and self-replicating mitosis in the Gray-Scott model.
XY Model Vortex Defects
Monte Carlo simulation of the 2D XY model reveals vortex/antivortex topological defects whose binding/unbinding drives the BKT phase transition.
Discrete Quantum Walk
A Hadamard coin drives ballistic spreading ~t with twin-peaked distributions, contrasting sharply with the Gaussian √t spread of a classical random walk.
Landau-Zener Transition
Watch non-adiabatic tunneling probability at an avoided crossing as sweep rate and coupling strength are tuned, verifying P_LZ = exp(−2πΔ²/α).
RG Fixed Point Flow
Interactive renormalization group flow diagram showing stable and unstable fixed points, critical surfaces, and basins of attraction for Wilson-Fisher and KT models.
Hopf Fibration S³→S²
Stereographic projection reveals the Hopf fibration's fiber circles as linked tori in 3D — every pair of fibers carries linking number one.
Stochastic Resonance
An optimal noise level boosts the SNR of a subthreshold periodic signal at a threshold detector — noise paradoxically enhances signal detection.
Ergodic Hard Disk Gas
2D hard-disk billiard dynamics show that time-average speed distributions converge to the Maxwell-Boltzmann ensemble, demonstrating the ergodic hypothesis.
Joukowski Airfoil
The Joukowski conformal map w = z + λ²/z transforms a shifted circle into a cambered airfoil with a sharp trailing edge, with interactive streamlines.
Lindblad Master Equation
Bloch sphere dynamics for an open qubit: damped precession under T₁ relaxation and T₂ dephasing via the GKSL master equation.
Kolmogorov Turbulence Cascade
K41 energy spectrum E(k)∝k^(−5/3) in the inertial range, the exact 4/5 law, and She-Leveque intermittency corrections.
Cavity Optomechanics
Radiation-pressure coupling and sideband cooling of a mechanical resonator inside a Fabry-Pérot cavity toward the quantum ground state.
DMFT Mott Transition
Local spectral function and quasiparticle weight in the Hubbard model, showing the Mott metal-insulator transition as U/W increases.
Virial Theorem & Gravitational Collapse
N-body gravitational simulation virializing toward 2⟨K⟩+⟨V⟩=0, demonstrating negative heat capacity and collapse dynamics.
Asymptotic Freedom QCD
QCD running coupling αs(μ) decreasing logarithmically at high energy scales — the discovery that won Gross, Politzer, and Wilczek the 2004 Nobel Prize.
Magnetic Reconnection
Sweet-Parker current sheet geometry: inflow at v_in ∝ S^(−½)v_A with outflow jets at the Alfvén speed, setting the solar flare timescale problem.
Anderson Localization 1D
Transfer matrix computation showing all 1D tight-binding eigenstates are exponentially localized by any disorder, via the Lyapunov exponent.
Keldysh Nonequilibrium Green's Functions
Retarded, advanced, and Keldysh Green's functions for a biased quantum dot, with real-time spectral function and Landauer current formula.
Chaos Synchronization
Two unidirectionally coupled Lorenz systems: the slave synchronizes to the master when coupling exceeds the largest Lyapunov exponent.
Information Bottleneck
Tishby's IB trade-off: compress X into T while preserving relevance about Y, sweeping the β curve interactively.
Replica Trick & Spin Glass
Sherrington-Kirkpatrick model: the replica method computes free energy, RS order parameter q_EA, and the spin-glass phase transition.
Contact Process
Directed-percolation universality: 1D contact process transitions between active and absorbing phases at λc ≈ 1.649.
Phononic Crystal Bandgap
1D elastic metamaterial: transfer-matrix band structure reveals Bragg-scattering bandgaps and evanescent wave decay.
Dyson-Schwinger Propagator
Self-consistent Dyson equation: self-energy Σ(ω) dresses the propagator, shifting the quasiparticle pole and broadening the spectral function.
Spectral Flow & Atiyah-Singer
Dirac operator eigenvalues flow through zero as θ varies; the net spectral flow equals the topological index Q.
Minority Game
N bounded-rational agents compete for the minority side; volatility is minimized at the critical complexity α* ≈ 0.34.
Coupled Map Lattice
Kaneko's CML of logistic maps displays spatiotemporal chaos, frozen domains, and a real-time Lyapunov exponent across the lattice.
Saffman-Taylor Fingering
DLA-based Hele-Shaw simulation: Laplacian growth produces fractal viscous fingers with dimension D_f ≈ 1.71.
Quantum Error Correction
Stabilizer codes (Shor, Steane, toric): apply Pauli errors, measure syndromes, and watch corrections restore the logical qubit.
XXZ Spin Chain Bethe Ansatz
Exact Bethe ansatz magnon dispersion for the anisotropic Heisenberg chain across ferromagnetic, critical, and Néel phases.
Kitaev p-wave Chain
Topological phase diagram of the Kitaev chain with Majorana edge modes emerging at the topological phase boundary |μ|=2t.
Hubbard Mott Transition
Spectral function of the Hubbard model showing the quasiparticle peak collapse and Mott gap opening as U/t exceeds the critical value.
Yang-Mills Vacuum & Instantons
Topological vacuum landscape of Yang-Mills theory with instanton tunneling between winding number sectors and θ-vacuum energy.
Driven-Dissipative BEC
Real-time simulation of polariton condensation via the complex Gross-Pitaevskii equation with pump gain and cavity loss.
Grand Canonical Ensemble
Fermi-Dirac, Bose-Einstein, and Maxwell-Boltzmann occupation functions with density of states and particle number as μ varies.
Anomalous Hall Berry Curvature
Berry curvature map over the Brillouin zone for a two-band Chern insulator with quantized Hall conductivity σ_xy = Ce²/h.
SUSY QM & Witten Index
Partner potentials V± from superpotential W(x) illustrating isospectrality, zero modes, and the topological Witten index Δ.
Kauffman Bracket & Jones Polynomial
Animated knot diagrams with skein relation visualization computing the Kauffman bracket and Jones polynomial for classic knots.
Manna Sandpile Model
Stochastic toppling automaton self-organizing to criticality with power-law avalanche statistics in the Manna universality class.
Frustrated Magnetism: Kagome Antiferromagnet
Monte Carlo simulation of corner-sharing triangle spins revealing macroscopic degeneracy and spin liquid behavior.
Kerr Effect & Self-Phase Modulation
SPM spectral broadening via B-integral with real-time FFT showing characteristic multi-peak structure at high nonlinearity.
Entropy Production: Irreversible Thermodynamics
Onsager reciprocal relations and entropy production rate σ = J·X ≥ 0 visualized as a force-flux phase diagram.
Ecological Neutral Theory: Hubbell Forest
Zero-sum community dynamics with immigration and speciation producing log-series species abundance distributions.
Topological Magnons: Honeycomb Dirac Points
Spin-wave bands on the honeycomb lattice with DM-induced gap opening and Berry curvature at K/K' Dirac points.
Quantum Otto Cycle Efficiency
Two-level qubit heat engine showing finite-time efficiency tradeoff against power via incomplete thermalization.
OGY Chaos Control Method
Stabilizing unstable periodic orbits in the logistic map with tiny parameter perturbations using the Ott-Grebogi-Yorke algorithm.
Continuum Percolation: Random Disk Model
Boolean disk percolation with spanning cluster detection and probability scan revealing the universal threshold φ_c ≈ 0.676.
Turbulent Boundary Layer: Law of the Wall
Interactive visualization of the universal u⁺=f(y⁺) profile from viscous sublayer through log region to Coles wake.
Symplectic Integrators: Verlet vs Euler
Störmer-Verlet preserves the symplectic 2-form and shadow Hamiltonian while Euler drifts — shown in phase space and energy error plots.
DMRG & Entanglement Entropy
Visualize area law vs CFT logarithmic entanglement, Schmidt spectrum, and bond dimension in the transverse-field Ising chain.
Spin-Boson Model: Ohmic Bath
Explore quantum dissipation, J(ω)=2παω ohmic spectral density, and the Kosterlitz-Thouless quantum phase transition at α=1.
Quantum Spin Liquid: Spinon Fractionalization
Investigate RVB states, fractionalized S=½ spinons, diffuse neutron scattering continuum, and topological entanglement entropy on the Kagome lattice.
MIPS: Motility-Induced Phase Separation
Simulate run-and-tumble self-propelled particles spontaneously phase-separating into dense and dilute regions with purely repulsive interactions.
Mean Field Games: Nash Equilibrium
Solve the coupled HJB + Fokker-Planck system for crowd congestion dynamics and visualize convergence to the mean-field Nash equilibrium.
CFT: Operator Product Expansion
Explore OPE coefficients, conformal blocks, Virasoro Kac table, and 4-point correlators across Ising, free boson, and Liouville CFTs.
Hawking Radiation & Pair Production
Visualize virtual pair creation near the black hole horizon, Planckian spectrum at T_H, evaporation timeline, and the Page information curve.
SLE_κ: Stochastic Loewner Evolution
Generate random conformal curves for κ=2–8 covering loop-erased random walks, Ising interfaces, critical percolation, and space-filling UST.
Random Field Ising Model: Avalanches
Sweep the external field through quenched disorder and observe Barkhausen avalanche statistics, hysteresis, and the power-law size distribution near Rc.
Benjamin-Feir Modulational Instability
Watch NLS sideband growth, BF instability criterion, and the Peregrine soliton rogue wave form from a perturbed Stokes wave train.
Saffman-Taylor Viscous Fingering
DLA simulation of finger instabilities as less-viscous fluid invades a Hele-Shaw cell
Kibble-Zurek Defect Formation
Topological vortex defects nucleate during rapid symmetry-breaking quenches in TDGL simulation
Rayleigh-Plateau Jet Instability
Surface tension breaks a fluid jet into droplets; watch the dispersion spectrum and pinch-off
Ising Transfer Matrix Exact Solution
Onsager's 1944 exact free energy vs. Metropolis MC simulation with critical point Tc=2.269
SNIPER Bifurcation
Saddle-node on invariant circle: the theta neuron transitions from quiescence to infinite-period firing
Continuous-Time Random Walk
Anomalous diffusion with power-law waiting times; MSD ~ t^α subdiffusion tracked in real time
Marchenko-Pastur Spectrum
Spectral density of large Wishart matrices converges to the Marchenko-Pastur law; BBP spike transition
Topological Insulator Z₂ Invariant
BHZ model of HgTe/CdTe quantum well: bulk-edge correspondence with helical edge states
Kuramoto Synchronization Transition
Order parameter r(K) grows beyond critical coupling; self-consistency theory vs. N-oscillator simulation
Andreev Reflection (BTK Theory)
Electron retroreflects as hole at N-S interface; sub-gap conductance doubling and BTK G(V) curve
Fano Resonance
Asymmetric lineshape from interference between discrete resonance and continuum; tunable Fano parameter q
Bogoliubov Transformation (BCS)
BCS quasiparticle dispersion E_k=√(ξ_k²+Δ²); coherence factors u_k, v_k and the pairing amplitude
Rule 184 Traffic Flow
Cellular automaton model of highway traffic; free-flow vs congested phases and the fundamental diagram J(ρ)
Geodesic Flow — Hyperbolic Plane
Geodesics in the Poincaré disk (K=−1); exponential divergence, Anosov dynamics and ergodic mixing
Spatial Evolutionary Prisoner's Dilemma
Cooperation survives via cluster formation on a spatial lattice; Nowak-May imitation dynamics
Vicsek Model — Flocking Transition
Self-propelled particles align under noise; order parameter φ shows first-order flocking phase transition
Crooks Fluctuation Theorem
Forward and reverse work distributions cross at ΔF; Jarzynski equality and nonequilibrium free energy
Andreev Bound States — Josephson
Sub-gap ABS energies E±(φ)=±Δ√(1−τsin²φ/2) in short junctions; supercurrent vs phase
Passive Tracer Turbulent Advection
Tracer particles in random Fourier velocity field; Richardson dispersion, FSLE and Batchelor spectrum
Lévy Flight & Superdiffusion
Heavy-tailed jumps from α-stable distributions; MSD ~ t^(2/α) anomalous diffusion vs Gaussian walks
Lieb-Liniger Bose Gas
Exact Bethe ansatz solution for 1D bosons with contact interactions — quasi-momenta and momentum distribution
Takens Embedding Theorem
Reconstruct strange attractors from scalar time series using delay embedding — Lorenz, Rössler, Duffing
PT-Symmetric Gain-Loss
Non-Hermitian physics with real spectra — exceptional points, PT phase transitions, and complex eigenvalues
Schlögl Chemical Bistability
Prototype chemical bistability via cubic autocatalysis — potential landscape, deterministic and Gillespie dynamics
Edwards-Wilkinson Interface Growth
Surface roughening with EW and KPZ nonlinear noise — scaling exponents, roughness W(t) ~ t^β
Topological Defect Coarsening
XY model vortex-antivortex annihilation — BKT transition, defect density decay, and coarsening laws
Kicked Top Quantum Chaos
Floquet unitary on spin-j — Husimi Q-function, entanglement entropy growth, and quantum-classical correspondence
Oseen Tensor Hydrodynamics
Long-range 1/r fluid coupling between particles via Stokeslet velocity fields — collective Stokes flow
DMT Contact Mechanics
Adhesive contact between nanoscale sphere and substrate — DMT vs JKR models, Tabor parameter, pull-off forces
Hegselmann-Krause Opinion Dynamics
Bounded confidence model — consensus, polarization, and fragmentation with interactive ε and spacetime trajectories
FitzHugh-Nagumo Spirals
Reaction-diffusion spiral waves in an excitable medium — click to stimulate
First Passage Time
1D random walk hitting an absorbing barrier — inverse Gaussian FPT distribution
Maier-Saupe Nematic
Mean-field liquid crystal theory — isotropic-to-nematic phase transition via MC
SSH Chain Topology
Su-Schrieffer-Heeger model — winding number, edge states, bulk-boundary correspondence
Yard-Sale Wealth Model
Random pairwise trades → oligarchy; Gini coefficient, Lorenz curve, taxation effects
Chua Double-Scroll
First experimental chaos circuit — piecewise-linear double scroll strange attractor
Thomas-Fermi Atom
Semiclassical DFT — electron density from ODE shooting for elements H through U
Lattice Boltzmann Flow
D2Q9 LBM simulation — Poiseuille flow, draw obstacles, visualize velocity and vorticity
QCGL Dissipative Solitons
Quintic complex Ginzburg-Landau — stable localized pulses from gain/loss balance
Axelrod Culture Model
Social influence on a grid — cultural homogenization vs frozen mosaic phase transition
Huygens Synchronization
Two pendula on a shared beam spontaneously synchronize through weak mechanical coupling — Huygens' 1665 discovery.
Rabi Oscillations
A driven two-level quantum atom on the Bloch sphere — population oscillates at the generalized Rabi frequency Ω̃=√(Ω²+Δ²).
Perron-Frobenius Transfer Operator
Evolve probability densities under chaotic maps and watch them converge to SRB invariant measures.
Thermodynamic Uncertainty Relation
Precision costs entropy: the TUR bound Var(J)/⟨J⟩²·Ṡ ≥ 2k_B explored on a driven Markov chain.
Viscous Gravity Current
Lubrication theory: dense fluid spreading under gravity with self-similar nose radius scaling as t^(1/2).
Pólya Urn Reinforcement
Path-dependent randomness: drawing and reinforcing colors in an urn converges to a Beta-distributed random variable.
Topological Mechanics
Kane-Lubensky isostatic spring networks with topological soft modes and states of self-stress localized at edges.
Double Pendulum & Poincaré Section
Chaotic double pendulum with live Poincaré section revealing KAM tori breaking into chaos as energy increases.
Oregonator BZ Chemical Waves
Belousov-Zhabotinsky reaction-diffusion: Oregonator model producing target waves, spirals, and oscillatory excitation.
Bipartite Network Correlations
Degree correlations and nestedness in bipartite graphs with projections and assortativity measures.
Metabolic Scaling — Kleiber's Law
B ∝ M^(3/4): why metabolic rate scales with the 3/4 power of body mass, with heart rate, lifespan, and physiological scaling.
Hyperbolic Discounting
Temporal preference reversals: why we prefer $10 now over $11 tomorrow but switch when both rewards are shifted into the future.
Dyson's Coulomb Gas — Random Matrices
Eigenvalue repulsion as a Coulomb gas: Wigner semicircle law, level spacing statistics, and GOE/GUE/Poisson ensembles.
Cellular Potts — Cell Sorting
Steinberg's differential adhesion hypothesis: cells sort by minimizing surface energy in a Monte Carlo Potts model simulation.
Drude Model — AC Conductivity
Classical electron gas: σ(ω) and ε(ω) spectra, plasma frequency, skin depth, and electron motion under an oscillating field.
Majority Vote Model — Sociophysics
Opinion dynamics phase transition on a 2D lattice: order-disorder transition at noise threshold q_c with Ising universality class.
Cusp Catastrophe — Codimension Two
Normal form ẋ = −x³ + bx + a: the cusp catastrophe surface, fold curves, hysteresis, and preference reversal in 2-parameter space.
Baker's Map — Fluid Mixing
Chaotic advection via stretch-and-fold: Smale horseshoe, Lyapunov exponents, and exponential mixing in a unit square.
Hadamard Quantum Walk
Ballistic spreading σ~t versus classical diffusion σ~√t: interference-driven quantum transport with decoherence and coin angle control.
WKB Semiclassical Quantization
Bohr-Sommerfeld condition ∮p·dq=(n+½)h: WKB bound state energies vs exact quantum solutions across harmonic, Morse, and double-well potentials.
Multimode Laser Competition
Rate equations for N competing laser modes with gain saturation and cross-coupling — watch winner-take-all dynamics as modes compete for shared gain medium.
Hertzian Solitary Waves
Compact solitons in granular bead chains via Hertz contact law F=Aδⁿ. Fire a striker and watch finite-width pulses propagate through Newton's cradle at the nano-scale.
NK Fitness Landscape
Kauffman NK model: tune epistasis K from smooth (one peak) to fully rugged (exponential local optima). Watch evolving populations get trapped at local maxima.
Phonon Normal Modes
Harmonic crystal lattice with monatomic and diatomic bases. Visualize individual phonon modes and the acoustic/optical dispersion relation ω(q).
Abelian Sandpile SOC
Bak-Tang-Wiesenfeld sandpile self-organizes to criticality via toppling cascades. Watch power-law avalanches and the fractal identity element emerge.
Chaotic Advection
Aref's journal bearing: Stokes flow between eccentric rotating cylinders creates chaotic particle trajectories despite laminar flow — broken KAM tori mixing fluid.
AGN Accretion Disk SED
Shakura-Sunyaev multi-temperature disk around a spinning black hole. Tune mass, accretion rate, and spin to see how the UV bump and X-ray corona shape the SED.
Quorum Sensing Bistability
Bacterial collective switching via autoinducer molecules. Hill function cooperativity creates bistable hysteresis — the colony remembers past AI exposure.
Feed-Forward Loop Gene Circuit
Four FFL motifs (C1, C2, I1, I4) from E. coli gene networks. C1-FFL with AND gate filters noise and creates sign-sensitive delay — apply signal pulses to explore.
Gray-Scott Reaction-Diffusion
Turing patterns from the Gray-Scott model: spots, stripes, worms, coral, and solitons emerge from two coupled reaction-diffusion equations. Eight preset parameter regimes.
Birth-Death Markov Chain
Stationary distribution π_n ∝ (λ/μ)^n via detailed balance. Watch a random walker converge to the analytic steady state as histograms fill in real time.
Joukowski Airfoil
Conformal mapping w = z + c²/z transforms a circle into a lifting airfoil. Streamlines warp analytically; adjust eccentricity and angle of attack to tune lift.
Random Sequential Adsorption
Rods adsorb irreversibly on a 1D line — jamming freezes coverage at Rényi's constant 1−e^(−2) ≈ 0.8647. Watch the empirical curve converge to the analytic limit.
Jaynes-Cummings Collapse & Revival
Two-level atom in a coherent cavity field: ⟨σ_z(t)⟩ collapses then revives as photon-number components dephase and re-phase. Adjust mean photon number to see timescales.
Turing Instability — Gray-Scott PDE
Activator-inhibitor reaction-diffusion: spots, stripes, and labyrinths self-organize from noise. Dispersion relation shows the unstable band of spatial modes.
Stochastic Gene Expression
ON/OFF gene switching with Gillespie SSA: bursty mRNA transcription gives super-Poissonian noise. Fano factor theory vs. simulation tracked live.
Rayleigh Surface Acoustic Wave
Retrograde elliptical particle motion decaying exponentially with depth. Adjust wavevector and Poisson ratio; dispersion relation shown below the animation.
Leidenfrost Effect
Above 220°C a water droplet levitates on its own vapor cushion. Interactive Nukiyama boiling curve shows the transition from nucleate to film boiling.
SIS Epidemic on Scale-Free Network
Barabási-Albert network with SIS dynamics: hub super-spreaders push the epidemic threshold to zero. Phase diagram tracks LD/HD/MC regimes in real time.
TASEP Driven Diffusive System
Totally asymmetric exclusion process: three phases (LD, HD, MC) separated by boundary rates α and β. Density profiles, phase portrait, and current-density J(ρ) live.
Hong-Ou-Mandel Effect
Two indistinguishable photons entering a beamsplitter always bunch together — the HOM dip. Tune time delay and coherence length to watch coincidences vanish at zero delay.
Mean-Field Game: Crowd Motion
Rational agents minimize cost that depends on the crowd density, forming Nash equilibria via HJB + Fokker-Planck. Three modes: scatter, exit bottleneck, circular flow.
φ⁴ Kink Soliton
Topological solitons in the φ⁴ field theory: kinks, antikinks, breathers, and collisions. Exact tanh profile, energy density visualization, and kink-antikink annihilation.
Diffusion-Limited Aggregation
Random walkers stick to a growing cluster, producing fractal dendrites with dimension df ≈ 1.71. Tune sticking probability, color by age or distance, estimate fractal dim live.
Nonreciprocal Coupling & Sync
Asymmetric Kuramoto oscillators with K₊ ≠ K₋ break time-reversal symmetry, enabling traveling waves and exceptional-point bifurcations. Order parameter and phase velocity tracked live.
3-Qubit Quantum Error Correction
The three-qubit bit-flip code encodes α|000⟩+β|111⟩. Flip any qubit, measure syndromes (Z₀Z₁, Z₁Z₂), and watch majority-vote correction restore the logical state.
Polymer Brush Steric Repulsion
End-grafted polymer chains simulated with bead-spring Langevin dynamics. Alexander-de Gennes steric repulsion force grows as a wall compresses the brush: f ∝ (H/D)^(9/4).
Fractional Brownian Motion & Hurst
Generate fBm paths for any Hurst exponent H and estimate it via DFA and R/S analysis. H>0.5 gives persistent trends; H<0.5 gives anti-persistent mean-reverting paths.
Alfvén Wave MHD Dispersion
Magnetohydrodynamic wave modes — shear Alfvén, fast and slow magnetosonic — visualized as perturbed field lines. Dispersion relation ω(k) plotted for any propagation angle θ.
SIR Metapopulation Epidemic
Coupled SIR patches with inter-city mobility. Spatial heterogeneity drives traveling epidemic waves. Tune β, γ, and mobility; vaccinate cities; seed outbreaks and watch R₀ govern spread.
Navier-Stokes Vorticity-Streamfunction
2D incompressible flow via vorticity-streamfunction formulation. Add vortices, watch ∂ω/∂t + u·∇ω = ν∇²ω evolve. Vortex pairs orbit, merge, and cascade — Poisson solver closes the system.
Quantum Hall Composite Fermions
Fractional quantum Hall effect visualized through composite fermions. Jain sequences ν=p/(2p±1), flux attachment, and Landau level filling animate the emergence of σ_xy = νe²/h.
Abelian Sandpile & SOC
Drop grains; watch toppling avalanches cascade across the grid. The abelian group of recurrent configurations produces fractal self-organized criticality — toppling order never matters.
Chimera States · Nonlocal Kuramoto
Identical phase oscillators on a ring with nonlocal coupling spontaneously split: half synchronized, half chaotic. Space-time plot reveals the coexisting coherent and incoherent domains.
Shockwave & Rankine-Hugoniot
1D compressible flow with normal shock. Tune Mach number and γ; watch density, velocity, and pressure jump discontinuously. Exact Rankine-Hugoniot ratios computed in real time.
Avoided Crossing & Landau-Zener
Two-level quantum system swept through avoided crossing. Landau-Zener formula P=exp(−2πΔ²/v) governs adiabatic vs diabatic transitions. Bloch sphere tracks the quantum state live.
Metabolic Network Flux Balance
Animated central carbon metabolism network. Flux balance analysis finds steady-state fluxes (S·v=0) that maximize biomass. Glycolysis, TCA cycle, ETC, and fermentation overflow respond to enzyme constraints.
Rossby Waves & Beta-Plane Dispersion
Atmospheric Rossby waves on a beta-plane. Dispersion ω=−βk/(k²+l²+λ²) gives westward phase velocity but potentially eastward group velocity. Tune wavenumber and Rossby radius.
Stochastic Resonance · IF Neuron
Sub-threshold signal plus noise drives an integrate-and-fire neuron. At optimal noise the SNR peaks — stochastic resonance. Scan noise σ to trace the full SNR curve and find the sweet spot.
Heisenberg Spin Chain & Bethe Ansatz
1D XXZ Heisenberg chain solved exactly via Bethe Ansatz (1931). Magnon spin waves propagate with dispersion E(k)=J(1−cos k). Animate magnons, see quantized k-values and the eigenenergy spectrum.
Forest Fire SOC — Drossel-Schwabl
Self-organized criticality with no tuning needed. Trees grow, lightning sparks fires, cascades burn. Watch power-law fire-size distributions emerge in real time on a live grid.
Quantum Walk — Hadamard Coin
Discrete quantum walk on a line using the Hadamard coin operator. Ballistic spread σ∝t vs classical diffusion σ∝√t. Toggle decoherence to watch interference dissolve into classical noise.
Hertz Contact Mechanics
Two elastic spheres in contact. Interactive load control shows the semi-ellipsoidal pressure profile p(r)=p₀√(1−r²/a²), contact radius a∝F^(1/3), and subsurface stress field in real time.
FitzHugh-Nagumo Excitable Waves
2D reaction-diffusion on a grid. Trigger spiral waves, target patterns, and excitable pulses — a simplified neuron model that shows how cardiac waves propagate without decay.
Geodesic Deviation — Jacobi Equation
Neighboring geodesics on sphere, hyperbolic, torus, and flat surfaces. Curvature K drives convergence (K>0), linear spread (K=0), or exponential chaos (K<0) via the Jacobi field J′′+KJ=0.
1D Ising — Exact Transfer Matrix
The 1D Ising chain solved exactly via transfer matrix Z=Tr(T^N). Plot free energy, magnetization, and correlation length ξ=−1/ln(λ₋/λ₊) as functions of T. No phase transition — only at T→0.
Coupled Logistic Map Lattice — Kaneko
Kaneko's CML: lattice of coupled logistic maps showing spatiotemporal chaos, frozen patterns, and coherent structures. Space-time diagram reveals the turbulence analogy in (r, ε) parameter space.
HP Lattice Protein Folding
Lau-Dill HP model: hydrophobic (H) and polar (P) residues on a 2D lattice. Monte Carlo with Metropolis criterion folds the chain. Hydrophobic core collapses to minimize HH contact energy.
Thouless Pump — Quantized Charge
Rice-Mele model with adiabatic cycling. Exactly one quantum of charge pumped per cycle — a topological invariant (Chern number C=1). Watch Wannier centers march across the lattice.
Granular Flow — Bagnold Scaling
Particles rolling down an incline with hard-disk collisions. Bagnold's inertial regime: stress σ∝(dv/dy)² instead of σ∝dv/dy. Tune incline angle, restitution, and viscosity to see regime transitions.
Symplectic Integrator: Störmer-Verlet
Compare symplectic vs non-symplectic integrators for Hamiltonian systems — energy drift, phase space area preservation
Entropy Production: Nonequilibrium Steady State
Two-reservoir particle system; measure entropy production rate and verify Gallavotti-Cohen fluctuation theorem
Quantum Tunneling: WKB Barrier Decay
Wavepacket tunneling through a potential barrier; transmission coefficient T ≈ exp(−2∫κdx) vs barrier height and width
Active Matter: Vicsek Flocking Transition
Self-propelled particles aligning with neighbors; polar order parameter vs noise — flocking, disorder, and band formation
Fractal Basin Boundaries: Duffing Oscillator
Color-coded basins of attraction for the driven Duffing oscillator — fractal Wada boundaries and sensitive dependence
Topological Insulator: SSH Model Edge States
Su-Schrieffer-Heeger chain — topological phase transition, winding number, protected zero-energy edge states
Lévy Walk: Anomalous Superdiffusion
Power-law step lengths produce superdiffusive MSD ~ t^α; compare with Brownian motion and measure α vs μ
Hyperbolic PDE: Method of Characteristics
Wave equation solved via d'Alembert's formula — characteristic lines, domain of dependence, interactive initial conditions
Spin Squeezing: Quantum Metrology
One-axis twisting on the Bloch sphere — quantum Fisher information, Wineland parameter, metrological gain beyond SQL
Excitable Medium: Spiral Wave Breakup
Barkley model — spiral waves, tip trajectory, and transition to spatiotemporal chaos at high excitability
Directed Percolation: Contact Process
Absorbing-state phase transition on a lattice — DP universality class, critical infection threshold, order parameter, and spacetime diagram
Quantum Eraser: Which-Path Interference
Double-slit complementarity — which-path information destroys fringes; erasing it restores them. Englert-Greenberger duality V²+D²≤1
Colloidal Glass Transition
2D soft-core particles — packing fraction drives glass transition, cage trapping, MSD plateau, non-Gaussian parameter, dynamic heterogeneity
Neural Field Theory: Amari Bump
Amari equation with Mexican-hat connectivity — bump attractor states, traveling fronts, and working memory dynamics in a continuous neural field
MHD Alfvén Waves: Solar Corona
Transverse MHD waves on magnetic field lines — Alfvén speed v_A=B/√(μ₀ρ), shear vs compressional modes, coronal heating context
Gillespie Algorithm: Stochastic Kinetics
Exact SSA for chemical reactions A→B→C — stochastic vs ODE trajectories, exponential waiting times, propensity-based event selection
XY Model: Kosterlitz-Thouless Transition
2D XY model vortex visualization — topological KT transition, bound vortex-antivortex pairs below T_KT, helicity modulus, free vortex proliferation
Erdős-Rényi: Giant Component
G(n,p) random graph — giant component emerges at p=1/n, order parameter S/n vs c, Poisson degree distribution, animated phase transition
Wigner Function: Quantum Phase Space
Phase-space quasi-distribution — coherent, Fock, cat, and squeezed states, negative regions signal non-classicality, Moyal bracket dynamics
Crystal Growth: Kinetic Monte Carlo
KMC surface growth — deposition, diffusion, desorption; step-flow vs island nucleation modes, roughness exponent, KPZ universality
KdV Soliton Collision
Two-soliton exact solution: faster wave passes through slower one unchanged — only a phase shift remains. Nonlinear balance of steepening and dispersion.
Stadium Billiard Quantum Scars
Eigenfunctions of the chaotic Bunimovich stadium show enhanced density along unstable periodic orbits — quantum scars imprinted by classical chaos.
SIR on Heterogeneous Networks
Scale-free networks have epidemic threshold β_c → 0 as N→∞: hubs act as super-spreaders enabling persistence at vanishingly small transmission rates.
Rule 110 — Universal CA
Elementary cellular automaton proven Turing complete by Cook (2004). Gliders, collisions, and complex structures emerge from a single 3-cell lookup table.
Spin-Boson Quantum Dissipation
Two-level system in a bosonic bath: quantum-to-classical crossover, decoherence, and localization transition at coupling α=1 for the Ohmic spectral density.
Rayleigh-Bénard Convection
Thermal instability onset at Ra_c=1708: buoyancy overcomes viscosity and diffusion, forming convection rolls at the critical wavenumber k_c=π/√2.
Information Bottleneck
The I(T;Y) vs I(T;X) Pareto frontier: optimal compression that retains maximal relevance. Blahut-Arimoto iteration traces the trade-off curve for any β.
Magnetic Reconnection
Harris current sheet: X-point formation changes field-line topology, releases magnetic energy, and jets plasma outward at the Alfvén speed. Solar flares in miniature.
Moran Process Fixation
Birth-death dynamics in a finite population: ρ=(1−1/r)/(1−1/rᴺ). Fixation probability, time distributions, and the drift-selection crossover visualized.
Standard Map — KAM Theory
Chirikov standard map phase portrait: KAM tori, resonance islands, and chaotic seas. Last invariant torus breaks at k_c≈0.9716 — click to trace orbits.
Jarzynski Equality
Drag a particle through a potential at different speeds — verify ⟨e^{−βW}⟩=e^{−βΔF} and watch work distributions.
Berry Curvature & Chern Number
Visualize Berry curvature F_xy(k) in the Brillouin zone of the QWZ model and compute topological Chern numbers.
Brusselator Oscillator
Chemical limit cycles, Hopf bifurcation, and Turing spatial patterns from the canonical Brusselator reaction network.
Quantum Decoherence & Einselection
Watch off-diagonal density matrix elements decay on the Bloch sphere as environment selects pointer states.
Sandpile Identity Element
Compute and visualize the fractal additive identity of the abelian sandpile group — a self-similar harmonic structure.
Swift-Hohenberg Patterns
Stripe, hexagon, and defect dynamics from the Swift-Hohenberg PDE — tune ε through the pattern-forming bifurcation.
Voter Model & Zealots
Opinion coarsening on a 2D lattice — zealot minorities block consensus and pin non-trivial steady states.
Plasma Wave Dispersion
Bohm-Gross dispersion of Langmuir waves, ion-acoustic modes, and Landau damping from wave-particle resonance.
Wright-Fisher Genetic Drift
Allele frequency trajectories under drift and selection — fixation probability equals initial frequency (neutral case).
Kelvin-Helmholtz Instability
Vortex sheet roll-up at a shear flow interface — watch cat's-eye vortices form with growth rate σ=k|ΔU|/2.
Coupled Oscillator Normal Modes
N-mass spring chain: normal mode decomposition, beat phenomena, and energy exchange between modes. Interactive coupling and initial conditions.
Kitaev Honeycomb Spin Liquid
Exactly solvable quantum spin liquid on a honeycomb lattice — Majorana fermions, vison flux excitations, and the A/B phase diagram.
Diffusion & Green's Function
Heat equation on a 1D rod: watch the Gaussian Green's function G(x,t)=e^{−x²/4Dt}/√(4πDt) evolve, superposition, and entropy growth.
Gene Network Feedforward Loop
Type-1 coherent FFL: sign-sensitive delay, pulse generation, and mRNA/protein dynamics for gene regulatory network motifs.
Atmospheric Gravity Waves
Internal gravity waves in a stratified atmosphere — group velocity perpendicular to phase velocity, wave beams, and buoyancy frequency N.
Ostwald Ripening & Coarsening
Cahn-Hilliard phase separation: small droplets shrink, large ones grow. LSW coarsening law ⟨R⟩~t^{1/3} visualized in real time.
Jaynes-Cummings Vacuum Rabi
Two-level atom in a cavity: vacuum Rabi oscillations, collapse and revival from photon number distribution, Fock state dynamics.
Selection-Drift Balance
Kimura diffusion equation — allele frequency trajectories, stationary distribution, fixation probability, and the Ns criterion for evolution.
Turbulent Boundary Layer
Log law of the wall: u⁺=(1/κ)ln(y⁺)+B with κ≈0.41. Viscous sublayer, buffer layer, log region — compare laminar vs turbulent profiles.
Stochastic Block Model
Planted partition communities and the Kesten-Stigum detectability threshold — spectral gap, belief propagation, and community phase transition.
Adiabatic Elimination
Fast/slow variable decomposition and center manifold reduction — slaving principle, compare full vs reduced dynamics, watch 2D system collapse to slow manifold.
Quantum Phase Estimation
Kitaev QPE algorithm — controlled-U^{2^k} gates, inverse QFT, Dirichlet kernel probability distribution sharpening with more ancilla qubits.
Cell Polarity Wave-Pinning
Mori-Jilkine-Edelstein-Keshet wave-pinning model — bistable RD system where a wave pins at the cell boundary, creating stable front polarity.
Box-Counting Fractal Dimension
Measure D_f via N(ε) ~ ε^{-D} for Sierpinski triangle, Koch curve, Cantor dust, and dragon curve — animated log-log regression and live box overlay.
Point Vortex Dynamics
Kirchhoff Hamiltonian equations for N point vortices — leapfrogging pairs, chaotic 3-vortex problem, symmetric rings, and streamline visualization.
Landau Theory & Phase Transitions
Free energy F=am²+bm⁴+cm⁶ — 2nd vs 1st order transitions, metastability, hysteresis loop, Maxwell construction, and interactive phase diagram.
Adaptive Network Coevolution
Opinion-topology coevolution — agents rewire or adopt opinions, fragmentation transition to echo chambers, real-time network spring layout.
Parametric Oscillator & Arnold Tongues
Mathieu equation ẍ+[δ+ε·cos(2t)]x=0 — stability tongue map, parametric amplification, phase portrait, and resonance band visualization.
Information Geometry
Fisher-Rao metric on the Gaussian manifold — KL divergence heatmap, geodesics as hyperbolic half-plane arcs, information vs Euclidean distance.
Epidemic Wavefront — Fisher-KPP
Spatial SIR wave with diffusion — Fisher-KPP minimum speed c*=2√(Dr), pulled vs pushed waves, live wavefront tracking and speed measurement.
Quantum Graph State Computation
Cluster state measurement-based quantum computing — entanglement graph, single-qubit measurements propagating logical operations, teleportation gadget animated.
Eden Model & KPZ Roughening
Bacterial colony growth via Eden model — KPZ universality class, roughness exponent α≈0.5, growth exponent β≈1/3, live log-log scaling measurement.
Chemical Chaos: Oregonator & Rössler
Belousov-Zhabotinsky reaction chaos — 3D phase portrait, Poincaré section revealing fractal structure, Lyapunov exponent, compare with Rössler attractor.
Collective Motion & Flocking Transition
Minimal flocking model with noise threshold — polar order parameter, sharp phase transition, susceptibility peak, asynchronous Vicsek-like dynamics.
Sine-Gordon Topological Solitons
φ_tt − φ_xx + sin(φ) = 0 — kink, antikink, breather formation, kink-kink repulsion, exact analytical solutions animated with energy density.
Interdependent Network Cascade Failure
Buldyrev 2010 catastrophic collapse — coupled networks fail mutually causing first-order discontinuous transition vs isolated network gradual percolation.
Second Harmonic Generation
χ⁽²⁾ nonlinear crystal optics — coupled wave equations, phase matching Δk=0, conversion efficiency vs mismatch, quasi-phase matching with periodic poling.
MAPK Cascade Bistability
Goldbeter-Koshland zero-order ultrasensitivity — MAPK dual phosphorylation bistability, Hill coefficient >> 1, signal-response curves and bifurcation diagram.
Quantum Many-Body Scars (PXP)
Rydberg atom PXP model — scar states with persistent fidelity revivals, entanglement entropy growth, eigenvalue spectrum showing ETH violation.
Dendrite Crystal Growth
Snowflake solidification via phase-field model — Mullins-Sekerka instability, six-fold anisotropy, interactive undercooling and anisotropy strength.
Kitaev Honeycomb Spin Liquid
Exactly solvable spin liquid with Majorana fermions, Z₂ fluxes, and non-Abelian anyons.
OTOC: Quantum Scrambling
Out-of-time-order correlators measuring quantum chaos, scrambling time, and the MSS Lyapunov bound.
Fokker-Planck Equation
Probability density evolution with drift and diffusion toward Boltzmann stationary distributions.
Lieb-Liniger Bose Gas
Exactly solvable 1D Bose gas via Bethe ansatz: from BEC to Tonks-Girardeau fermionization.
Poincaré Disk: Hyperbolic Geometry
Hyperbolic geodesics, {p,q} tilings, and horocycles in the Poincaré disk model.
Tropical Max-Plus Algebra
Max-plus semiring, tropical curves, dequantization, and shortest-path matrix multiplication.
Dehn Functions & Word Problem
Filling area vs word length in groups: linear (hyperbolic), quadratic (Euclidean), to exponential.
Rossby Wave Propagation
Planetary-scale atmospheric waves on the beta-plane with phase speed, group velocity, and blocking.
Vortex Reconnection & Helicity
Topological helicity change during 3D vortex tube reconnection: writhe-twist exchange.
Marcus Electron Transfer
Marcus theory rates, reorganization energy, activation barrier, and the inverted region.
Fiber Bundle Holonomy
Parallel transport and holonomy on principal bundles: Berry phase as geometric curvature.
KPZ Directed Polymer
Free energy fluctuations of a directed polymer converge to the Tracy-Widom GUE distribution.
Quantum Hall Edge States
Chiral edge modes, Landauer conductance quantization, and skipping orbits in a magnetic field.
BKT Vortex Transition
Topological phase transition via vortex-antivortex unbinding in the 2D XY model.
Anderson Localization
1D transfer matrix, Lyapunov exponents, and exponential localization in random potentials.
Steepest Descent Method
Complex saddle points, asymptotic integrals, and the steepest descent contour deformation.
Kapitza Pendulum
Parametric stabilization of an inverted pendulum by rapid pivot oscillation and effective potential.
Bogoliubov Quasiparticles
BCS gap equation, Bogoliubov transformation, and the density of states with coherence peaks.
Interval Exchange Maps
Ergodicity and Rauzy induction for piecewise isometric interval exchange transformations.
Orr-Sommerfeld Stability
Viscous flow stability, critical Reynolds number, and Tollmien-Schlichting instability eigenmodes.
Bethe Lattice Percolation
Exact mean-field percolation on the Bethe lattice: giant component, branching process, critical exponents.
Topological Defects & Homotopy
Vortices, strings, monopoles, and Skyrmions classified by homotopy groups π₁, π₂, π₃ of order-parameter spaces.
Fisher-KPP Wavefront
Epidemic wave propagation: pulled vs pushed fronts, minimum speed selection c*=2√(rD), spacetime diagram.
XXZ Spin Chain
Phase diagram from ferromagnet to Néel antiferromagnet, Luttinger liquid, correlation functions via Bethe ansatz.
Geodesic Deviation & Tidal Forces
Jacobi equation, spaghettification near black holes, gravitational wave tidal stretching, Riemann tensor components.
FTLE & Lagrangian Coherent Structures
Finite-time Lyapunov exponents reveal hidden transport barriers in unsteady flows: double gyre, Bickley jet.
GUE Eigenvalue Spacing
Wigner surmise, level repulsion, and the semicircle law for GOE/GUE/GSE random matrix ensembles.
Magnetic Reconnection
Sweet-Parker current sheet, Alfvén speed, Lundquist number scaling, and why solar flares need faster reconnection.
Diffusion-Limited Aggregation
DLA cluster growth via random walk aggregation, fractal dimension ~1.71, Witten-Sander Laplacian growth model.
Hénon-Heiles Poincaré Section
KAM tori vs chaotic sea in Hénon-Heiles potential and Chirikov standard map, phase-space structure at varying energy.
Replica Trick & Spin Glass
Disorder-averaged free energy via replica method, Parisi overlap matrix, RSB vs RS phases.
Motility-Induced Phase Separation
Run-and-tumble particles self-trap into dense clusters via MIPS — no attraction needed.
Topological Band Inversion
Berry phase winding, Z₂ invariant, and edge state emergence across a topological transition.
Kohn-Sham DFT: SCF Loop
Self-consistent field iterations converging electron density with LDA exchange-correlation.
Non-Hermitian Skin Effect
Asymmetric hopping collapses all eigenstates to one edge; bulk spectrum winding number detects it.
Granular Gas & Haff's Law
Inelastic hard-disk gas cools as T ~ (1+t/τ)⁻², with clustering instability and non-Gaussian tails.
Cross-Newell Phase Equations
Phase dynamics of stripe patterns, Eckhaus instability, defect formation, and grain boundaries.
Gaussian Chain Field Theory
End-to-end distribution, Debye structure factor, and Rouse mode relaxation spectrum of polymers.
Vicsek Model: Topological Flocking
k-nearest-neighbor alignment in the Vicsek model; order-disorder transition, polar vs nematic.
Active Brownian Particle Alignment
Polar and nematic order in ABP systems with alignment, defect dynamics, and motility contrast.
Jarzynski & Crooks Fluctuation Theorems
Non-equilibrium work distributions, free energy estimation, and the Jarzynski equality from Langevin trajectories.
Catch Bonds in Cell Adhesion
Force-dependent bond lifetime: Bell slip bonds vs Dembo catch-bond models and receptor cluster dynamics.
Kinematic Dynamo
Magnetic field amplification by chaotic flow — Roberts and Ponomarenko dynamos, growth rate vs Rm.
Rule 110 — Turing-Complete CA
Wolfram Class IV cellular automaton with gliders, persistent structures, and proven Turing completeness.
Erdős-Rényi Giant Component
Phase transition at c=np=1: giant component emergence, component size distribution, and percolation threshold.
Rayleigh-Bénard Convection Onset
Convection rolls above critical Ra, Nusselt number scaling, and the viscous-to-turbulent transition.
Gibbs-Duhem & Chemical Potential
Thermodynamic consistency in mixtures: chemical potential surfaces, spinodal decomposition, and the Gibbs-Duhem constraint.
Turbulent Boundary Layer Log-Law
Von Kármán constant, viscous sublayer, buffer layer, and outer region velocity profiles in wall-bounded turbulence.
Quantum Discord vs Entanglement
Classical vs quantum correlations in two-qubit mixed states: mutual information, concurrence, and discord beyond entanglement.
Spectral Graph Bisection
Fiedler vector, algebraic connectivity λ₂, and spectral bisection quality vs random cuts across graph types.
Information Bottleneck
Rate-distortion tradeoff and the IB curve: compressing X while preserving relevant information about Y, with β-controlled Pareto frontier.
Hyperbolic Cayley Graphs
Cayley graphs of hyperbolic groups in the Poincaré disk: word metric, Gromov boundary, and exponential volume growth of balls.
Hamilton-Jacobi-Bellman Equation
Optimal stochastic control via dynamic programming: value function iteration, optimal feedback policy, and the HJB PDE.
Worm Algorithm
Efficient cluster updates via open worldlines in the XY loop model: winding numbers, superfluid stiffness, and the KT transition.
3D Turing Instability on Sphere
Schnakenberg reaction-diffusion on a sphere surface: spot and stripe competition, spherical harmonic modes, and Turing bifurcation.
Critical Slowing Down
Relaxation time divergence near the 2D Ising critical point: autocorrelation decay, integrated time τ_int, and dynamic scaling z≈2.
Living Polymers & Micelles
Wormlike micellar kinetics: exponential length distribution, scission-recombination equilibrium, and Maxwell viscoelastic relaxation.
Minority Game
Market ecology with adaptive agents: attendance oscillations, phase transition at α_c≈0.34, and volatility minimum at optimal complexity.
Quantum Zeno Effect
Measurement-induced freezing of quantum decay: survival probability, effective decay rate, and the anti-Zeno acceleration regime.
Chinese Remainder Theorem
Simultaneous congruences visualized as grid intersections: CRT reconstruction, modular inverses, and RSA-CRT cryptographic speedup.
Quantum Zeno Effect
Frequent measurement freezes quantum decay: watch how observation rate changes survival probability through the Zeno and anti-Zeno regimes.
Mpemba Effect
Hot water sometimes freezes faster than cold — explore Newton cooling with evaporation to see when and why this paradox occurs.
SSH Topological Insulator
The Su-Schrieffer-Heeger chain: edge states, winding number, and the bulk-boundary correspondence of the simplest 1D topological phase.
Rule 110 — Turing Complete CA
The simplest known Turing-complete system: three trivial update rules generate universal computation from any initial seed.
Bloch Oscillations
Electrons in a periodic lattice under a uniform field oscillate rather than accelerate — the Wannier-Stark ladder and Bloch period T_B = h/eFa.
Kicked Quantum Rotator
Quantum resonance vs. dynamical localization: periodic delta-kicks cause ballistic energy growth at T=4πn, Anderson localization elsewhere.
Landau-Zener Transition
Sweep a two-level system through an avoided crossing: slow sweeps follow the adiabatic path; fast sweeps cause diabatic transitions P=e^{-2πΩ²/α}.
Quantum Tunneling
Wave packets penetrate classically forbidden barriers: visualize evanescent decay, transmission coefficients, and resonance above the barrier.
Holevo Bound
The quantum limit on classical information: χ = S(ρ̄) − Σpᵢ S(ρᵢ) as a function of state overlap, prior, and depolarizing noise.
Jaynes-Cummings Model
Atom-cavity quantum optics: vacuum Rabi oscillations, collapse and revival, and atom-field entanglement entropy from a single photon mode.
Fano Resonance
Discrete-continuum interference produces the asymmetric Fano lineshape σ ∝ (q+ε)²/(1+ε²) — from atomic physics to nanophotonics.
Schwinger Pair Production
Strong electric fields spontaneously create electron–positron pairs from the QED vacuum near the critical Schwinger field E_S ≈ 1.3×10¹⁸ V/m.
Magnon Dispersion
Quantized spin waves in a ferromagnet follow ω(k) = Dk², a quadratic Goldstone dispersion arising from broken rotational symmetry.
Lieb–Robinson Bound
Non-relativistic quantum lattices have an effective speed of light — the Lieb–Robinson bound defines a causal cone without special relativity.
Superconductor Proximity Effect
Cooper pairs leak from a superconductor into an adjacent normal metal, inducing an exponentially decaying gap Δ(x) ∝ e−x/ξ.
Peierls Instability & Dimerization
A 1D metal spontaneously dimerizes to open a band gap at k = ±π/2a, lowering electronic energy at the cost of elastic distortion.
Shilnikov Saddle-Focus Chaos
When ρ/λ < 1 near a homoclinic orbit, Shilnikov's theorem guarantees infinitely many unstable periodic orbits and chaotic dynamics.
Kitaev Honeycomb Model
Kitaev's exactly solvable honeycomb spin model hosts Z₂ flux excitations (visons) and Majorana fermions with topological phases.
Abelian Sandpile Identity
The identity element of the abelian sandpile group has a striking fractal structure, emerging from iterative toppling rules on a 2D grid.
Yu–Shiba–Rusinov States
A magnetic impurity in a superconductor creates sub-gap YSR bound states at ±ε = ±Δ(1−α²)/(1+α²), with a quantum phase transition at α=1.
Composite Fermions & FQHE
Jain's composite fermion theory maps fractional QHE at ν = p/(2p±1) to integer QHE of electrons bound to magnetic flux quanta.
Kondo Impurity Scattering
Magnetic impurities cause a logarithmic upturn in resistivity at low temperature, competing with phonon suppression to produce the famous Kondo resistance minimum.
Rashba Spin-Orbit in 2DEG
Structural inversion asymmetry splits the Fermi surface into two spin-helical rings where electron spin locks perpendicular to momentum — the Rashba effect.
Weyl Node Fermi Arcs
Weyl semimetals host topologically protected surface states called Fermi arcs that connect projections of bulk Weyl nodes of opposite chirality in the surface BZ.
Majorana Fermions in Topological Wire
The Kitaev chain model predicts unpaired Majorana zero modes at wire ends when |μ| < 2|t|, offering a route to topological quantum computing qubits.
Casimir Effect — Quantum Vacuum
Two parallel conducting plates attract each other via quantum vacuum fluctuations — only discrete modes fit between the plates, creating a radiation pressure imbalance F/A = −πħc/240d⁴.
Skyrmion Topological Hall Effect
A magnetic skyrmion's topologically nontrivial spin texture (winding number Q=−1) creates an emergent magnetic field that deflects conduction electrons — the topological Hall effect.
Magneto-Optical Kerr Effect
Light reflected from a magnetized surface undergoes polarization rotation by the Kerr angle θ_K ∝ M, enabling optical imaging of magnetic domains and reading of magnetic storage.
Spin Seebeck Effect
A temperature gradient across a magnetic insulator drives a magnon spin current detected via the inverse spin Hall effect, converting heat directly into a transverse spin voltage.
Quantum Anomalous Hall & Axion
Magnetically doped topological insulators exhibit quantized Hall conductance σ_xy = Ce²/h without external field, governed by the axion term θ = π of topological electrodynamics.
Mott-Hubbard Transition
When Coulomb repulsion U exceeds bandwidth W, the half-filled Hubbard band splits into upper and lower Hubbard bands, driving a correlation-driven metal-insulator transition.
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Sandpile Avalanche Scaling
Self-organized criticality in the abelian sandpile: avalanche sizes follow a power law P(s) ∝ s⁻¹·²⁵ — no tuning required, the system organizes itself to the critical point.
Kuramoto Synchronization Transition
N coupled oscillators with Lorentzian frequency spread undergo a second-order phase transition at K_c = 2γ — the order parameter r jumps from zero to collective synchrony.
Viscous Fingering — Saffman-Taylor
A less viscous fluid displacing a more viscous one in a Hele-Shaw cell develops branching fingering instabilities governed by Darcy's law and surface tension selection.
XOR-SAT Phase Transition
Random 3-XOR-SAT has an exact satisfiability threshold at α_c = 1 (clauses/variables), provable via linear algebra over GF(2) — a sharp computational phase transition.
Floquet Topological Anomalous Hall
Periodic driving engineers topologically non-trivial Floquet bands with Chern number C = ±1 and quantized Hall conductance σ_xy = Ce²/h — topology from time-periodic Hamiltonians.
KPZ Surface Growth & Tracy-Widom
The Kardar-Parisi-Zhang equation governs stochastic interface growth; stationary fluctuations follow the Tracy-Widom GUE distribution with roughness exponent α = 1/2, z = 3/2.
Random Walk on Sierpiński Gasket
Diffusion on fractal lattices is anomalous: ⟨r²⟩ ~ t^(2/d_w) with d_w = log5/log2 ≈ 2.32 and spectral dimension d_s ≈ 1.365 — subdiffusion from fractal topology.
Hubbard Dimer Mott Crossover
Exact diagonalization of the two-site Hubbard model reveals the Mott metal-insulator crossover: double occupancy collapses and a charge gap opens as U/t increases past ~3.
Stochastic Resonance
Optimal noise enhances weak signal detection in a double-well potential: the SNR peaks at a non-zero noise level σ*, demonstrating that noise can improve signal transmission.
Frustrated Magnets — Kagome Spin Liquid
Antiferromagnetic interactions on the kagome lattice are geometrically frustrated, producing extensive ground-state degeneracy and a quantum spin liquid with fractionalized excitations.
Maxwell's Demon
Szilard engine and Landauer's principle: sorting particles costs information entropy.
Wishart Ensemble
Marchenko-Pastur law for random covariance matrix eigenvalues and GUE semicircle.
Zeta Zeros & Montgomery
Riemann zeta non-trivial zeros and their GUE pair correlation — random matrix universality.
Replica Trick & RSB
Parisi's replica symmetry breaking solution for spin glass free energy and disorder.
Turbulence Intermittency
K41 energy cascade, She-Leveque multifractal exponents, and anomalous scaling.
Ecological Regime Shifts
Fold bifurcation, alternative stable states, and critical slowing down near tipping points.
Taylor-Couette Flow
Hydrodynamic stability, Taylor vortices, and the route to turbulence between cylinders.
CFT Ising Correlators
c=1/2 Ising CFT two-point functions, conformal blocks, and OPE structure.
Quantum Discord
PPT separability criterion, concurrence, mutual information and discord hierarchy.
GL Vortex Nucleation
Abrikosov vortex lattice, Type-I/II superconductors, and magnetic flux quantization.
Kauzmann Paradox
Supercooled liquid entropy races toward zero at the Kauzmann temperature — a thermodynamic crisis averted only by the glass transition.
Arrow of Time
Entropy production drives irreversibility: watch gas particles diffuse and explore time-reversal violation through the Boltzmann H-theorem.
Protein Folding Funnel
Proteins navigate a rugged free-energy funnel landscape; the native state sits at the global minimum accessible without Levinthal's astronomical search.
Neutrino Oscillations
Quantum interference between mass eigenstates causes neutrino flavors to oscillate over cosmic distances — tunable via the PMNS mixing matrix.
Cosmic String Network
One-dimensional topological defects from symmetry-breaking phase transitions form a scale-invariant network that radiates gravitational waves.
Kibble-Zurek Mechanism
Quenching through a phase transition inevitably creates topological vortex defects; the density follows a universal power law in the quench rate.
Entanglement Monogamy
Entanglement is monogamous: the Coffman-Kundu-Wootters inequality bounds how quantum correlations can be shared across multiple parties.
Bose Glass Phase
Disordered bosons on a lattice exhibit three phases: Mott insulator, Bose glass, and superfluid — the glass always intervenes between the other two.
Percolation Giant Component
At the Erdős-Rényi threshold p_c = 1/N, a giant connected component emerges abruptly — mean-field critical exponents govern this phase transition.
Hyperuniform Sphere Packing
Jammed sphere packings suppress long-wavelength density fluctuations S(k→0)→0 — a hidden order shared with crystals but without periodicity.
DMFT: Mott Transition
Dynamical mean-field theory maps correlated lattice electrons onto a self-consistent impurity, capturing the Mott metal-insulator transition and Hubbard bands in the spectral function.
Floquet Driven Qubit
A periodically driven two-level system exhibits Rabi oscillations, dressed states, and Autler-Townes splitting — watch the Bloch vector precess as you tune frequency and amplitude.
Quantum Spin Liquid
Geometric frustration on the triangular lattice prevents magnetic order at T=0. Explore the spin structure factor's diffuse scattering — hallmark of a quantum spin liquid ground state.
Active Gel: Cytoskeleton
Actin filaments driven by myosin motors form an active gel that generates contractile stress, flows spontaneously, and self-organizes — the physics behind cell motility and shape change.
Rigidity Percolation
Random spring networks undergo a rigidity transition at the Maxwell isostatic point: below threshold, floppy mechanisms dominate; above, the network becomes an elastic solid.
Thermal Hall: Magnon-Phonon
In magnetic insulators, coupled magnon-phonon modes under broken time-reversal generate a transverse heat current — the thermal Hall effect — via Berry curvature of hybridized bands.
Topological Magnon Bands
Dzyaloshinskii-Moriya interaction on a honeycomb ferromagnet gives magnon bands a non-zero Chern number, producing chiral edge modes immune to backscattering — magnon Hall effect.
Fibonacci Cantor Spectrum
The Fibonacci chain — a 1D quasicrystal — has an energy spectrum forming a Cantor set of measure zero with fractal dimension φ−1 ≈ 0.618 and critical (multifractal) eigenstates.
Quasicrystal Diffraction
Penrose and octagonal quasicrystals produce sharp X-ray diffraction peaks with 5- or 8-fold symmetry forbidden in periodic crystals — sharp order without periodicity.
Phononic Crystal Band Gap
Periodic bilayer materials create acoustic band gaps where sound cannot propagate — explore dispersion, transmission, and wave attenuation using transfer matrix theory.
Topological Soliton — Kink in φ⁴ Theory
Stable kink solutions connecting degenerate vacua, topological charge, kink–anti-kink collisions
WKB Tunneling — Double Well
Quantum tunneling through classically forbidden barriers, WKB transmission coefficient, instanton action
Voter Model with Zealots
Opinion dynamics on 2D lattice with committed minorities — when do zealots prevent consensus?
FitzHugh-Nagumo Pulse Propagation
Excitable pulse propagation in 1D, neuron-like refractory periods, click to inject stimuli
Rouse Polymer Normal Modes
Gaussian chain dynamics, normal mode decomposition, τ_p ∝ N²/p² relaxation spectrum
Critical Opalescence — Light Scattering
Ornstein-Zernike structure factor, diverging correlation length ξ ~ |T−Tc|^−ν near critical point
Standard Map — Hamiltonian Chaos
Chirikov standard map Poincaré section — KAM tori dissolve into chaos as kick strength K increases
Quantum Zeno Effect
Frequent measurements suppress quantum decay — the watched pot that never decays
Thermodynamic Cycles — Carnot, Otto, Stirling
P-V and T-S diagrams for heat engine cycles, efficiency comparison against Carnot limit
Zitterbewegung — Dirac Free Particle
Rapid trembling oscillation of a relativistic electron from positive/negative energy interference
SSH Chain — Topological Edge States
Su-Schrieffer-Heeger model, winding number topology, protected zero-energy edge modes
Shilnikov Saddle-Focus Chaos
Homoclinic chaos near a saddle-focus equilibrium — Smale horseshoes and the Shilnikov condition λ<ρ
Kitaev Honeycomb — Majorana Fermions
Honeycomb spin model with three exactly solvable phases — gapless B-phase hosts Majorana quasi-particles
Andreev Reflection & BTK Theory
Blonder-Tinkham-Klapwijk conductance of N-S junctions — electron-to-hole conversion at a superconductor
Stochastic Gene Toggle Switch
Gardner-Collins mutual repressor model — noise-driven switching between stable states creates bimodal cell populations
Kolmogorov Complexity & Lempel-Ziv
LZ76 compressibility as a proxy for algorithmic complexity — compare random, periodic, and fractal sequences
Fiber Bundle & Parallel Transport
Holonomy from parallel-transporting a vector around a closed loop — Gauss-Bonnet theorem made visible
Abelian Sandpile Identity Fractal
The identity element of the abelian sandpile group has intricate fractal self-similarity on the integer lattice
Benjamin-Feir Modulational Instability
NLS envelope solitons and the breakup of uniform wave trains — rogue waves and Akhmediev breathers
Percolation Clusters & SLE₆ Hull
Site percolation on ℤ² near p_c≈0.5927 — cluster hulls converge to SLE₆ (Smirnov 2001, Fields Medal)
Surface Code — Quantum Error Correction
Kitaev's topological surface code, stabilizer measurements, syndrome extraction, and minimum-weight decoding
Magnon Dispersion — Heisenberg Ferromagnet
Spin-wave quasiparticle dispersion ℏω(k)=4JS(1−cos ka)+2DS² on a 1D lattice; thermal occupation and anisotropy gap
Mie Scattering — Dielectric Sphere
Exact electromagnetic scattering from a sphere; polar diagrams, extinction/scattering efficiencies Q_ext and Q_sca vs size parameter x
Thouless Topological Pump
Adiabatic cycling of Rice-Mele model pumps exactly C₁ electrons per cycle — quantized charge transport as a 1D analogue of IQHE
Kuramoto-Sakaguchi Phase-Lag Oscillators
Phase synchronization with coupling lag β; chimera states, order parameter r(t), and the transition from incoherence to locking
Stochastic Block Model — Belief Propagation
Random graphs with planted communities; Kesten-Stigum detectability threshold and BP-based community inference
Polymer Adsorption & Surface Depletion
Flexible chain concentration profile near a surface — adsorption layer vs entropy-driven depletion zone; Edwards equation theory
Quantum Walk — Discrete Grover Coin
Ballistic σ~t spreading vs classical σ~√t; bimodal probability distribution and coin-state interference on a 1D line
Rule 110 — Turing-Complete CA
The simplest known Turing-complete system (Cook 2004); persistent glider-like structures emerge from elementary CA rules
Hyperbolic Space — Poincaré Disk Geodesics
Click to place points and draw geodesics in constant negative curvature K; {7,3} tessellation and hyperbolic distance measurements
Spin-Torque Nano-Oscillator (STNO)
LLGS magnetization dynamics — spin-transfer torque drives sustained GHz precession above a critical current; tunable via field and damping
Floquet-Bloch Driven Lattice Micromotion
Periodically driven quantum lattice: quasienergy bands, micromotion within each Floquet period, and dynamic localization at zeros of the Bessel function J₀(A/ω)
Wigner Crystal Formation
Classical 2D electron lattice: Coulomb repulsion vs thermal disorder; bond-orientational order parameter ψ₆ tracks crystallization as r_s grows in Langevin dynamics
Tipping Points — Fold Catastrophe
Saddle-node bifurcation: potential landscape flips abruptly at μ=±2√3/3; hysteresis loop and bifurcation diagram reveal irreversible state transitions with noise
Active Matter — Vicsek Flocking
Self-propelled particles align with neighbors; noise-driven order-disorder transition at η* produces flocks with giant number fluctuations — no equilibrium analog
Quantum Metric Tensor & Berry Curvature
Quantum geometric tensor of a two-band model: Berry curvature Ω(k) integrates to Chern number; quantum metric g(k) measures state distinguishability; topological transition at |m|=2
Viscous Fingering — Saffman-Taylor
Less-viscous fluid displacing a viscous one: Saffman-Taylor instability drives dendritic fingering; surface tension selects fastest-growing mode k*; analogue of DLA and electrodeposition
Neural Field Theory — Wilson-Cowan
Excitatory-inhibitory cortical dynamics on a 1D ring: spatially coupled sigmoidal neurons produce traveling waves, bump attractors, and Turing patterns tunable via coupling weights
Multipole Expansion — Charge Distribution
Place up to three charges and compare exact potential against monopole, dipole, and quadrupole truncations; equipotential lines reveal how each order dominates at large r
Sisyphus Laser Cooling
Atoms climbing optical potential hills lose kinetic energy to fluorescence; Langevin simulation shows sub-Doppler cooling to T∝I/|Δ| and entropy exported to the radiation field
Stochastic Resonance
Noise of optimal amplitude maximizes SNR for weak signal detection; threshold, bistable double-well, and FitzHugh-Nagumo systems all show the resonance peak — noise helps!
Entanglement Entropy & Area Law
Bipartite entanglement of a 1D spin chain: gapped phases obey the area law S=const while critical CFT points show logarithmic growth S=(c/3)ln[sin(πℓ/N)·N/π] with central charge c
Off-Lattice DLA — Fjord Cluster
Diffusion-limited aggregation without a grid lattice grows branching fractal clusters with D_f≈1.71; fjords become inaccessible as harmonic measure concentrates on tips
Adiabatic Elimination — Fast/Slow Manifold
When ε→0 a fast variable y slaves to the slow x via y*=x²; trajectories collapse onto the slow manifold in O(ε) time and the full system reduces to a single ODE
Pyrochlore Spin Ice
Ice-rule frustration on a 2-in 2-out lattice: violations create emergent magnetic monopoles connected by Dirac strings; Monte Carlo explores the degenerate ground-state manifold
Lid-Driven Cavity Flow
Vorticity-stream function solution to Navier-Stokes in a square cavity with a moving lid; as Reynolds number increases corner vortices appear alongside the primary recirculation
Path Integral Monte Carlo
Feynman's imaginary-time path integral maps a quantum particle onto a classical ring polymer; PIMC samples the Boltzmann distribution of polymer configurations to recover ⟨x²⟩ and energy
Drift-Selection Balance
Wright-Fisher model tracks allele frequency trajectories under genetic drift, selection, and mutation; Kimura fixation probability and mutation-selection equilibrium emerge from ensemble statistics
Josephson Junction — RCSJ IV Curve
Resistively-shunted junction model traces hysteretic IV curves on the washboard potential; overdamped junctions show no hysteresis while underdamped ones latch into resistive states
Rossby Waves — Beta-Plane
Barotropic Rossby waves propagate westward via the Coriolis gradient β=df/dy; interactive dispersion relation shows how phase and group velocities depend on wavenumber and latitude
Spectral Density — Colored Noise
Generate 1/f^α noise by shaping white noise in frequency domain; log-log power spectra, autocorrelation, and time series reveal how α governs long-range correlations and Hurst exponent
Wilson Loop Lattice Gauge
U(1) lattice gauge theory with Metropolis Monte Carlo — watch Coulomb confinement emerge from plaquette phases as you tune the coupling β.
Predator-Prey Turing Patterns
Spatial Lotka-Volterra with cross-diffusion spontaneously breaks symmetry into spots and stripes — the Turing instability in ecology.
Quantum Dot Coulomb Blockade
Single-electron transistor stability diagram — explore Coulomb diamonds, charging energy, and how temperature washes out blockade peaks.
Phononic Metamaterial Band Gap
Diatomic mass-spring chain dispersion: tune the mass ratio to open a phononic band gap and watch waves turn evanescent inside it.
Oregonator BZ Oscillator
Minimal model of the Belousov-Zhabotinsky reaction — chemical spiral waves, limit cycles, and excitable media from three kinetic parameters.
Newton Fractal Basin Boundaries
Newton's method in the complex plane creates fractal boundaries between basins of attraction — Hausdorff dimension exactly 2 (Shishikura 1994).
Amphiphile Lamellar Self-Assembly
Ohta-Kawasaki free energy drives microphase separation — tune amphiphile fraction and segregation to explore lamellar, hexagonal, and micellar phases.
Transfer Entropy & Granger Causality
Directional information flow between coupled stochastic processes — measure who drives whom with transfer entropy and Granger's F-test.
Eigenstate Thermalization (ETH)
Quantum chaos and the ETH: tune chaos level to see level statistics transition from Poisson to Wigner-Dyson and observables thermalize.
Riemannian Sectional Curvature
Visualize Gaussian curvature on spheres, saddles, and tori — geodesics converge (K>0) or diverge (K<0) as Gauss-Bonnet holds globally.
Born-Oppenheimer Potential Energy Surface
Morse potential wells, quantized vibrational levels, and anharmonic wavefunctions — the electronic energy landscape that governs molecular dynamics.
Gaussian Channel Capacity
Shannon-Hartley theorem: C = W·log₂(1+SNR) — visualize the joint input-output distribution and watch channel capacity respond to signal and noise power.
Drucker-Prager Yield Surface
Pressure-sensitive plasticity for soils, rocks, and concrete — explore yield criteria in the π-plane and meridional plane with adjustable friction and cohesion.
Mean-Field Game Nash Equilibrium
Infinite-player rational agents balance crowd aversion against goal-seeking — watch the population density evolve toward a Nash equilibrium configuration.
Cosmic String Network
One-dimensional topological defects from cosmological phase transitions — strings reconnect, shed loops, and decay by gravitational radiation as the universe expands.
Laughlin Wavefunction (FQHE)
Fractional quantum Hall states at filling ν = 1/m — Laughlin's correlated electron liquid, anyonic quasi-particles, and exactly quantized Hall resistance R_H = mh/e².
Amyloid Aggregation Kinetics
Nucleation-elongation mechanism behind Alzheimer's and prion diseases — tune primary nucleation, elongation, and surface-catalyzed secondary nucleation to shape the sigmoidal curve.
Network Epidemic Heterogeneous Mixing
SIR dynamics on scale-free contact networks — super-spreader hubs lower the epidemic threshold and targeted vaccination outperforms random strategies.
Goldstone Mode & Spontaneous Symmetry Breaking
Mexican-hat potential, vacuum manifold, and the massless Goldstone boson versus the massive Higgs mode — explicit breaking generates pseudo-Goldstone mass.
Chaotic Scattering Fractal Deflection
Sensitive dependence on impact parameter produces fractal singularities in the deflection function — the fingerprint of a chaotic saddle in classical scattering.
Bloch Oscillations & Esaki-Tsu NDR
Electrons in a periodic lattice under DC field — Bloch oscillations, negative differential resistance, and the Esaki-Tsu formula.
Onsager Reciprocal Relations
Linear irreversible thermodynamics: L₁₂=L₂₁ enforced by time-reversal symmetry, Seebeck/Peltier duality, and thermoelectric efficiency ZT.
Topological Magnon Bands — Honeycomb
Spin-wave topology on the honeycomb lattice (CrI₃-type): DMI-driven Chern bands, Berry curvature, and magnon Hall effect edge states.
Quantum Chaos & Brody Distribution
Level spacing statistics crossing from Poisson (integrable) to Wigner-Dyson (chaotic) via the Brody interpolation parameter β.
Nematic Defect Annihilation ±½
Active liquid crystal dynamics: half-integer topological defects attract, self-propel, and annihilate with coarsening law ⟨N⟩∝t⁻¹.
Turbulent Boundary Layer
Law of the wall: viscous sublayer U⁺=y⁺, log layer (1/κ)ln(y⁺)+B, eddy viscosity, Reynolds stress, and turbulent kinetic energy profiles.
Cooperative Binding & Hill Equation
Hemoglobin oxygen dissociation curves, Hill plot, MWC allosteric model, Bohr effect, and free energy landscape of cooperative binding.
Epidemic Control & Vaccination
SEIR dynamics, herd immunity threshold, final size vs vaccine coverage, optimal intervention timing, and network heterogeneity effects.
Keldysh NEGF Quantum Transport
Non-equilibrium Green functions on the Keldysh contour: retarded/lesser GFs, Landauer transmission, I-V characteristics through a nanojunction.
Fracture Mechanics & Stress Intensity
LEFM crack tip stress field, Mode I/II intensity factors K, plastic zone shape, Griffith energy release rate G, and fracture instability criterion.
Thermoacoustic Standing Wave Resonance
Rijke tube dynamics: heat source position drives acoustic instability — visualize pressure, velocity, and temperature fields in a resonant tube.
Skyrmion Magnetic Topological Texture
Topologically protected spin vortices in chiral magnets — explore winding number, Néel vs Bloch types, and multi-skyrmion configurations.
Frenkel-Kontorova C-IC Transition
Atoms in a periodic substrate — devil's staircase of commensurate lock-ins and topological kink (soliton) formation in incommensurate phases.
Stochastic Block Model Community Detection
Generate SBM random graphs and detect communities via spectral clustering — observe the Kesten-Stigum detectability phase transition.
Reaction-Diffusion Cross-Diffusion Turing
Cross-diffusion terms expand the Turing instability regime — Gray-Scott patterns with cross-coupling reveal new spots, stripes, and labyrinths.
Quantum Discord Geometric Measure
Beyond entanglement: explore the landscape of classical vs quantum correlations in two-qubit Werner and X-states — discord survives separability.
Active Matter Motility-Induced Phase Separation
Self-propelled particles slow in dense regions, spontaneously phase-separating without attractive forces — MIPS mechanism with Péclet number control.
Hamiltonian Truncation Variational Bounds
Ritz variational principle: truncated Hamiltonian eigenvalues converge from above to exact values — watch convergence for anharmonic and double-well potentials.
Exceptional Point Non-Hermitian Degeneracy
PT-symmetric gain-loss coupler: at the exceptional point, eigenvalues and eigenvectors coalesce — explore Riemann sheet topology and PT phase transition.
Ecological Stoichiometry — Droop Model
Phytoplankton nutrient dynamics with internal cell quota — Droop's model separates uptake from growth, capturing luxury uptake and stoichiometric constraints.
Kibble-Zurek Defect Formation
Topological defect nucleation during symmetry-breaking quenches through the XY model transition — vortex/antivortex pair production and winding number counting.
Vicsek Polar Active Flock
Collective motion and long-range polar order in 2D active matter — the Vicsek 1995 model where alignment beats Mermin-Wagner in non-equilibrium systems.
Anderson Localization 1D
Disorder-induced quantum wavefunction localization in a 1D tight-binding chain — IPR, localization length, and eigenstate visualization across the full spectrum.
Chemical Potential & Van der Waals
Maxwell equal-area construction, fugacity, and phase coexistence in the Van der Waals gas — P-V isotherms and chemical potential curves near the critical point.
Topological Defects & Winding Numbers
Place vortex and antivortex defects on an XY spin lattice, watch the winding number topology, and explore BKT-transition physics interactively.
Lévy Stable Random Walks
Heavy-tailed superdiffusion via the Chambers-Mallows-Stuck algorithm — tune stability α from Cauchy (α=1) to Gaussian (α=2) and observe the generalized CLT.
Plasma Debye Shielding & Langmuir Probe
Electrostatic sheath formation, Debye length λ_D, and I-V characteristic of a Langmuir probe — electron/ion saturation and floating potential in a plasma.
Quantum Tunneling — WKB
WKB transmission through square, Gaussian, triangular, and parabolic barriers — evanescent wavefunctions, resonances, and comparison with exact analytic results.
Firefly Synchronization
Mirollo-Strogatz pulse-coupled integrate-and-fire oscillators synchronizing on a dark night — Kuramoto order parameter, coupling strength, and frequency spread.
Percolation Cluster Hulls & Scaling
Site percolation on a square lattice near p_c ≈ 0.5927 — spanning cluster identification, fractal dimension d_f=91/48, hull tracing, and critical exponent signatures.
Shilnikov Chaos: Homoclinic Spiral
Saddle-focus equilibrium with a homoclinic orbit generates infinite Smale horseshoes when ρ < |λ| — spiral chaos in 3D, rotating phase portrait, and Shilnikov condition monitor.
Kitaev Honeycomb: Flux Sectors
Exactly solvable spin-1/2 model on a honeycomb lattice — click plaquettes to insert Z₂ π-flux vortices, toggle coupling strengths, and explore the A/B phase boundary.
Non-Affine Deformation in Glasses
Shear a disordered particle packing and watch local non-affine displacements D² light up — soft modes, shear transformation zones, and Falk-Langer STZ theory.
Wigner Crystal Melting: 2D Electrons
Low-density electrons freeze into a triangular Wigner crystal at rₛ ≫ 37 — KTHNY two-stage melting via dislocation and disclination unbinding, Lindemann ratio, and ψ₆ order.
Ostwald Ripening: Lifshitz-Slyozov Law
Large droplets grow at the expense of small ones via diffusion — ⟨R⟩³ ~ t coarsening law, self-similar LSW size distribution, and live histogram versus theory.
Quantum Zeno Effect: Repeated Measurement
Frequent measurements freeze an unstable quantum state — quantum trajectories, Zeno vs anti-Zeno regimes, effective decay rate, and Misra-Sudarshan survival probability.
Fractal Basin Boundaries: Newton Map
Newton's method on complex polynomials creates fractal boundaries of infinite dimension — zoom into Cayley's 1879 paradox where arbitrarily close starts converge to different roots.
Active Gel: Polar Filament Network
Motor proteins drive actin filaments into self-organized flows — extensile vs contractile activity, Toner-Tu flocking, polar order parameter, and active turbulence defects.
Spectral Flow: Chern Insulator Edge States
Qi-Wu-Zhang model bulk bands, Chern number computation, and chiral edge modes crossing the gap — bulk-edge correspondence and spectral flow under inserted flux.
Entropy Production & Information Erasure
Landauer's principle: erasing one bit dissipates at least k_BT ln2 as heat — Maxwell's Demon, double-well erasure dynamics, Jarzynski equality, and the thermodynamics of computation.
Calogero-Moser Integrable Particles
Exactly solvable many-body system of particles on a line with inverse-square repulsion and a harmonic trap — equilibrium positions are zeros of Hermite polynomials, spectrum is perfectly equidistant.
Jamming Transition & Random Close Packing
Hard disks compressed toward the jamming point φ_J ≈ 0.84 (2D) — motion freezes, pressure diverges, and a contact network spans the system: a geometric phase transition with no thermal fluctuations.
Magnon Spin-Wave Dispersion in a Ferromagnet
Collective spin precession propagating through a ferromagnetic chain — visualize the quadratic dispersion ω(k) ∝ k² (versus linear for antiferromagnets), Zeeman gap, and real-space spin dynamics.
Crystal Surface Roughening — KPZ Growth
Kardar-Parisi-Zhang nonlinear stochastic growth: interface width W ~ t^β with β=1/3 for KPZ, 1/4 for Edwards-Wilkinson. Watch universality classes emerge from different microscopic rules.
Quantum Revivals — Dispersive Wave Packet
A Gaussian wave packet spreads and seemingly vanishes due to anharmonic dispersion, then miraculously reassembles at T_rev = π/χ — fractional revivals at T_rev/2, T_rev/3 create Schrödinger cat states.
Contact Process — Directed Percolation
The canonical absorbing-state phase transition at λ_c/μ ≈ 1.649: below critical rate infections die, above they survive forever. Space-time diagram reveals the DP universality class critical exponents.
Hysteresis Loop — Preisach Ferromagnet
The Preisach model decomposes a ferromagnet into bistable hysterons with distributed switching fields — watch the M-H loop build from thousands of individual Barkhausen jumps on the Preisach plane.
Flocking — Topological Voronoi Neighbors
Starling murmurations use topological (k nearest) rather than metric (fixed radius) interaction — compare the two rules and see how k≈7 neighbors make the flock robust to density perturbations and predators.
Chaotic Mixing — Baker Map & Fractal Invariant Set
The baker map (stretch + fold) generates exponential mixing with Lyapunov exponent λ = ln2. Watch colored blobs shred into fractal filaments, or reveal the Cantor-set repeller of the dissipative version.
Two-Temperature Model — Ultrafast Laser Heating
Femtosecond laser pulses heat electrons to ~10,000 K while the lattice stays cold — the TTM equations describe electron-phonon equilibration over picoseconds, underpinning laser ablation and ultrafast demagnetization.
Aharonov–Bohm Effect — Flux Tube
A charged particle acquires a quantum phase Δφ = 2πΦ/Φ₀ from a magnetic flux tube even where B = 0 — shifting double-slit fringes via the vector potential alone, a purely topological quantum effect.
Epidemic on Heterogeneous Networks
SIS dynamics on scale-free networks via heterogeneous mean-field theory: as γ→2, the epidemic threshold β_c→0 — any infection spreads for any nonzero rate, a consequence of diverging ⟨k²⟩.
Stress-Stiffening Biopolymer Network
Semiflexible filament networks (actin, collagen) stiffen dramatically under strain as thermal undulations are pulled taut — the differential modulus K grows as σ^(3/2), orders of magnitude above linear elasticity.
Quantum Annealing — Transverse Field Ising
Adiabatic optimization by slowly turning off a transverse field: quantum tunneling can bypass classical energy barriers, but the minimum spectral gap Δ_min sets the hardness via T >> 1/Δ_min².
DLA Fjord Statistics
Diffusion-limited aggregation grows a fractal cluster (d_f ≈ 1.71) whose tips capture most of the harmonic measure while fjords are exponentially screened — multifractal statistics from Laplacian growth.
Ecological Resilience — Fold Catastrophe
Shallow lakes exhibit alternative stable states (clear vs. turbid) connected by hysteresis: past the tipping point, a sudden irreversible regime shift occurs — a fold catastrophe in Scheffer's 1993 model.
Optomechanical Sideband Cooling
A red-detuned laser drives anti-Stokes scattering in an optical cavity, removing one phonon per photon. In the resolved-sideband limit Ω_m >> κ, mechanical oscillators are cooled to their quantum ground state.
Random Matrix Level Repulsion — GOE
GOE eigenvalue spacings obey the Wigner surmise P(s) ~ s·exp(−πs²/4) with P(0) = 0 — level repulsion from a logarithmic Dyson gas. Contrast with Poisson statistics of integrable systems.
Turbulent Prandtl Number & Convection
Rayleigh-Bénard convection heat transport follows Nu ~ Ra^0.3 (Grossmann-Lohse theory). Explore how Prandtl number shapes thermal vs. viscous boundary layers and the path to the ultimate regime Nu ~ Ra^(1/2).
Topological Thouless Pump — Rice-Mele
Adiabatically cycling the Rice-Mele model parameters transports exactly one electron per cycle — quantized by the Chern number C = 1, the 1D analogue of the quantum Hall effect realized in cold-atom experiments.
Coupled Map Lattice — Spatiotemporal Chaos
A 1D lattice of logistic maps coupled diffusively generates rich spacetime patterns — uniform states, periodic waves, and fully chaotic regimes — as growth rate r and coupling ε are tuned across phase boundaries.
Marangoni Flow — Surface Tension Gradient
Temperature gradients drive fluid toward cooler, higher-tension regions. This Marangoni instability creates dramatic convective rolls visible in tears of wine, spreading droplets, and thin-film coating defects.
Quantum Walk — Discrete-Time Coin
A quantum particle's wavefunction spreads ballistically (σ~t) via a coin unitary, contrasting sharply with classical random walks (σ~√t). Explore how coin angle and phase sculpt the probability distribution.
Neural Field Theory — Wilson–Cowan
Two coupled neural populations (excitatory E, inhibitory I) on a 1D cortical sheet interact via Mexican-hat kernels. Tune couplings to generate traveling waves, stationary bumps, and cortical oscillations.
Thermodynamic Uncertainty Relation
The TUR (Barato & Seifert 2015) proves precision costs entropy: Var(J)/⟨J⟩² ≥ 2k_B/Σ. Watch Markov chain currents accumulate and verify the fundamental bound on current fluctuations vs. entropy production.
Kondo Effect — Spin-Flip Scattering
A magnetic impurity in a metal causes resistance to rise logarithmically as temperature falls — a many-body quantum effect where conduction electrons screen the impurity spin below the Kondo temperature T_K.
Elastic Snap-Through — Bistable Arch
A shallow elastic arch has two stable states. Applied force shifts the energy landscape until the barrier vanishes via saddle-node bifurcation — triggering catastrophic snap-through used in MEMS and deployable structures.
Invasion Percolation with Gradient
An invading fluid displaces defender by always choosing the weakest neighboring throat. With gravity gradient G, the invasion front transitions from fractal DLA-like clusters to compact flat displacement fronts.
Membrane Fluctuations — Helfrich Undulations
Lipid bilayer membranes fluctuate thermally. The Helfrich Hamiltonian with bending κ and tension σ sets each Fourier mode's amplitude via ⟨|h_q|²⟩ = k_BT/(κq⁴+σq²). High κ suppresses undulations.
Variational Autoencoder — Latent Space
A VAE encodes data into a Gaussian latent space z~N(μ,σ²) regularized by KL divergence. Navigate the 2D latent space to decode class interpolations — revealing how β-VAE β controls disentanglement vs. reconstruction.
Faraday Instability — Parametric Resonance
Faraday waves: vertical oscillation drives surface waves via the Mathieu equation. Each mode's natural frequency ω₀²=gk+σk³/ρ resonates at ω≈2ω₀, producing subharmonic standing waves at half the drive frequency.
Spiral Waves — Cardiac Arrhythmia
Rotating spiral waves in excitable media (FitzHugh-Nagumo style). A broken wavefront curls into a self-sustaining spiral — the mechanism behind re-entrant tachycardia and ventricular fibrillation.
Persistent Random Walk — Telegrapher Equation
A particle moves at speed v, reversing direction after exponential waiting times with mean τ. The density obeys τ∂²p/∂t²+∂p/∂t=v²τ∂²p/∂x² — interpolating between wave (short t) and diffusion D=v²τ/2 (long t).
Quantum Phase Estimation — Kitaev's Circuit
Kitaev's QPE uses n ancilla qubits, controlled-U^{2^k} gates, and inverse QFT to read eigenphase φ to n bits. Output probability peaks sharply at the best rational approximation — core of Shor's algorithm.
Cavity Method — Bethe Lattice & Belief Propagation
Exact inference on trees via cavity messages: h_{i→j}=(1/β)arctanh[tanh(βJ)tanh(βh_{k→i})]. Below Tc=J/arctanh(1/√(z−1)) ferromagnetic order emerges. The replica trick generalizes this to spin glasses.
Abel-Ruffini — Unsolvability of the Quintic
Why degree-5 polynomials have no radical formula: the Galois group of a generic quintic is S₅, which is not solvable (A₅ is the unique simple non-abelian group). Visualize roots and derived series.
Schwarzschild — Photon Sphere & Gravitational Lensing
Photon geodesics in Schwarzschild spacetime traced via d²u/dφ²+u=3Mu². The photon sphere sits at r=3M; critical impact parameter b_c=3√3 M separates captured from escaping rays. Deflection diverges logarithmically near b_c.
Holographic Complexity — CV Conjecture
Complexity=Volume: the quantum circuit complexity of the boundary CFT state equals the maximal Cauchy slice volume through the AdS-Schwarzschild wormhole. Late-time growth dC/dt=2M saturates the Lloyd bound.
Ergodic Decomposition — Pinsker Factor
Any invariant measure decomposes into ergodic components. The Pinsker factor is the largest zero-entropy factor. Irrational rotations (h=0) ARE their own Pinsker factor; doubling maps (h=log 2) are fully Bernoulli.
Vicsek Model — Flocking Transition
N self-propelled particles align with neighbors plus noise η. Order parameter φ=|⟨e^{iθ}⟩| undergoes a discontinuous (first-order) transition at critical noise η_c. Giant number fluctuations mark the flocking phase.
1D Ising Transfer Matrix
Exact solution via transfer matrix: free energy, magnetization & correlation length vs temperature
Deffuant Opinion Dynamics
Bounded-confidence model: agents interact only within tolerance ε; watch echo chambers form
KdV Soliton Collision
Two-soliton exact solution: shapes survive collision intact, only a phase shift remains
Polymer Coil-Globule Transition
Flory theory & Monte Carlo pivot: radius of gyration collapses at the theta point
Skyrmion Lattice
Magnetic spin textures with topological charge Q=±1; Bloch vs Néel types via DMI helicity
Stochastic Lotka-Volterra
Demographic noise drives extinction events; quasi-cycles persist at large N, vanish at small N
Quantum Walk Spreading
Discrete quantum walk: ballistic σ~t vs classical diffusive σ~√t from coin interference
Anderson Localization
All 1D states localize in random potential; exponential decay of ψ with disorder-controlled ξ
Floquet Prethermalization
Periodic drive: prethermal plateau before infinite-temperature heating; τ_pre ~ e^{Ω/J}
Fokker-Planck Double Well
Steady-state P~e^{−V/D} and Kramers escape rate over barrier; live particle simulation
Majority-Vote Model
Absorbing state transition on 2D lattice; order parameter vs noise q; q_c ≈ 0.075 critical point
Spatial Prisoner's Dilemma
Cooperation clusters on grid; Nowak-May model; tune temptation b to watch cooperation collapse
Lévy Walk · Anomalous Diffusion
Heavy-tail step lengths P(ℓ)~ℓ^{−μ}; superdiffusive MSD~t^α with α=(3−μ)>1
Brusselator Limit Cycle
Chemical oscillator phase plane; supercritical Hopf bifurcation when B>A²+1; stable limit cycle
Kuramoto Chimera State
Two coupled oscillator populations; one synchronized, one incoherent; symmetry-broken chimera
Quantum Double-Barrier Tunneling
Transfer matrix method; transmission resonances T→1 at quasi-bound energies; resonant tunneling diode
Diblock Copolymer Microphase
Ohta-Kawasaki dynamics; lamellae/cylinders/spheres morphologies; order-disorder transition at χN≈10.5
Rigidity Percolation · Pebble Game
Jacobs-Hendrickson pebble game; rigid vs floppy bond classification; transition at mean degree 2.4
SIS Epidemic · Heterogeneous Network
Scale-free BA network; degree-based mean field; vanishing epidemic threshold β_c→0 as N→∞
Hard Disk Packing · Jamming
2D random close packing; jamming transition at φ≈0.84; pressure divergence; rattler detection
Morse Potential · Diatomic Vibration
Anharmonic energy levels; dissociation vs harmonic approximation; classical turning points
Voter Model · 1D Coarsening
Interface density ~ 1/√t; domain coarsening; consensus time scaling T ~ N²
Clonal Selection · Affinity Maturation
B-cell shape space; somatic hypermutation; selection pressure drives exponential affinity improvement
HP Model · Protein Folding
2D lattice Monte Carlo; hydrophobic core formation; H-H contact energy minimization
Kink Soliton · φ⁴ Field Theory
Topological charge Q; kink-antikink collision and annihilation; Lorentz-contracted solitons
Gierer-Meinhardt · Activator-Inhibitor
Turing instability; spot and stripe pattern formation; D_h/D_a ratio controls morphology
Fiber Bundle · Fracture Avalanches
Equal load sharing; avalanche size distribution ~ s⁻⁵/²; catastrophic failure transition
Kinetic Ising · Glauber Dynamics
Autocorrelation function; critical slowing down near Tc; dynamic exponent z≈2.17
Quantum Morse Oscillator
Exact spectrum En=ℏωe(n+½)[1−χe(n+½)]; anharmonicity constant; level convergence at dissociation
Boolean Network · Attractor Basins
NK Kauffman model; order/chaos transition at K=2; attractor cycle length ~ √N
Sandpile · Self-Organized Criticality
Bak–Tang–Wiesenfeld sandpile; toppling rules; avalanche power law P(s)∝s⁻¹·²; SOC without parameter tuning
SIR Epidemic · Heterogeneous Network
Degree-based mixing; R₀=β/γ·⟨k²⟩/⟨k⟩; herd immunity threshold; scale-free vs random vs lattice
Chaotic Mixing · Fluid Advection
Double-gyre flow; passive tracer advection; Poincaré sections; KAM tori and chaotic seas
Game of Life · Methuselahs
Conway B3/S23; R-pentomino runs 1103 gens; Gosper glider gun; population statistics
Kuramoto · Order Parameter
Phase oscillator synchronization; Kc=2/πg(0); Ott–Antonsen exact reduction; r vs K/Kc
Fibonacci Quasicrystal
1D quasiperiodic chain; projection from 2D lattice; sharp Bragg peaks; Cantor spectrum
Jarzynski Equality
⟨e⁻ᵝᵂ⟩=e⁻ᵝᐩᶠ; nonequilibrium pulling simulations; free energy from irreversible work
Quantum Zeno Effect
Repeated measurement slows decay; survival P=[cos²(Ωτ/2)]^{t/τ}; Zeno vs anti-Zeno regime
Coupled Oscillators · Arnold Tongue
Two coupled phase oscillators; frequency entrainment; Arnold tongue in (Δω, K) space
Chip-Firing · Toppling Automaton
Abelian sandpile group; wave propagation on grid; identity element fractal; order-independence
Elastoplastic STZ Model
Shear transformation zones in amorphous solids; stress-strain curves; avalanche plasticity events
Population Genetics: Drift & Selection
Wright-Fisher model; genetic drift, selection, mutation; allele trajectories; Kimura fixation probability
Laser Mode Competition
Longitudinal modes competing via gain saturation; winner-take-all dynamics; mode-locking output field
SK Spin Glass
Sherrington-Kirkpatrick model; replica-symmetric order parameter q; de Almeida-Thouless instability
Carbon Nanotube Bands
Tight-binding zone-folding; armchair/zigzag/chiral tubes; metallic vs semiconducting selection rule
Rule 110 — Universal CA
Wolfram's Rule 110; Turing-complete cellular automaton; gliders on complex background; all 256 rules
Kagome Antiferromagnet
Geometric frustration; macroscopic ground-state degeneracy; spin-ice rules; residual entropy
Jaynes-Cummings Model
Vacuum Rabi splitting; dressed states; collapse and revival of Rabi oscillations; cavity QED
Edwards-Wilkinson Roughening
Surface growth scaling W ~ t^β L^α; Family-Vicsek hypothesis; EW vs KPZ universality classes
SSH Topological Chain
Su-Schrieffer-Heeger model; winding number; bulk-boundary correspondence; zero-energy edge states
Spectral Form Factor
K(t) dip-ramp-plateau in GOE/GUE random matrices — quantum chaos fingerprint vs integrable Poisson
Eigenvalue Repulsion
GOE/GUE/GSE level spacing P(s)∝s^β — Wigner surmise vs Poisson; repulsion and universality classes
Ruppeiner Geometry
Thermodynamic curvature R diverges at phase transitions — Fisher metric on van der Waals state space
Stochastic Resonance
Noise enhances signal detection in a bistable potential — optimal SNR at non-zero noise amplitude
Thouless Charge Pump
Adiabatic pumping of charge per cycle equals the Chern number — SSH model with time-dependent hopping
Quantum Discord
Discord = total − classical correlations; non-zero for separable Werner states; resource beyond entanglement
Information Bottleneck
Compress X→T while preserving Y; Blahut-Arimoto traces the I(T;X) vs I(T;Y) tradeoff curve
Network Controllability
Minimum driver nodes via maximum matching — Liu et al. 2011; Kalman rank condition on directed graphs
Wetting Transition
Young equation cos θ = (γ_SV−γ_SL)/γ_LV; spreading coefficient S; complete vs partial wetting
Stochastic Block Model
Spectral community detection in planted partition graphs; detectability threshold; Bethe Hessian eigenvectors
Capstan Equation
T_load = T_hold · e^(μθ) — exponential mechanical advantage from rope wrapping friction; Euler 1747
Diffusion-Limited Aggregation
Witten-Sander (1981) fractal cluster growth: random walkers stick on contact, building branching dendrites with D≈1.71
Thomas Reaction-Diffusion
Substrate-inhibition kinetics with Turing instability: two chemicals diffuse at different rates, spontaneously forming spots and stripes
Poisson Point Process
Complete spatial randomness, Matérn clusters, hard-core inhibition, Cox processes — Ripley's K-function distinguishes each
Ricci Flow
∂g/∂t = −2 Ric(g) — Hamilton's metric evolution drives surfaces toward constant curvature; basis of Perelman's Poincaré proof
Geodesic Deviation & Jacobi Fields
D²J/ds² + R(J,γ')γ' = 0 — neighboring geodesics converge on spheres, diverge on hyperbolic space, stay parallel on flat manifolds
Shilnikov Chaos
Spiral homoclinic orbit to a saddle-focus: when |Re(λ)| < Im(λ), infinitely many horseshoes guarantee chaos (1965)
Kitaev Honeycomb Model
Exactly solvable spin-½ model yielding Majorana fermions and non-Abelian anyons; phase diagram with Chern number ν=±1
Navier-Stokes Vorticity-Stream
∂ω/∂t + (u·∇)ω = ν∇²ω — 2D incompressible flow via stream function; lid-driven cavity, channel flow, cylinder wake
Quantum Walk — Discrete Coin
Ballistic spread σ∝t vs classical σ∝√t; Hadamard, Grover, Fourier coins; interference creates double-peaked distributions
Fokker-Planck Drift-Diffusion
∂P/∂t = -∂(μP)/∂x + D∂²P/∂x² — probability density evolution for stochastic processes; bistable potentials, Kramers escape, stationary distributions
Chern-Simons Field Theory
Purely topological gauge theory with no metric — anyonic braiding phases e^(2πiq²/k), Wilson loops, and the mathematics behind topological quantum computing
Abelian Sandpile Toppling
Self-organized criticality: each cell topples when ≥4 grains, generating power-law avalanches; stable configurations form an abelian group under addition
Stochastic Resonance
Noise enhances detection of subthreshold signals in bistable systems — SNR peaks at optimal noise amplitude; used in neuroscience and sensory biology
Quantum Zeno Effect
Frequent measurement suppresses quantum decay: P_survive = cos²ⁿ(Γτ) → e^(-Γ²τT) — the watched pot never decays; compare free, measured, and Zeno regimes
Hyperbolic Tiling: Poincaré Disk
Regular {p,q} tessellations of hyperbolic space — geodesics as circular arcs, (p-2)(q-2)>4 condition, and the exponential growth of hyperbolic area
Topological Defects & Vortices
Click to place vortex defects in 2D order parameter fields — winding numbers n=±1,±2 classify defects topologically; Kosterlitz-Thouless transition physics
Epidemic Network Percolation
SIR spreading on Erdős-Rényi, scale-free, and small-world networks with contact tracing — R₀=p⟨k⟩/γ controls giant-component percolation threshold
Gravitational Wave Inspiral
Binary black hole merger: Peters orbital decay, chirp mass M_c=(m₁m₂)^(3/5)/(m₁+m₂)^(1/5), rising GW frequency and strain — LIGO-style detection
Worm-Like Chain Polymer
Semiflexible polymer conformations: tangent-tangent correlations ⟨u(s)·u(0)⟩=e^(-s/ℓp), ensemble statistics, and Marko-Siggia force-extension (DNA, actin)
Morse Theory Height Function
Sweeping level sets across a manifold: critical points (min/saddle/max) encode topology, Euler characteristic χ = C₀−C₁+C₂, animated height plane on torus and sphere.
Feigenbaum Period-Doubling
Universal period-doubling cascade to chaos in unimodal maps: bifurcation diagram, Feigenbaum constant δ ≈ 4.6692 as ratio of successive interval widths, logistic/sine/tent maps.
Ising Kramers–Wannier Duality
Self-duality of the 2D Ising model: sinh(2/T)·sinh(2/T*)=1 maps high-T to low-T, fixing T_c=2/ln(1+√2) — dual lattice Monte Carlo with live spin configurations.
KdV Two-Soliton Collision
Exact two-soliton solution of the KdV equation via Hirota's method: taller solitons travel faster, pass through smaller ones, emerging with only a phase shift — elastic integrability.
Vicsek Polar Flocking
Self-propelled particles aligning with neighbors: non-equilibrium phase transition to long-range polar order at low noise, breaking continuous symmetry forbidden by Mermin-Wagner in equilibrium.
Fibonacci Quasicrystal Diffraction
1D quasiperiodic tiling from substitution A→AB, B→A: sharp diffraction peaks at k=(2π/a)(m+nφ) — dense but non-periodic spectrum, with cut-and-project construction visualization.
Newton Fractal Basin Boundaries
Newton's method on z³−1 in the complex plane: basins of attraction for the three roots form a fractal boundary (Julia set) with Hausdorff dimension 2, switchable to z⁴ and z⁵.
Chirikov Standard Map
Canonical Hamiltonian chaos: Poincaré section of p→p+K sin(θ), θ→θ+p shows KAM tori breaking at K_c≈0.9716 (Greene 1979) — from integrable circles to global chaotic diffusion.
Smoluchowski Coagulation
Cluster growth via coagulation kernels (constant, additive, product): Smoluchowski rate equations, gelation transition at t_c=1 for product kernel, evolving power-law size distributions.
Abelian Sandpile Identity
Self-organized criticality via toppling rules: grains≥4 fire to neighbors, avalanche sizes follow a power law, and the group identity element forms a remarkable fractal-symmetric pattern.
Rule 110 Universality
The simplest known Turing-complete system: elementary cellular automaton Rule 110 generates complex gliders and interactions from a 3-cell neighborhood rule, proved universal by Matthew Cook in 2004.
Non-Hermitian Skin Effect
In the Hatano-Nelson model, asymmetric hopping collapses all bulk eigenstates to one boundary; the winding number of the PBC spectrum predicts OBC skin mode count via non-Bloch band theory.
SBM Spectral Community Recovery
The stochastic block model's planted communities are detectable when SNR exceeds the Kesten-Stigum threshold, opening a spectral gap in the normalized Laplacian that k-means can exploit.
Gilbert Connectivity Threshold
Random geometric graph G(n,r): n points in the unit square connected within radius r undergo a sharp connectivity transition at r_c = √(ln n / πn), driven by the disappearance of isolated nodes.
Turbulent Pipe Flow Transition
Osborne Reynolds' 1883 discovery: pipe flow transitions from laminar Poiseuille parabola to turbulent log-law profile at Re≈2300–4000, via intermittent puffs and slugs in a subcritical bifurcation.
Quantum Eraser
The Englert-Greenberger-Yasin duality K²+V²≤1 quantifies the trade-off between which-path distinguishability and fringe visibility; erasing path information restores interference in coincidence counts.
Kramers Escape Rate
Thermal activation over a potential barrier: Kramers' 1940 formula Γ = (ω_min·ω_sad/2πγ)·exp(−ΔV/kT) gives the Arrhenius rate for overdamped Langevin dynamics, visualized via stochastic simulation.
Lévy Walk Anomalous Diffusion
Heavy-tailed step lengths P(ℓ)~ℓ^{−α−1} break normal diffusion: MSD scales as t^ν with ν=4−α for 1<α<2, producing superdiffusion relevant to animal foraging and human mobility.
Explosive Synchronization
Kuramoto oscillators on scale-free networks with frequency-degree correlation show an abrupt first-order synchronization transition with hysteresis — explosive synchronization discovered by Gómez-Gardeñes (2011).
Crooks Fluctuation Theorem
P_F(W)/P_R(−W) = exp[(W−ΔF)/kT]: forward and reverse work distributions cross at W=ΔF, enabling free energy measurement from nonequilibrium pulling and encoding the arrow of time in trajectory statistics.
Adiabatic Invariant Pendulum
For a pendulum with slowly varying length, J=E/ω stays constant even as energy changes — the classical adiabatic invariant underlying quantum number conservation in slowly-varied Hamiltonians.
Thouless Pump
Adiabatic cycling of the Rice-Mele Hamiltonian pumps exactly one electron per cycle — an integer set by the Chern number of the occupied band over the (k, φ) torus.
Cross-Diffusion Turing Patterns
The Shigesada-Kawasaki-Teramoto model: cross-diffusion (one species avoiding the other) drives spatial segregation patterns even when self-diffusivities are equal — bypassing classical Turing constraints.
Ferroelectric Hysteresis Loop
Landau-Devonshire double-well free energy drives polarization switching in ferroelectrics — the P-E hysteresis loop area equals energy dissipated per cycle, the signature of memory in capacitors.
Pseudospectrum & Non-Normality
Trefethen's ε-pseudospectrum reveals where eigenvalues can move under ε-perturbation — non-normal matrices like the Grcar or Jordan block have enormous pseudospectra despite innocuous eigenvalues.
Rössler Attractor — Funnel Chaos
Rössler's 1976 minimal chaotic chemical oscillator: a 3D flow with a single nonlinear term that spirals, resets via a z-spike, and produces the simplest strange attractor with Lyapunov exponent λ₁≈0.07.
Percolation & Cardy Formula
At p_c≈0.5928 critical site percolation is conformally invariant (Smirnov 2001) — Cardy's crossing formula predicts the spanning probability exactly using the hypergeometric function ₂F₁.
Spin Glass — Parisi Overlap
The SK model below T_c exhibits replica symmetry breaking: the overlap distribution P(q) spreads continuously from 0 to q_EA, encoding the ultrametric structure of an exponentially rugged energy landscape.
Double Barrier Resonant Tunneling
Two quantum barriers create a quasi-bound well state — at resonance energies, transmission T→1 despite E<V₀, the operating principle of resonant tunneling diodes computed by transfer matrices.
Information Bottleneck
Tishby's bottleneck principle: min I(X;T)−β I(T;Y) traces the Pareto frontier of compression vs. relevance — the IB curve unifies rate-distortion theory, sufficiency, and deep learning representation analysis.
Heteroclinic Orbit Cycle
Dynamical system cycling between saddle equilibria via heteroclinic connections — slow-fast dynamics, Z_N symmetry, and noise-induced escape from the cycle.
Magnon Dispersion: Heisenberg Chain
Spin waves in ferromagnetic and antiferromagnetic chains: ω(k) dispersion, quadratic vs. linear Goldstone modes, group velocity, and animated precessing spins.
ASEP Density Profile
Asymmetric simple exclusion process: non-equilibrium phases (low density, high density, maximum current) and the exact density profile emerging from boundary rates α and β.
Topological Kink — φ⁴ Field
Kink and antikink solitons in the double-well φ⁴ potential: exact tanh solutions, kink-antikink scattering, topological charge Q=±1, and the Lorentz-boosted mass.
Continuum Percolation: Boolean Model
Random disk placement on the plane — cluster formation, spanning transition at η_c≈1.128, coverage fraction 1−e^{−λπr²}, and real-time union-find cluster detection.
Deffuant Bounded Confidence
Opinion dynamics where agents only interact if |x_i−x_j|≤ε — consensus, polarization, and fragmentation phases with ~1/(2ε) expected clusters.
Turbulent Boundary Layer Log-Law
von Kármán's law of the wall: viscous sublayer u⁺=y⁺, log region u⁺=(1/κ)ln(y⁺)+B, buffer layer transition, roughness effects, and animated particle streaks.
Quantum Phase Estimation (Kitaev)
QPE circuit with n ancilla qubits: controlled-U^(2^k) gates, inverse QFT, eigenphase probability distribution, and precision scaling δφ=1/2ⁿ.
Hastings-Powell Food Chain Chaos
Three-trophic-level system (prey/predator/top-predator) exhibiting deterministic chaos — strange attractor in population phase space, first ecological chaos model (1991).
Classical Nucleation Theory
Free energy barrier ΔG*=16πγ³Ω²/(3Δμ²), critical radius r*=2γΩ/Δμ, Arrhenius nucleation rate, and animated crystal embryo growth and dissolution.
Anderson Localization in 1D
Quantum wavefunction localization by random disorder — transfer matrix method shows all states localized for any W>0, with exponential envelope and Lyapunov exponent.
Störmer–Verlet Symplectic Integrator
Phase-space-area-preserving Kepler orbit integration contrasted with energy-drifting Euler — symplectic structure gives bounded long-term error vs secular growth.
Cahn–Morral 3-Component Phase Separation
Spinodal decomposition in a ternary alloy — Flory–Huggins free energy drives multicomponent microstructure formation through coupled Cahn–Hilliard diffusion.
Network Motif Triad Census
Count all 13 directed triad types and compute Z-scores vs Erdős–Rényi null model — reveals biologically and socially enriched subgraph signatures (Milo et al. 2002).
Quantum Stadium Billiard Wavefunctions
Bunimovich stadium eigenstates — quantum chaos, Berry's random wave conjecture, quantum scars concentrated on unstable periodic orbits, and Wigner–Dyson level repulsion.
Helfrich Bending Elasticity — Vesicle
Lipid bilayer shape fluctuations governed by F=∫κ(2H)²dA — thermal undulation modes with q⁻⁴ Helfrich spectrum, flicker spectroscopy, and persistence length visualization.
Benjamin–Feir Modulation Instability
NLS sideband growth from Stokes wave — Akhmediev breather recurrence, space-time amplitude map, and rogue wave genesis from focusing nonlinear modulation instability.
Kleiber's Law — Metabolic Scaling
B∝M^(3/4) across 27 orders of magnitude from bacteria to blue whales — West–Brown–Enquist fractal vascular network theory explains the universal ¾ exponent.
Limit Order Book Dynamics
Zero-intelligence agent market simulation — Poisson limit/cancel/market order arrivals generate realistic spread, price impact, and volume profiles (Cont–Stoikov–Talreja).
Chromatic Polynomial via Deletion–Contraction
Interactive graph builder computing P(G,k) exactly — deletion–contraction recursion, chromatic number detection, greedy coloring, and conflict highlighting.
Triatomic Normal Modes
Symmetric stretch, antisymmetric stretch, and bending modes of a linear triatomic molecule.
Subcritical Bifurcation & Fold
Bistability, hysteresis, and catastrophic jumps in a subcritical pitchfork with saddle-node folds.
Quantum Hall — Laughlin Argument
Topologically exact conductance plateaux σ_xy = νe²/h via Laughlin's gauge-threading argument.
Household Epidemic & SAR
Reed-Frost chain binomial model of within-household transmission and secondary attack rates.
Complex Threshold Cascades
Watts (2002) threshold model: global social cascades from local adoption thresholds on random networks.
Phononic Crystal Bragg Stopband
Elastic wave dispersion and Bragg band gaps in a periodic layered acoustic medium.
Winding Number of Random Walk
How many times does Brownian motion wind around the origin? Spitzer's Cauchy-log theorem visualized.
Anisotropic Dendritic Growth
Phase-field simulation of snowflake-like dendrites: anisotropy selects tip speed and symmetry.
Cosmic String Network
Kibble-Zurek mechanism: topological defects from symmetry-breaking transitions form a scaling string network.
Biological Clock Oscillators
Goodwin oscillator and repressilator: genetic feedback loops generating circadian and synthetic rhythms.
Sticky Particle Adhesion
Pressureless gas with inelastic collisions — Burgers shockwaves and cosmic structure.
Spectral Graph Clustering
Fiedler vector of graph Laplacian reveals optimal partitions via eigenvector geometry.
Rayleigh-Taylor Instability
Heavy fluid atop light: small perturbations grow into turbulent fingers and mushroom caps.
Master Stability Function
Network synchronizability via MSF: eigenvalue spectrum determines if coupling achieves sync.
Quantum Annealing Landscape
Quantum tunneling vs thermal hopping through energy barriers to find global minima.
Mean-Field Game Theory
Rational agents optimizing against the aggregate distribution — Nash equilibrium from McKean-Vlasov.
LWR Traffic Flow
Lighthill-Whitham-Richards model: shockwaves and rarefaction fans from the fundamental diagram.
Hyperbolic Discounting
Present-bias and preference reversal: why humans discount the future hyperbolically, not exponentially.
Activator-Substrate Patterns
Gierer-Meinhardt model: activator-substrate dynamics producing Turing spots and stripes.
Non-Equilibrium Flux Network
Directed probability fluxes in non-equilibrium steady states — entropy production and detailed balance breaking.
Laughlin Quasihole
Fractional quantum Hall effect: Laughlin wavefunction with e/3 quasihole excitations.
Network Motif Enrichment
Gene regulatory motifs — bifan, feedforward loops — enriched beyond random null models.
Fractional Brownian Motion
Hurst exponent H controls sub/super-diffusion: MSD ~ t^(2H) with long-range correlations.
Red Queen Coevolution
Host-parasite arms race: cycling allele frequencies in match-allele and gene-for-gene models.
Schrödinger Bridge
Optimal stochastic interpolation between distributions — entropy-regularized transport paths.
Anosov Geodesic Flow
Exponential trajectory divergence on negatively curved surfaces — the paradigm of chaos.
Persistence Barcode TDA
Vietoris-Rips filtration reveals H0 and H1 topological features as a point cloud grows.
Belief Propagation
Cavity method on sparse graphs: messages converge to exact marginals on trees, approximate elsewhere.
Vicsek Flocking Transition
Noise-driven order-disorder transition in self-propelled particles — collective motion emergent.
Renewal Theory & Aging
Hazard rate functions shape survival curves: constant, increasing (aging), and decreasing (infant mortality).
Random Matrix β-Ensembles
GOE/GUE/GSE level spacing statistics — Wigner surmise vs Poisson for β=1,2,4.
Rough Volatility
Log-volatility as fractional Brownian motion with H≈0.1 — rougher than Brownian, skewing vol surfaces.
Blinking Vortex Chaos
Alternating point vortices create chaotic advection — KAM tori and Poincaré sections in 2D flow.
Gō Model Protein Folding
Native-contact-based energy landscape drives lattice chain folding — funnel topology emerges.
Bounded Confidence Opinions
Deffuant-Weisbuch model: confidence bound ε determines consensus vs fragmentation.
SBM Community Detection
Spectral recovery of planted communities in stochastic block model — eigenvectors reveal structure.
Phononic Crystal Bandgap
Alternating mass-spring chain creates forbidden frequency bands — elastic waves reflected at interfaces.
Complex Contagion Cascades
Watts threshold model: social reinforcement creates all-or-nothing global cascades from local seeds.
Quantum Zeno Effect
Frequent measurement freezes quantum evolution — survival probability approaches 1 as interval → 0.
Dryland Vegetation Patterns
Klausmeier model: water-plant feedback creates migrating vegetation stripes in arid ecosystems.
Ergodicity Breaking
Time-average vs ensemble-average MSD diverge in CTRW subdiffusion — non-ergodic anomalous diffusion.
Cell Polarity Wave-Pinning
Mass-conserved Turing instability breaks symmetry — stable polarity fronts in the Mori-Jilkine model.
Transcriptional Bursting
On/off promoter switching generates negative-binomial mRNA distributions — noise beyond Poisson.
Quantum Discord
Quantum correlations beyond entanglement: discord = total minus classical correlations in two-qubit states.
Viscous Fingering
Saffman-Taylor instability in Hele-Shaw cell: less viscous fluid fingers into more viscous — DLA-like branching.
Neutral Theory of Biodiversity
Hubbell's ecological drift: species abundance distributions from stochastic birth-death-speciation.
Lévy Stable Flights
Power-law jump distributions with stability index α — superdiffusive paths with fat tails.
Motility-Induced Phase Separation
Self-propelled particles self-organize into dense clusters — MIPS without attractive forces.
Dynamical Mean Field Theory
DMFT self-energy and spectral function reveal the Mott metal-insulator transition as U/W increases.
Population Neural Code
Von Mises tuning curves and population vector decoding — Fisher information quantifies coding precision.
Interdependent Network Cascades
Coupled networks exhibit abrupt first-order collapse — mutual dependence makes systems fragile.
1D Ising Transfer Matrix
Exact solution via transfer matrix eigenvalues — free energy, magnetization, and susceptibility analytically.
Geodesic Deviation & Curvature
Jacobi equation: nearby geodesics converge on spheres, diverge on saddles — curvature made visible.
Shannon Capacity & Water-Filling
C = B log₂(1+SNR) and optimal power allocation across parallel channels via water-filling.
Active Gel & Cortex Flow
Actin-myosin active stress drives spontaneous cortical flows — non-equilibrium mechanics of the cell.
DNA Melting Bubbles
Poland-Scheraga model: heterogeneous GC content creates complex denaturation curves and breathing bubbles.
Tipping Points & Early Warnings
Critical slowing down near fold bifurcations — variance and autocorrelation signal approaching collapse.
Cultural Evolution
Conformity bias, payoff bias, and random copying shape trait frequency dynamics in cultural transmission.
Magnons & Spin Waves
Holstein-Primakoff bosons: ferromagnetic dispersion ω(k)=4JS(1−cos ka) and thermal magnon population.
BEC Finite-Size Effects
Bose-Einstein condensation in finite systems — ground-state fraction, shifted Tc, and level occupation.
Adaptive Network Coevolution
Node states and topology coevolve — rewiring away from infected neighbors drives fragmentation.
Optical Solitons & SPM
Nonlinear Schrödinger equation: dispersion and self-phase modulation balance to form stable solitons.
Explosive Percolation
Achlioptas product rule delays giant component — near-discontinuous transition vs standard percolation.
Rock-Paper-Scissors Dynamics
Replicator dynamics on the simplex — cyclic dominance creates perpetual spiraling coexistence.
Stochastic Resonance
Optimal noise enhances detection of subthreshold signals — SNR peaks at intermediate noise intensity.
Magnetic Skyrmions
Topologically protected spin whirls with integer charge Q — candidate for ultra-dense magnetic memory.
Social Force Crowd Model
Helbing's social forces: desired velocity, repulsion, and walls — lane formation and bottleneck jamming.
Spots vs Stripes Selection
Two-mode competition in Turing systems selects pattern symmetry — spots, stripes, and labyrinthine patterns.
Hubbard Dimer Correlations
Two-site exact diagonalization: U/t tunes metal-to-Mott crossover and electron correlation.
Heterogeneous SIS Epidemic
Quenched mean-field on scale-free networks: λc→⟨k⟩/⟨k²⟩→0 — hubs eliminate epidemic threshold.
Gerchberg-Saxton Phase Retrieval
Iterative constraint projection recovers phase from intensity measurements in image and Fourier planes.
TSP Simulated Annealing
Thermal annealing escapes local optima in the traveling salesman problem — energy landscape navigation.
EW vs KPZ Surface Growth
Nonlinear KPZ term changes roughening exponents — β=1/3 vs EW β=1/4 in one dimension.
Neural Field Bumps & Waves
Amari neural field: localized activity bumps for working memory and traveling waves from E/I coupling.
Slow Manifold Reduction
Fast-slow systems collapse onto a slow manifold — center manifold theory reduces dimension dramatically.
Behavioral Epidemic Coupling
Awareness of infection reduces contact rates — behavior-disease feedback flattens epidemic curves.
Turbulent Multifractal Spectrum
f(α) singularity spectrum from structure function scaling — anomalous exponents beyond K41.
Neutral vs Niche Biodiversity
Log-series (neutral drift) vs log-normal (niche partitioning) — two theories of forest species abundance.
Jarzynski & Crooks Relations
Work fluctuation theorems: ⟨e^{-βW}⟩=e^{-βΔF} verified from non-equilibrium switching distributions.
Protein Aggregation Kinetics
Nucleation-dependent polymerization: lag phase, seeding shortcuts nucleation barrier — amyloid growth model.
Bias-Variance Tradeoff
Decomposing prediction error: underfitting vs overfitting and the sweet spot of model complexity.
Optimal Foraging Theory
Marginal value theorem: optimal patch departure when instantaneous gain equals average environment rate.
Oscillator Death
Diffusive coupling drives amplitude death — oscillators quench each other into stable silence.
BB84 Quantum Key Distribution
Quantum bases and eavesdropping detection — information-theoretic security from measurement disturbance.
DLA Fractal Growth
Random walkers aggregate into branching fractal clusters — diffusion-limited growth with dimension ~1.71.
Chimera States
Nonlocal coupling on oscillator rings: synchronized and incoherent domains coexist spontaneously.
Hertzsprung-Russell Diagram
Stellar evolution tracks across the HR diagram — main sequence, red giant branch, and white dwarf cooling.
Ant Colony Optimization
Pheromone reinforcement and evaporation guide ant swarms to shortest paths — emergent collective intelligence.
SSH Topological Edge States
Su-Schrieffer-Heeger model: winding number predicts in-gap edge states in the topological phase.
Blasius Boundary Layer
Flat plate boundary layer: Blasius ODE solution, δ~√(νx/U∞), and wall shear stress profile.
Quantum Hall: Chalker-Coddington
Network model of plateau transitions and conductance quantization
STDP Synaptic Plasticity
Spike-timing dependent plasticity and Hebbian learning windows
Wada Basin Fractals
Three interleaved basins of attraction meeting at every boundary point
Bose-Hubbard: Mott Lobes
Superfluid-Mott insulator quantum phase transition mean-field diagram
Epidemic Network Percolation
SIR spreading on random networks near the bond percolation threshold
Rényi Entropy & Multifractals
Multifractal spectrum f(α) and generalized dimensions D_q
Fiber Bundle Holonomy
Parallel transport on curved surfaces accumulates geometric phase
Diffusion-Limited Aggregation
DLA clusters grown from random walkers with fractal dimension ~1.71
Quantum Zeno Effect
Frequent measurement freezes quantum evolution — watched pot never boils
Kondo Resonance
Magnetic impurity screened by conduction electrons below T_K
SOC Forest-Fire Clusters
Self-organized criticality with power-law cluster size distributions
Fisher-Rao Information Geometry
Hyperbolic geometry on the manifold of Gaussian distributions
Hyperbolic Poincaré Tiling
Tessellations of the hyperbolic plane via Möbius transformations
φ⁴ Kink-Antikink Solitons
Topological soliton collisions in phi-fourth scalar field theory
Stochastic Epidemic Final Size
Bimodal outbreak distribution near the epidemic threshold R₀=1
Wilson-Cowan Neural Mass
Excitatory-inhibitory neural population dynamics and alpha rhythms
Phase-Field Crystal Growth
Dendritic solidification patterns from phase-field simulation
Quantum Weak Values
Anomalous weak measurement values via pre- and post-selection
Full Counting Statistics
Electron transfer distributions and cumulants in mesoscopic transport
Finite-Size Scaling Collapse
Data collapse for 2D Ising magnetization using critical exponents
Modulational Instability
Benjamin-Feir instability breaks a uniform wave into soliton trains
Magnon Spin-Wave Dispersion
Spin wave propagation and ferromagnetic resonance in exchange-coupled chains
Ecological Regime Shifts
Fold bifurcation, hysteresis, and catastrophic tipping points in ecosystems
WKB Quantum Tunneling
Semiclassical transmission through potential barriers via WKB approximation
Feed-Forward Loop Motif
Coherent FFL gene circuit acts as sign-sensitive delay and pulse generator
Point Vortex Dynamics
Hamiltonian point vortex system: integrable pairs and chaotic triplets
Carnot Cycle & Entropy
Entropy production in reversible vs irreversible thermodynamic cycles
Stochastic Resonance
Optimal noise enhances weak signal detection via stochastic resonance
SSH Topological Chain
Su-Schrieffer-Heeger model: winding number and topological edge states
Chemical Master Equation
Stochastic birth-death kinetics and probability distributions beyond the ODE
Boltzmann Entropy S=k·ln(Ω)
Phase space volume and microcanonical entropy for an ideal gas
Baker's Map Chaotic Mixing
Stretch-and-fold horseshoe dynamics creating fractal tracer patterns
Coulomb Blockade Quantum Dot
Single-electron tunneling and Coulomb oscillations in a quantum dot
Active Brownian Particles MIPS
Motility-induced phase separation in self-propelled particle suspensions
Lloyd's Algorithm & Voronoi
Iterative centroidal Voronoi tessellation for optimal quantization
DMD Koopman Spectrum
Dynamic mode decomposition extracts coherent structures from flow data
Heterogeneous Mean-Field SIR
Degree-stratified epidemic dynamics on scale-free networks
Standard Map: KAM & Chaos
Chirikov standard map Poincaré section: tori, islands, and chaotic sea
Jarzynski Equality
Work fluctuation theorem: free energy from nonequilibrium pulling experiments
Auxin Canalization PIN
Self-organizing auxin vein patterning via PIN transporter feedback
Kitaev Honeycomb Majoranas
Majorana fermions and vison excitations in the Kitaev honeycomb model
Sandpile Identity Fractal
Self-similar fractal pattern of the abelian sandpile group identity
Activator-Substrate Patterns
Substrate-depletion reaction-diffusion producing Turing-like patterns
First-Passage Time Distribution
Lévy-Smirnov 1/t^{3/2} distribution for random walk hitting an absorber
Kelvin-Helmholtz Instability
Shear flow vortex sheet rolling up into Kelvin-Helmholtz billows
Red Queen Coevolution
Host-parasite allele frequency cycles under fluctuating selection
QEC Repetition Code Threshold
Logical vs physical error rates for the quantum repetition code
Spectral Graph Clustering
Fiedler vector and spectral embedding reveal graph community structure
Mean-Field Game Crowd Motion
Nash equilibrium crowd flow balancing travel time and congestion cost
Second Harmonic Generation
Phase-matched SHG in nonlinear crystals: coherent frequency doubling
Preisach Magnetic Hysteresis
Memory-dependent hysteresis loops from distribution of bistable relays
Superfluid Vortex Defect
Order parameter profile and phase winding around a topological vortex
MDP Value Iteration
Value propagation and optimal policy arrows in a grid-world MDP
Quantum Phase Estimation
Phase kickback circuit for eigenvalue readout with ancilla precision
Protein Elastic Network Modes
Low-frequency collective motions of proteins via elastic network model
Oregonator BZ Clock
Belousov-Zhabotinsky chemical oscillator via the Oregonator ODE model
Lévy Stable Distributions
Heavy-tailed generalized CLT attractors parameterized by stability index α
Moody Diagram Pipe Flow
Friction factor vs Reynolds number for laminar, transitional, and turbulent flow
Wright-Fisher Genetic Drift
Allele fixation and loss in finite populations under drift and selection
Entanglement Area vs Volume Law
Ground state area-law vs excited state volume-law entanglement entropy scaling
Lattice Boltzmann D2Q9
D2Q9 lattice Boltzmann fluid simulation with cylinder wake and streamlines
Chern-Simons Knot Invariants
Topological quantum field theory computing Jones polynomials via skein relations
Clonal Selection Affinity
B cell affinity maturation in germinal centers via selection and hypermutation
Pendulum Phase Portrait
Nonlinear pendulum libration, rotation, and separatrix in full phase space
Complex Contagion Cascades
Social behavior spreading requires multiple exposures — unlike simple contagion
Quantum Annealing
Transverse field tunneling through energy barriers for combinatorial optimization
Granular Gas Cooling
Inelastic Maxwell model: Haff's law cooling and non-Gaussian velocity tails
NK Rugged Fitness Landscape
Epistatic NK model: local optima density and adaptive walk trapping
Stadium Billiard Chaos
Bunimovich stadium: chaotic geodesic flow and exponential trajectory divergence
Vicsek Flocking Coarsening
Polar domain coarsening in the Vicsek model of flocking active matter
Vacuum Rabi Splitting cQED
Cavity QED polariton avoided crossing and vacuum Rabi oscillations
Milling & Schooling Motion
Collective motion phases: disordered, milling rings, and polarized schools
SBM Detectability Transition
Planted partition phase transition: when community detection becomes impossible
Euler Column Buckling Modes
Critical loads and deflection shapes for elastic column buckling modes
Labyrinthine RD Patterns
Stripe-to-spot transition and labyrinthine pattern coarsening in reaction-diffusion
Continuous-Time Quantum Walk
Ballistic quantum spreading vs diffusive classical random walk on graphs
Kingman Coalescent
Backward-in-time gene genealogy coalescence and time to MRCA
Rayleigh-Bénard Convection
Onset of convection rolls above critical Rayleigh number Ra_c≈1708
k-Core Network Decomposition
Iterative k-core peeling reveals network's dense inner cores
Deffuant Bounded Confidence
Opinion clustering when agents only interact within confidence bound ε
Wigner Semicircle Law GOE
Random matrix eigenvalue density converging to the semicircle distribution
LV Competition Isoclines
Two-species Lotka-Volterra competition: coexistence, exclusion, and bistability
CTRW Subdiffusion
Power-law waiting times produce subdiffusive ⟨x²⟩~t^α with α<1
Topological Magnon Hall Effect
Berry curvature and transverse magnon drift in a topological magnon insulator
Kerr Cavity Bistability
Driven-dissipative phase transition with hysteresis in a nonlinear optical cavity
Wisdom of Crowds
Bayes-optimal aggregation: collective accuracy improves as 1/N with crowd size
Viscous Gravity Current
Self-similar Barenblatt spreading solution for a viscous gravity current
Gamma Neural Oscillations
Interneuron network PING/ING mechanisms generating 30-80 Hz gamma rhythms
Quantum S-Matrix Resonances
Breit-Wigner transmission resonances and S-matrix poles in 1D scattering
Turing Digit Patterning
Reaction-diffusion model of limb digit formation: domain size controls finger count
Patchy Particle Self-Assembly
Directional patch interactions drive colloidal particles to crystalline order
Causal Dynamical Triangulation
Monte Carlo simplicial geometry and Hausdorff dimension in quantum gravity
Spiral Wave Breakup
Excitable media spiral to fibrillation transition via wave breakup
Kernel SVM Decision Boundary
RBF and polynomial kernel SVMs with margin visualization and C/γ tuning
Master Stability Function
Network synchronization onset via Laplacian eigenvalue spectrum and MSF
Moist Convection Parcel
CAPE, CIN, and conditional instability from atmospheric temperature soundings
Rubber Elasticity Affine Model
Neo-Hookean stress-strain from affine deformation of polymer network strands
Shannon Channel Capacity
Noisy channel coding theorem: capacity C=1-H(p) for binary symmetric channel
Metacommunity Patch Dynamics
Species colonization-extinction and sorting across heterogeneous habitat patches
Quantum Chaos Level Statistics
Poisson vs Wigner-Dyson level spacing: diagnosing quantum chaos via r-statistic
Stokes Creeping Flow
Low-Reynolds streamlines and the Stokes paradox for creeping flow past a cylinder
Critical Casimir Effect
Thermal fluctuation-induced force between plates near a critical point
Integrate-and-Fire Sync
Coupled leaky integrate-and-fire neurons synchronizing via collective firing
Kauffman Boolean Network
NK random Boolean networks: ordered, chaotic, and critical K=2 dynamics
Gravitational Wave Chirp
Binary inspiral chirp waveform: amplitude and frequency rising to merger
Yang-Mills Instanton
BPST instanton profile and topological charge density in Yang-Mills theory
SIR Contact Tracing
Network epidemic control via contact tracing and ring quarantine
Pyragas Chaos Control
Delayed feedback stabilizes unstable periodic orbits within chaotic attractors
Polymer Brush Extension
Alexander-de Gennes polymer brush: steric repulsion and compression osmotic pressure
Mandel Q Photon Statistics
Coherent, sub-Poissonian, and super-Poissonian photon number distributions
Nonlinear Wave Envelopes
NLS equation: envelope solitons, modulation instability, and soliton collisions
Contact Process & Directed Percolation
Absorbing-state phase transition on a 1D lattice — DP universality class
Protein Folding HP Model
Dill's hydrophobic-polar lattice model: MCMC search for minimum-energy conformations
Condorcet Voting Cycles
Arrow's impossibility: majority preference cycles and intransitive collective choice
Resonant Tunneling Diode
Double-barrier transfer matrix: transmission resonances and quasi-bound states
Saffman-Taylor Fingering
Laplacian growth: viscous fingering in a Hele-Shaw cell (DLA regime)
Gene Network Motifs
Autoregulation and coherent/incoherent feed-forward loops in transcription networks
Fokker-Planck Double Well
Probability density dynamics and Kramers escape in a bistable double-well potential
XY Model Vortex Dynamics
Kosterlitz-Thouless transition: vortex-antivortex pairs and topological defect annihilation
Stochastic Lotka-Volterra
Demographic noise drives population cycles, extinction risk, and phase-portrait spirals
Hyperbolic Tiling — Poincaré Disk
Regular {p,q} tilings of the hyperbolic plane: geodesics as circular arcs orthogonal to the boundary disk
Age-Structured Epidemic (SEIR)
SEIR epidemic with POLYMOD contact matrix — age groups mix asymmetrically, driving heterogeneous disease spread
Resonance — Forced Oscillator
Explore amplitude peaks and phase lag as driving frequency sweeps through resonance in a damped harmonic system
Perron-Frobenius Markov Eigenvalue
Power iteration converges to the unique stationary distribution guaranteed by Perron-Frobenius for irreducible chains
2D Ising Model — Metropolis MC
Ferromagnetic phase transition at Onsager's exact critical temperature T_c ≈ 2.269 via Metropolis sampling
Active Matter — MIPS Transition
Motility-induced phase separation: self-propelled particles cluster with no attractive forces at high density
DLA Fractal Growth
Diffusion-limited aggregation, Eden model, and ballistic deposition — fractal dimension d_f ≈ 1.71 emerges from random walks
Quantum Coherent State
Minimum-uncertainty Gaussian wavepackets oscillate like classical particles in the quantum harmonic oscillator potential
Stochastic Block Model
Community detection near the Kesten-Stigum information-theoretic threshold in planted-partition random graphs
Vicsek Flocking Model
Phase transition from disordered motion to global flocking as angular noise crosses the Vicsek critical threshold η_c
Calogero–Moser Particles
Exactly integrable N-body system with inverse-square repulsion — Lax matrix yields N conserved quantities; phase-space trajectories visualized
Epidemic Bond Percolation
Edges open with probability p — at the critical threshold p_c = 1/⟨k⟩ a giant connected component suddenly spans the network
Wave Turbulence: Four-Wave Resonance
Weakly nonlinear wave interactions via k₁+k₂=k₃+k₄ resonance; energy cascades toward the Kolmogorov-Zakharov k⁻³ spectrum
Metapopulation Patch Dynamics
Levins model: patches colonize and go extinct; persistence requires c/e > 1; spatial patterns and extinction cascades emerge
Coulomb Blockade Staircase
Single-electron transistor: charging energy quantizes electron addition, producing equally spaced conductance peaks in gate voltage
Phononic Crystal Stopband
Bragg scattering in a periodic elastic medium opens a bandgap — waves in the stopband become evanescent and cannot propagate
Chimera Death
Coupled Stuart-Landau oscillators: chimera states (sync/async coexistence) collapse into oscillation death as coupling crosses a threshold
Cross-Diffusion Turing Instability
Cross-diffusion drives pattern formation even when self-diffusion alone is stable — spontaneous spatial segregation beyond the classical activator-inhibitor
Sandpile Avalanches (BTW)
Bak-Tang-Wiesenfeld 1987: toppling cascades follow a power law P(s) ~ s⁻¹·⁵ — the canonical demonstration of self-organized criticality
Amari Neural Field Bump
Mexican-hat lateral inhibition creates localized activity bumps — a continuous attractor for spatial working memory and head-direction coding
KPZ Interface Roughening
Kardar-Parisi-Zhang universality: nonlinear stochastic growth with exact exponents β=1/3, χ=1/2, z=3/2 — log-log scaling measured live
Epidemic on Heterogeneous Networks
Degree-stratified SIR mean-field theory on scale-free networks — vanishing epidemic threshold when γ≤3 and ⟨k²⟩ diverges
Anderson Localization in 1D
All eigenstates localize exponentially in 1D random potentials — transfer matrix reveals Lyapunov exponent and localization length ξ~1/W²
2D Turbulence & Vortex Merger
Inverse energy cascade: same-sign vortices merge into larger coherent structures via Biot-Savart interactions — Kraichnan's 1967 prediction made visible
Minority Game Phase Transition
Grand canonical minority game — adaptive agents competing to be in the minority show a phase transition at α_c separating herd from efficient-market regimes
Coupled Map Lattice Defect Turbulence
Logistic CML spatiotemporal chaos — defects nucleate and propagate through the lattice, preventing pattern formation in the space-time diagram
First Passage Hitting Times
1D random walk first passage distribution — Lévy-Smirnov P(T)~t^(-3/2) tail with divergent mean; bias shifts the threshold dramatically
Stochastic Gene Expression Bimodality
Gillespie SSA of bistable gene circuits — intrinsic noise drives switching between ON/OFF expression states, producing bimodal protein distributions
SSH Model Winding Number
Su-Schrieffer-Heeger chain: bulk winding number W=0 (trivial) vs W=1 (topological) — bulk-boundary correspondence and zero-energy edge states
Directed Percolation Phase Transition
2D contact process at the DP universality class — absorbing-state transition at p_c≈0.705 with exact exponents β=0.584, ν=0.734
Hénon-Heiles Poincaré Section
Surface-of-section mapping reveals KAM tori, island chains, and chaotic seas in a 2D Hamiltonian — at E→1/6 the entire phase space becomes stochastic
Topological Flocking (Voronoi)
Metric-free Vicsek-style flocking where each agent aligns with its k topological neighbors — replicating starling murmurations with 8-fold coordination
Glauber Dynamics — Ising Coarsening
Single-spin-flip dynamics at and near Tc — domain coarsening follows Allen-Cahn ξ~t^(1/2) growth law with Glauber detailed-balance acceptance rates
Kelvin-Helmholtz Instability
Vortex sheet rollup at a shear interface — perturbations grow at rate σ=k|ΔU|/2 then roll into billows, as seen in clouds, oceans, and planetary bands
Rayleigh-Taylor Instability
Heavy fluid above light fluid: fingers grow exponentially with rate √(Agk) then form mushroom-cap plumes — key in ICF implosions and supernovae
Entropy Rate — Hidden Markov Model
Forward algorithm traces the instantaneous entropy rate H(Xₙ|X₁…Xₙ₋₁) converging to the true HMM entropy rate — driven by hidden state uncertainty
Master Equation: Birth-Death Process
Exact probability distribution P(n,t) integrated from the birth-death master equation, relaxing toward Poisson(λ/μ) stationary distribution
Kuramoto-Sakaguchi Chimera States
Non-local coupling with phase lag α creates chimera states — coexisting synchronized and incoherent populations in identical oscillators (Kuramoto & Battogtokh, 2002)
Ammann-Beenker Quasiperiodic Tiling
8-fold symmetric quasicrystal tiling by deflation — squares and 45° rhombi with silver ratio λ=1+√2 inflation, dense Bragg peaks without periodicity
Thouless Topological Pump (Rice-Mele)
Adiabatic charge transport quantized by the Chern number — one full pump cycle of the Rice-Mele Hamiltonian moves exactly one electron across the chain
Gray-Scott Reaction-Diffusion v2
U+2V→3V with tunable feed/kill rates produces spots, stripes, mazes, and chaos — canonical spatiotemporal pattern formation with four interactive presets
Magnetic Skyrmion Hopf Texture
Topologically protected spin vortices stabilized by Dzyaloshinskii–Moriya interaction — winding number Q computed live as DMI, exchange, and field vary
Lévy Stable Distributions
GEV attractor for heavy-tailed sums — Chambers-Mallows-Stuck sampling with α, β, c, μ controls; compare Gaussian, Cauchy, and Lévy tails live
Voter Model — Coarsening to Consensus
Agents copy random neighbors: coarsening time scales as L² log L in 2D, interface density decays as 1/(t log t), consensus reached by fluctuations alone
Continuous-Time Quantum Walk
U(t)=exp(−iγAt) on cycle, complete, path, hypercube, and star graphs — amplitude interference visualized via Jacobi eigendecomposition of the adjacency matrix
Actin Treadmilling & Molecular Motors
ATP-actin adds at the barbed end, ADP-actin falls off the pointed end — net treadmilling flux drives cell motility as myosin motors walk toward the barbed end
Brusselator 2D Pattern Formation
When B exceeds the Turing threshold the homogeneous state spontaneously breaks into spots, stripes, and labyrinths — RD instability from differential diffusion
Extreme Value: Gumbel, Fréchet, Weibull
Block maxima converge to GEV(μ,σ,ξ) — shape ξ governs light, heavy, or bounded tails; return level plot shows 1-in-T-year event magnitudes
Real-Space Renormalization Group
Decimation RG on the 1D Ising chain: K′=½ ln cosh(2K) cobweb diagram flows to K*=0 or ∞ — fixed points, RG flow, and correlation length visualized
Wright-Fisher Model & Coalescent
Stochastic allele frequency drift, selection, and mutation; Kingman's coalescent traces lineages backward to MRCA — fixation probability equals initial frequency
Nonlinear Pendulum Phase Portrait
Driven damped pendulum: libration, rotation, and chaos in phase space. Tune driving amplitude to trigger chaotic trajectories.
Spectral Clustering Laplacian
Graph Laplacian eigenvectors reveal cluster structure invisible to linear methods. Separate rings, moons, and blobs spectrally.
Bell Inequality Violation
CHSH test of quantum entanglement: sample correlated photon pairs and watch |S|>2 violate classical bounds, reaching 2√2.
Burgers Equation Shocks
Inviscid Burgers equation forms shocks from smooth initial data. Add viscosity to regularize. Watch characteristics cross.
2D Spin Glass Frustration
Edwards-Anderson model with random ±J bonds. Frustrated plaquettes trap the system in glassy metastable states.
Allometric Scaling West Networks
WBE theory: metabolic rate scales as M^(3/4) from space-filling fractal vascular networks. Tune branching geometry to change β.
Blahut-Arimoto Channel Capacity
Iterative algorithm converges to Shannon channel capacity. Watch alternating optimization find the optimal input distribution.
Turbulent Pipe Flow Log Law
Von Kármán log-law of the wall: viscous sublayer, buffer layer, and inertial region. Reynolds stress budget profiles.
Bicoid Morphogen Gradient
Reaction-diffusion patterning in Drosophila: exponential bicoid gradient encodes positional information via threshold crossings.
Adaptive Network Coevolution
SIS epidemic on a rewiring network: susceptibles avoid infected neighbors. Competing timescales produce endemic, extinction, and fragmentation phases.
Cavity QED Vacuum Rabi
Jaynes-Cummings model: a two-level atom in a cavity undergoes vacuum Rabi oscillations. Collapse and revival reveal the quantum nature of the field.
Smoluchowski Coagulation
Cluster mass distributions evolving under coagulation kernels. Constant, sum, and product kernels exhibit distinct gelation behaviors and power-law tails.
Fibonacci Quasicrystal Diffraction
The Fibonacci substitution lattice L→LS, S→L generates a quasiperiodic structure whose diffraction reveals sharp Bragg peaks indexed by two integers.
Topological Insulator Dirac Cone
3D topological insulators host surface Dirac cones with spin-momentum locking. Explore the band structure, Fermi surface, and Z₂ topological invariant.
Branching-Annihilating Walk
Particles branch and annihilate pairwise. Parity conservation separates two universality classes — DP and PC — with distinct critical exponents.
Totalistic CA Rule Space
Outer totalistic 1D cellular automata. Navigate Wolfram's rule space from Class I fixed points to Class IV universal computation (Rule 110).
Hertz Contact Stress Field
Subsurface stress distributions in elastic contact mechanics. Von Mises, normal, and shear stresses reveal why failure initiates below the surface.
Kuramoto Finite-N Synchronization
N Lorentzian-distributed phase oscillators coupled all-to-all. Spontaneous synchronization above K_c with finite-N fluctuations in the order parameter.
Stochastic Resonance Paleoclimate
Double-well climate model: weak Milankovitch forcing plus optimal noise enables resonant glacial-interglacial transitions. SNR peaks at the ideal noise level.
Preisach Ferromagnetic Hysteresis
Superposition of bistable hysterons over the Preisach (α,β) plane traces realistic M-H hysteresis loops. Explore how the distribution shapes coercivity and remanence.
Vesicle Shape & Helfrich Energy
Lipid bilayer vesicles minimize Helfrich bending energy under fixed area and volume. Tune reduced volume ν and spontaneous curvature to watch sphere→prolate→stomatocyte transitions.
QAOA Energy Landscape
Quantum Approximate Optimization Algorithm: visualize the ⟨H_C⟩(γ,β) landscape for MaxCut. Watch gradient descent find approximate solutions as circuit depth p increases complexity.
Kerr Self-Phase Modulation
Intensity-dependent refractive index broadens laser pulse spectra via SPM. Explore how φ_max = γP₀L generates new frequency components and spectral splitting.
Bayesian Social Learning & Herding
Sequential agents update beliefs using private signals + social observation. Cascades lock in wrong answers when public history overwhelms private information.
Kagome Antiferromagnet Frustration
Corner-sharing triangles prevent simultaneous satisfaction of all antiferromagnetic bonds. Monte Carlo on the kagome lattice reveals spin-liquid ground states with massive degeneracy.
Epidemic Spatial Patch Model
Multi-patch SIR dynamics on a heterogeneous network with mobility fluxes. Spatial structure raises or lowers epidemic thresholds depending on hub connectivity.
Mechanochemical Pattern Formation
Murray-Oster model: morphogen concentration couples to tissue strain, driving spatial patterns via mechanical instability — a Turing-like mechanism without reaction-diffusion.
Chebyshev Spectral Collocation
Chebyshev–Gauss–Lobatto nodes enable exponential convergence for smooth functions. Compare approximation error, spectral coefficients, and convergence rates vs. polynomial degree N.
Carbon Cycle Reservoir Box Model
Five-box global carbon model: atmosphere, surface ocean, deep ocean, biosphere, sediments. Emit CO₂ pulses and watch multi-timescale equilibration across centuries to millennia.
Mackey-Glass Delay Chaos
Single time-delay τ drives transitions from steady state → limit cycle → period-doubling → chaos in this blood-cell production model. Observe delay embedding phase portraits.
Entropy & Arrow of Time
Boltzmann H-theorem: particle gas relaxing to Maxwell–Boltzmann distribution, H(t) decreasing.
Neutrino Oscillations
PMNS matrix three-flavor mixing: survival and appearance probabilities as a function of baseline.
Magnetic Skyrmion
Topological charge Q=±1 spin texture with Néel/Bloch helicity and topological charge density map.
Activator-Substrate Patterns
Meinhardt–Gierer Gray-Scott reaction-diffusion: spots, stripes, labyrinthine, and hole patterns.
Quantum Decoherence
Lindblad master equation on the Bloch sphere: energy relaxation and dephasing erasing coherence.
Active Matter Flocking
Vicsek model polar flock: noise-driven phase transition from disordered gas to ordered collective motion.
Kondo Effect
Impurity spin screening cloud and resistivity minimum: Kondo temperature, Hamann formula, Friedel oscillations.
Jamming Force Chains
Granular packing at the J-point: force chain networks emerging above random close packing fraction.
Stochastic Resonance
Optimal noise enhances neural signal detection: SNR peak, ensemble threshold crossings, SNR sweep.
NLS Envelope Solitons
Nonlinear Schrödinger equation: bright/dark solitons, two-soliton elastic collisions, modulational instability.
Fokker-Planck: Overdamped Landscape
Probability density evolving under drift and diffusion across a double-well potential.
Sierpiński Gasket — IFS Attractor
Chaos game: three contractive maps converge to the fractal attractor with dim ≈ 1.585.
Topological Defect Pair Annihilation
XY-model vortex-antivortex pairs attract and annihilate via relaxational spin dynamics.
Ferroelectric Hysteresis Loop
Landau free energy drives polarization switching, tracing out the P-E hysteresis loop.
SIR Epidemic on Heterogeneous Network
Scale-free hubs lower the epidemic threshold — even tiny β triggers outbreaks.
Rayleigh-Bénard Convection Rolls
Fluid heated from below becomes unstable at Ra > 1707 and forms convection rolls.
Entanglement Entropy & Area Law — 1D
Compare area-law, CFT logarithmic, and volume-law entanglement entropy across block sizes.
Lévy Stable Distributions
Heavy-tailed Lévy stable laws with power-law tails |x|^(−1−α) — generalizing the Gaussian.
Müller's Ratchet — Mutation Accumulation
Deleterious mutations irreversibly accumulate in asexual populations, eroding mean fitness.
2D Ising — Metropolis & Wolff Cluster
Square-lattice Ising model at Onsager's T_c ≈ 2.269J with Wolff cluster updates.
Quantum Zeno Effect
Pulsed measurements freeze quantum decay — frequent observation suppresses state evolution.
BTW Sandpile Avalanches
Self-organized criticality: grains topple in power-law avalanches without tuning any parameter.
Chimera States
Identical coupled oscillators spontaneously split into synchronized and incoherent clusters.
Saffman–Taylor Fingering
Viscous fingering instability in a Hele-Shaw cell — less-viscous fluid invades in fractal fingers.
First Passage Time
Random walkers absorbed at a boundary follow a heavy-tailed inverse Gaussian distribution.
Saddle-Node Bifurcation
Normal form ẋ = r + x² — equilibria collide and annihilate at the fold point r = 0.
Protein Folding Funnel
Bryngelson–Wolynes funnel landscape with ruggedness — folding time scales as exp(ε²/T²).
Lattice Boltzmann Cavity Flow
D2Q9 LBM simulates lid-driven cavity flow — recirculating vortices at tunable Reynolds number.
Epidemic Threshold on Networks
On scale-free networks the epidemic threshold vanishes — any infection spreads for N→∞.
Coherent State Wigner Function
QHO coherent states orbit classically in phase space — Wigner quasi-probability in real time.
Superfluid Vortex Quantization
Rotating BEC: Abrikosov triangular vortex lattice emerges as angular momentum is quantized in units of h/m.
Anderson Localization 1D
All eigenstates localize exponentially in 1D disorder — transfer matrix reveals wavefunction decay and localization length.
Coupled Map Lattice Chaos
Spatiotemporal chaos and synchronization in a 1D lattice of coupled logistic maps, visualized as a spacetime diagram.
Voter Model Coarsening
Opinion domains coarsen and consensus eventually emerges — interface density decays as 1/log(t) in 2D.
Capillary Wave Roughening
Thermal fluctuations roughen a liquid-gas interface; Fourier modes follow S(k) ~ T/(γk²+ρg) with logarithmic width growth.
Stochastic Block Model
Community detection phase transition: below SNR=1 threshold no algorithm can recover planted partitions better than chance.
Resonant Double-Barrier Tunneling
Exact transfer matrix transmission through two barriers shows perfect resonant tunneling T=1 at quasi-bound state energies.
Competitive Exclusion Principle
Lotka-Volterra competition: nullcline geometry determines whether coexistence or competitive exclusion wins — explore all four outcomes.
Geodesic Deviation & Tidal Forces
Riemann curvature drives separation of nearby geodesics: a ring of test particles stretches radially and squeezes tangentially near a black hole.
SSH Model Topological Edge States
Su-Schrieffer-Heeger chain: tune hopping ratio to cross the topological transition and watch protected edge states appear in the gap.
Alfvén Wave Propagation
Transverse waves along magnetic field lines in a conducting plasma: watch v_A = B/√(μ₀ρ) carry perturbations with resistive damping.
Replica Symmetry Breaking & Parisi q(x)
Spin glass order parameter q(x) and overlap distribution P(q) — Parisi's Nobel-winning solution to the SK model.
Frustrated Magnetism: Triangular & Kagome
Antiferromagnetic spins on triangular plaquettes cannot simultaneously satisfy all bonds — watch frustration and spin-liquid signatures emerge.
Extreme Value Statistics: GEV Universality
Fisher-Tippett-Gnedenko theorem: block maxima converge to Gumbel, Fréchet, or Weibull depending on tail behavior of the parent distribution.
Ecological Resilience & Tipping Points
Scheffer's lake model: hysteresis, catastrophic shifts between clear and turbid states, and early-warning signals near bifurcation.
Hyperbolic Geometry: Poincaré Disk Tiling
Regular {p,q} tilings of the hyperbolic plane in the conformal Poincaré disk model — geodesics as circular arcs, K = −1 curvature.
Quantum Weak Value Amplification
Aharonov-Albert-Vaidman: post-selected weak measurements yield A_w = ⟨f|A|i⟩/⟨f|i⟩, amplifying tiny signals beyond the eigenvalue spectrum.
Nonequilibrium Steady State: Driven Lattice Gas
Katz-Lebowitz-Spohn model: external field breaks detailed balance, generating persistent current and entropy production σ = J·E > 0.
Contact Process: Directed Percolation
Absorbing-state phase transition at λ_c ≈ 3.298 (1D): spacetime diagram of infection spreading near the DP universality class critical point.
Topological Fluid: Vortex Helicity
Moffatt helicity H = ∫ v·ω dV counts vortex linkage — a topological invariant of ideal Euler flow, from Hopf links to trefoil knot vortices.
Magnon Spin-Wave Dispersion
Holstein-Primakoff transformation: ferromagnet dispersion ω(k), coherence factors, and DOS across lattice types.
Stochastic Epidemic (Gillespie SSA)
Exact stochastic SIR simulation via Gillespie algorithm — multiple trajectories show variability around the mean-field ODE.
Nonlinear Pendulum Separatrix
Phase portrait of the driven, damped pendulum: separatrix between libration and rotation, onset of chaos with driving.
BCS Bogoliubov Quasiparticles
Bogoliubov transformation mixing electrons and holes: quasiparticle dispersion Eₖ = √(ξₖ² + Δ²) and BCS density of states.
Allen-Cahn Coarsening
Phase-ordering kinetics via Allen-Cahn PDE: domains grow as L(t) ~ t^(1/2) after a quench into the miscibility gap.
Random Matrix Level Repulsion
GOE/GUE/Poisson nearest-neighbour spacing statistics reveal universality classes — Wigner surmise vs Poisson.
Active Brownian Particles & MIPS
Self-propelled particles with rotational diffusion: motility-induced phase separation driven by swim pressure.
Random Walk → Diffusion Equation
The continuum limit a→0, τ→0 at fixed D = a²/2τ: histogram of N walkers converges to Gaussian solution of ∂P/∂t = D∇²P.
Arnold Tongues — Frequency Locking
Circle map Arnold tongues in (Ω, K) space: frequency locking, winding numbers, and the Devil's staircase at K=1.
Geometric Frustration — Spin Ice
Kagome spin ice with ice rules: Monte Carlo dynamics showing residual Pauling entropy from geometric frustration.
Majorana Fermions — Kitaev Chain
Topological wire hosts unpaired Majorana zero modes at its ends when |μ| < 2|t|.
Shannon Noisy Channel Coding — BSC
Capacity C = 1−H(p) of the binary symmetric channel; reliable comms iff rate R < C.
Reynolds Boids — Flocking Simulation
Separation, alignment, and cohesion rules produce emergent flocking behavior.
Cayley Graph — Abelian Groups
Directed graph of group elements connected by generator actions; ℤₙ × ℤₘ products form torus grids.
Chemical Master Equation — Birth-Death
Gillespie simulation of stochastic birth-death reaction network converges to Poisson steady state.
Microcanonical Entropy — Boltzmann Counting
S = k_B ln Ω: entropy from counting microstates of N two-level spins.
Hopf Fibration — S³ → S²
Every fiber of the Hopf map is a circle; any two distinct fibers are linked in 3-space.
Stochastic Resonance — Array SR
Optimal noise level maximizes SNR for subthreshold signal detection through threshold arrays.
Ecological Stoichiometry — Redfield Ratio
Phytoplankton maintain C:N:P = 106:16:1; Liebig's Law determines which nutrient limits growth.
Quantum Phase Estimation Algorithm
QPE estimates eigenphase φ of unitary U to precision 1/2ᵗ using t ancilla qubits and the QFT.
Witten Index & SUSY QM
Tr[(-1)^F e^{-βH}] counts topologically protected ground states in supersymmetric quantum mechanics.
Rule 110 Universality
Cook's proof that Wolfram's Rule 110 is Turing-complete via cyclic tag system embedding.
Gutzwiller Trace Formula
Semiclassical density of states from periodic orbits: quantum-classical correspondence in billiards.
Adiabatic Invariant J = ∮ p dq
Action variable conservation when a pendulum's length changes slowly — basis of the quantum adiabatic theorem.
Noise-Induced Transitions
Multiplicative white noise creates new stable states absent in the deterministic system — bimodal PDF from unimodal drift.
Quantum Discord
Separating quantum from classical correlations: D = I(A:B) − J(A:B) in bipartite mixed states.
Age-Structured Epidemic
McKendrick–Von Foerster PDE with age-dependent infection rates and contact mixing matrices.
Vorticity Stretching
Helical vortex tubes intensified by strain: Dω/Dt = (ω·∇)u driving the turbulent energy cascade.
Car Parking & RSA Jamming
Rényi's parking constant θ* ≈ 0.7477 — random sequential adsorption reaching the jamming limit.
O(3) Sigma Model Skyrmions
Belavin-Polyakov topological solitons with integer winding number Q and Bogomolny energy bound E ≥ 4π|Q|.
Erdős–Rényi Giant Component
Watch a giant connected component emerge as edge probability crosses 1/n.
Frenkel-Kontorova: C-IC Transition
Atoms on a substrate potential lock into commensurate phases or slide as soliton trains.
Marangoni Thermocapillary Flow
Surface tension gradients drive fluid motion — the physics of tears of wine.
Bloch Oscillations & Wannier-Stark
Electrons in a tilted periodic lattice oscillate rather than accelerate.
Gillespie Algorithm: Exact Stochastic Kinetics
Exact simulation of chemical reactions as discrete stochastic events.
Self-Avoiding Walk & Flory Exponent
Polymers swell in good solvent — end-to-end distance scales as N^ν with ν≈3/4 in 2D.
Young-Laplace: Wetting & Contact Angle
Surface energy balance determines whether a droplet spreads or beads up.
Rayleigh-Bénard Convection Onset
Convection rolls emerge above the critical Rayleigh number Ra = 1708.
Magnetic Skyrmion & Topological Charge
A topologically protected spin texture with winding number Q = −1.
Stochastic SIS Epidemic: Master Equation
Finite-size fluctuations cause extinction below the deterministic epidemic threshold.
Kuramoto Order Parameter
Coupled oscillators spontaneously synchronize above a critical coupling strength.
Debye-Hückel Electrostatic Screening
Ions in solution screen electric fields exponentially over the Debye length λ_D.
Anderson Localization
Disorder localizes quantum wavefunctions — diffusion stops in 1D and 2D for any disorder.
Stochastic Resonance: Double-Well
Optimal noise level amplifies weak periodic signals in a bistable system.
Adiabatic Elimination of Fast Variables
Slow manifold reduction collapses high-dimensional dynamics to an effective equation.
Chern-Simons & Hall Conductance
Topological Chern number quantizes the Hall conductance in integer multiples of e²/h.
Smoluchowski Coagulation & Gelation
Aggregating clusters form a spanning gel in finite time for multiplicative kernels.
Saddle-Node Bifurcation Ghost
After a saddle-node collision, trajectories slow dramatically near the ghost — critical slowing down.
KPZ Interface Roughening
Growing interfaces develop correlated roughness with universal KPZ scaling exponents.
Casimir Effect: Vacuum Fluctuations
Zero-point quantum fluctuations between conducting plates create an attractive force F ~ 1/d⁴.
Shilnikov Chaos: Spiral Attractor
A homoclinic orbit to a saddle-focus creates infinitely many horseshoes — Shilnikov's theorem.
Takens Embedding: Attractor Reconstruction
A scalar time series contains the full attractor geometry — Takens' embedding theorem.
First Passage Time: Random Walk
The time for a random walker to first reach an absorbing barrier has a heavy-tailed distribution.
Lévy Flight: Truncated Stable
Truncating the heavy tail of a Lévy distribution creates tempered stable motion with finite moments.
Ising 1D: Exact Transfer Matrix
The transfer matrix method solves the 1D Ising model exactly — no phase transition at finite T.
Doppler Cooling & Optical Molasses
Counter-propagating laser beams create a viscous force that slows atoms to microkelvin temperatures.
Quantum Zeno Effect
Frequent measurements freeze quantum evolution — the watched pot never boils.
Cross-Diffusion Turing Instability
Cross-diffusion terms broaden the parameter space for Turing pattern formation.
Voter Model: Coarsening Dynamics
Opinions coarsen toward consensus — interface density decays as 1/√t in 1D.
Hawkes Process: Self-Exciting Events
Each event increases the rate of future events — earthquake aftershocks, viral cascades.
Membrane Fluctuations & Helfrich Rigidity
Lipid bilayer shape fluctuations are governed by bending rigidity κ — soft membranes fluctuate wildly.
Fractional Brownian Motion & Hurst Exponent
Correlated random walks with H>1/2 are persistent, H<1/2 antipersistent — memory in noise.
Coupled Map Lattice: Spatiotemporal Chaos
Locally coupled chaotic maps display traveling waves, frozen chaos, and synchronization.
Thermodynamic Uncertainty Relation
Precision costs entropy: relative fluctuations × entropy production ≥ 2kB — a fundamental bound.
Epidemic on Networks: Heterogeneous Mean Field
Hubs with high degree lower the epidemic threshold to zero in scale-free networks.
Photon Bunching & Antibunching: g²(τ)
Classical light bunches (g²(0)>1), coherent light is Poissonian (g²(0)=1), quantum light antibunches (g²(0)<1).
Continuum Percolation: Boolean Model
Randomly placed discs form a spanning cluster above a critical area fraction η_c ≈ 0.676.
Moran Process: Genetic Drift & Fixation
In finite populations, neutral mutations fix with probability 1/N — drift overpowers selection when Ns≪1.
Elastoplastic Depinning: Avalanche Statistics
Amorphous solids yield through avalanches with mean-field exponents — stress drops as τ^{-3/2}.
Thouless Pump: Adiabatic Charge Pumping
Adiabatic cycling of a 1D periodic potential pumps exactly one charge per cycle — quantized by topology.
Fibonacci Chain: Quasiperiodic Spectrum
The Fibonacci quasicrystal has a Cantor-set energy spectrum — infinitely fragmented, measure zero.
Maxwell Model: Viscoelastic Rheology
A spring and dashpot in series gives elastic response at short times, viscous flow at long times.
Neural Field: Amari Bump Attractor
A continuous neural field with local excitation and lateral inhibition forms stable localized activity bumps.
Topological Entropy & Horseshoe Map
Smale's horseshoe map has topological entropy ln 2 — each iterate doubles the number of periodic orbits.
Kibble-Zurek: Defects from a Quench
Quenching through a phase transition freezes in topological defects — their density scales as the quench rate.
Ricci Flow: Geometric Heat Equation
Ricci flow evolves a metric by its curvature — the geometric analogue of heat diffusion, used to prove Poincaré.
Jarzynski Equality: Work & Free Energy
⟨e^{−W/kT}⟩ = e^{−ΔF/kT} — free energy differences from nonequilibrium work measurements.
Kinesin: Hand-Over-Hand Stepping
Kinesin walks on microtubules in 8 nm steps, alternating feet — a chemomechanical ratchet.
Scale-Free Networks: Robustness vs Attack
Scale-free networks are robust to random failures but fragile to targeted hub removal.
Nematic Defects: ±½ Charge Annihilation
Half-integer topological defects in liquid crystals attract and annihilate — heads attract tails.
Kauffman NK Network: Edge of Chaos
Random Boolean networks are frozen (K<2), chaotic (K>2), or critical (K=2) — life lives at the edge.
Lattice Boltzmann: D2Q9 Fluid
Mesoscopic kinetic simulation of fluid flow — particles stream and collide on a lattice.
Cosmic String Network: Defect Dynamics
Vortex defects formed after a phase transition form a scale-invariant network — one string per Hubble volume.
Arrhenius: Activated Rate Theory
Reaction rates follow k = A·exp(−Eₐ/kT) — a line on an Arrhenius plot reveals the activation energy.
Spin Waves & Magnon Dispersion
Small oscillations in a Heisenberg ferromagnet are quantized as magnons with ω ~ k² dispersion.
Nonlinear Pendulum: Exact Period
The pendulum period diverges logarithmically near the separatrix — T(θ₀) = 4√(L/g)K(sin²(θ₀/2)).
Complex Ginzburg-Landau: Spiral Waves
The CGLE hosts rotating spiral waves, targets, and defect-mediated turbulence.
Stochastic Block Model: Community Detection
Communities in networks arise from dense within-group connections — detectability has a sharp threshold.
Wigner-Dyson: Level Repulsion in Random Matrices
Eigenvalues of random matrices repel — nearest-neighbor spacing follows the Wigner surmise P(s) ~ s·e^{−πs²/4}.
Swift-Hohenberg: Stripe & Hexagon Selection
The Swift-Hohenberg equation models pattern-forming instabilities — stripes or hexagons selected by nonlinearity.
Mean-Field Ising: Critical Slowing
Relaxation time τ diverges as T→Tc — watch the order parameter decay and scan the divergence curve
Abrikosov Vortex Lattice
Flux quanta forming triangular lattice in Type-II superconductors — field, order parameter, and supercurrent maps
KdV Soliton Elastic Collisions
Exact 2-soliton solution to Korteweg–de Vries — solitons pass through each other leaving only a phase shift
FitzHugh-Nagumo Pulse Propagation
Excitable nerve impulse waves in 1D — fire pulses, tune excitability, view space-time diagrams
Stochastic Resonance (Noise-Enhanced)
Optimal noise level boosts weak signal detection in bistable systems — scan SNR vs. noise curve
Quantum Tunneling: WKB Approximation
WKB barrier penetration for rectangular, Gaussian, triangular, and double-barrier potentials
Active Matter: Vicsek Flocking
Self-propelled particles align with neighbors — order-disorder phase transition driven by noise
Skyrmion Topological Charge Texture
Magnetic skyrmion spin textures with quantized topological charge Q — Néel, Bloch, and antiskyrmion modes
Percolation Cluster Hull Fractals
Site percolation at critical p≈0.5927 — cluster hulls with fractal dimension d_H=7/4, SLE₆ boundaries
Chaos Synchronization: Lorenz Systems
Pecora-Carroll master-slave coupling — two chaotic Lorenz attractors synchronize via x-coupling
DLA Fractal Growth v2
Random-walk particles stick to a cluster producing fractal branching structures with Hausdorff dimension ≈ 1.71.
Josephson Junction Washboard v2
RSJ model: phase particle on a tilted washboard transitions from zero-voltage to voltage state at the critical current.
GUE Eigenvalue Level Spacing
Wigner-Dyson spacing statistics of random Hermitian matrices — eigenvalue repulsion connecting quantum chaos to RMT.
Kinetic Ising Glauber Dynamics
Single-spin-flip Glauber dynamics near T_c = 2.269J/k_B reveals critical slowing down and scale-free domain coarsening.
Nonlinear Shallow Water Breaking
Saint-Venant equations: nonlinear steepening of water waves until characteristics cross and shocks form.
Information Geometry Fisher-Rao
The manifold of Gaussian distributions is the Poincaré half-plane; geodesics give the Fisher-Rao statistical distance.
Topological Kink φ⁴ Field Theory v2
φ⁴ kink solitons carry topological charge Q=±1 interpolating between vacua — watch kink-antikink collisions and annihilation.
SIR Epidemic Heterogeneous Network v2
SIR dynamics on scale-free networks: hub heterogeneity lowers the epidemic threshold via ⟨k²⟩/⟨k⟩ amplification.
Euler Beam Buckling Instability
Euler's critical load P_cr = n²π²EI/L²: above the threshold the straight column snaps to bent equilibrium — bifurcation made visible.
Quantum Entanglement Entropy Chain
Von Neumann entropy of 1D chain bipartitions: area law (gapped), logarithmic CFT scaling, and volume law for random states.
Abelian Sandpile Avalanches
Self-organized criticality and the abelian group structure of sandpile stable configurations — click to topple.
Rabi Oscillations
Coherent quantum oscillations of a two-level atom driven by a radiation field — Bloch sphere precession with detuning and decay.
Turing Activator-Substrate Patterns
Reaction-diffusion model where short-range activation and long-range inhibition spontaneously generate spots, stripes, and labyrinths.
Duffing Oscillator Fractal Basins
The boundary between coexisting attractors of the Duffing oscillator is fractal — chaotic transients produce infinite complexity at the basin frontier.
Density Matrix & Decoherence
Visualize quantum decoherence as the Bloch vector shrinks from the sphere surface (pure) to the interior (mixed) via T1/T2 relaxation.
Turbulent Boundary Layer Log-Law
The universal law of the wall: viscous sublayer, buffer layer, and logarithmic velocity profile in turbulent wall-bounded flow.
Epidemic Threshold Heterogeneous Mean-Field
On scale-free networks the epidemic threshold vanishes as degree variance diverges — HMF theory tracks infection per degree class.
SSH Chain Topological Edge States
The Su-Schrieffer-Heeger model hosts zero-energy edge states when intercell hopping dominates — Thouless pumping cycles the topological phase.
Brownian Ratchet — Flashing Potential
An asymmetric sawtooth potential switching on/off converts thermal noise into directed drift — the Feynman ratchet without violating thermodynamics.
Bacterial Quorum Sensing
Bacteria secrete autoinducer molecules that diffuse through space; crossing a threshold triggers colony-wide collective gene expression.
Hodgkin-Huxley Neuron
The 1952 biophysical model of action potential generation — voltage-gated Na⁺/K⁺ conductances, gating variables m, h, n, and stimulus-driven spiking with voltage clamp.
Lie Algebra so(3) Rotations
Visualize SO(3) generators Jₓ, Jᵧ, J_z, the Rodrigues exponential map, and how angular velocity traces geodesics on the rotation group manifold.
KLS Driven Lattice Gas
Katz-Lebowitz-Spohn driven diffusive system: particles driven by an external field break detailed balance, forming distinct nonequilibrium universality classes.
Spectral Clustering & Graph Laplacian
Normalized cut clustering via eigenvectors of the graph Laplacian — data embedded in spectral space reveals geometric cluster structure invisible to k-means.
Quantum Zeno Effect
Frequent measurement collapses the wavefunction and suppresses quantum decay — Misra & Sudarshan 1977. Compare free evolution vs. watched decay with tunable measurement rate.
Granular Gas & Haff's Cooling
Inelastic particle collisions dissipate kinetic energy following Haff's T∝(1+t/t₀)⁻² cooling law — no energy equipartition, clustering instabilities emerge at low restitution.
Kitaev Chain — Majorana Modes
The 1D p-wave superconductor hosts Majorana zero modes at chain ends when |μ|<2|t|. Compute BdG spectrum and track the Z₂ topological invariant across the phase transition.
Real-Space RG Flow — 2D Ising
Migdal-Kadanoff renormalization group flow in the (K, h) coupling space — fixed points at disorder, order, and the critical point Kc≈0.4407 controlling Ising universality.
Takens Delay Embedding
Reconstruct strange attractors from a single observable time series using delay coordinates — Takens 1981 theorem guarantees generic diffeomorphic embedding for dimension ≥2d_A+1.
Rayleigh-Bénard Convection
Boussinesq fluid heated from below develops convection rolls above Ra_c≈1708 — 2D finite-difference simulation of vorticity, streamfunction, and temperature field evolution.
Geodesic Flow on Hyperbolic Surface
Billiard trajectories in the Poincaré disk — negative curvature (K=−1) drives exponential divergence and chaotic mixing of geodesics.
Adiabatic Elimination — Fast/Slow Manifold
When ε≪1, fast variables relax instantly onto the slow manifold, reducing system dimension. Phase portrait and time series show the reduction in action.
Network Motif Frequency vs Random
Count recurring 3-node subgraph patterns in a directed network and compare to randomized ensembles — z-scores reveal functionally over-represented motifs.
Quantum Hall Chiral Edge Modes
In the Integer QHE, bulk states are localized while ν chiral edge channels carry quantized Hall conductance σ_xy = νe²/h. Backscattering is topologically forbidden.
Polymer Folding — Self-Avoiding Walk 2D
Pivot-algorithm sampling of self-avoiding walks on a square lattice. Flory exponent ν=3/4 governs end-to-end scaling R~N^ν. HP model adds hydrophobic folding forces.
Stochastic Block Model — Community Detection
SBM generates graphs with planted community structure. Detectability threshold: (p_in−p_out)²n/2 > p_out. Visualize adjacency matrix and force-directed layout.
Joukowski Airfoil — Conformal Mapping
The Joukowski transform maps a circle to an airfoil. Potential flow with circulation Γ (Kutta condition) generates lift L=ρUΓ — conformal invariance at work.
Phase-Locked Loop — Frequency Acquisition
PLL dynamics: phase error obeys a nonlinear pendulum ODE. Lock range, pull-in transients, and saddle-node bifurcation all visible in the phase portrait.
Active Gel — Polar Order Vortex Knot Reconnection
Two vortex rings collide and reconnect — a topological transition changing linking number. Euler fluid topology meets 3D visualization of the reconnection event.
Saddle-Node & Cusp Bifurcation
Normal form ẋ = r + ax + x²: watch stable and unstable fixed points collide and vanish. Explore the cusp catastrophe surface and hysteresis in parameter space.
Peregrine Soliton & Modulation Instability
NLS equation rogue waves: the rational Peregrine solution localizes in space and time. Modulation instability drives exponential growth of sidebands on a uniform background.
Adaptive Network Epidemic Coevolution
SIS epidemic on a network where susceptible nodes rewire away from infected neighbors. Coevolution between topology and disease state creates fragmentation and oscillations.
ST-FMR: Spin-Torque Resonance
Macrospin LLG dynamics with Slonczewski spin-transfer torque. AC current drives magnetization precession; sweep field to find FMR absorption peak and auto-oscillation threshold.
Maximum Entropy Ising Inference
Given observed spin means and correlations, gradient ascent finds the max-entropy Ising model — the inverse Ising problem at the heart of statistical inference.
Kramers Escape Rate — Double Well
Overdamped Langevin dynamics in V(x)=x⁴−2x². Thermal fluctuations drive particles over the barrier; Kramers formula predicts the rate k ∝ exp(−ΔV/kT).
Quantum Discord
Beyond entanglement: quantum discord D = I(A:B) − J(A:B) captures quantum correlations absent from classical mutual information. Explore the Werner state phase diagram.
Max-Flow Ford-Fulkerson
Visualize the Edmonds-Karp algorithm finding maximum flow from source to sink via BFS augmenting paths. Watch the min-cut theorem emerge as paths are saturated.
Metacommunity Patch Dynamics
Levins metapopulation model with two competing species across a patch landscape. Colonization-extinction tradeoffs and competition drive coexistence or competitive exclusion.
Hyperbolic Geodesic Flow
Geodesic flow on hyperbolic surface in the Poincaré disk: Anosov mixing, horocycles, and exponential trajectory divergence visualized in real time.
KdV 2-Soliton Collision
Exact 2-soliton solution to the KdV equation: elastic collision with phase shifts, inverse scattering transform, individual soliton traces.
Plateau's Minimal Surfaces
Interactive 3D minimal surfaces with zero mean curvature: catenoid, helicoid, Enneper surface, and Scherk's surface with depth-colored rendering.
Riemann Surface and Monodromy
Domain coloring of sqrt(z), cbrt(z), log(z): branch cuts, sheet structure, and animated monodromy paths showing sheet permutation on encircling branch points.
Toda Lattice Integrability
Toda lattice with exponential interactions: Lax pair conserved quantities, symplectic Verlet integration, space-time diagram of soliton-like pulses.
Joukowski Conformal Airfoil
Joukowski transform maps circle to airfoil: conformal streamlines, Kutta condition, lift via circulation, pressure map, circle and airfoil domain toggle.
Morse Theory Gradient Flow
Gradient flow on torus, sphere, double well, and monkey saddle: critical points by Morse index, sublevel set evolution, Euler characteristic computation.
Stormer-Verlet Symplectic
Compare symplectic Verlet vs Euler vs RK4 on pendulum, Kepler, Henon-Heiles: energy drift visualization shows long-time structure preservation.
Poisson-Voronoi Tessellation
Poisson point process to Voronoi cells with Delaunay dual: cell area and edge distributions, Lewis law mean 6 edges, animated point motion.
SU(2) Adjoint Orbits
SU(2) adjoint action on Bloch sphere: commutator flow generates rotations, particles trace latitude orbits, quaternion spin-1/2 dynamics visualized.
Double Pendulum Poincaré Section
Chaos visualized: watch the double pendulum's 4D phase space slice into fractal structure — regular KAM tori vs chaotic seas revealed by Poincaré section crossings.
Jaynes-Cummings Collapse & Revival
Two-level atom in a cavity: quantum Rabi oscillations collapse due to photon-number spread, then revive — a purely quantum interference effect with no classical analog.
Bond Percolation Giant Component
Erdős-Rényi random graph G(N,p): watch a giant connected component suddenly emerge at the critical threshold p_c = 1/N — a second-order phase transition in network connectivity.
Abelian Sandpile Identity Element
The identity element of the sandpile group is a fractal fixed point: it stabilizes to itself under toppling — a purely algebraic object with beautiful self-similar visual structure.
Swift-Hohenberg 2D Patterns
The canonical pattern-formation PDE near bifurcation: stripes, hexagons, and labyrinthine textures emerge as ε crosses zero — the (1+∇²)² operator selects a critical wavelength.
Rydberg Blockade Atomic Chain
Quantum many-body dynamics in a 1D chain: van der Waals blockade V∝1/r⁶ creates frozen zones producing Z₂ ordered states and collective Rabi oscillations.
Oregonator BZ Reaction-Diffusion
Field-Körös-Noyes model of the Belousov-Zhabotinsky reaction: spiral waves and target patterns emerge from chemical oscillation coupled to diffusion.
Stochastic Resonance Array
Noise can improve signal detection: an array of threshold units detects a subthreshold signal best at optimal noise σ* — SNR peaks at resonance, improving as √M with array size.
Kuramoto Mean-Field Synchronization
N coupled phase oscillators with Lorentzian frequency spread: order parameter r jumps from 0 to √(1−K_c/K) at the critical coupling — a continuous phase transition in the thermodynamic limit.
Hopf Algebra of Feynman Diagrams
Connes-Kreimer (1998): rooted trees encode renormalization. Admissible cuts define the coproduct Δ(T); the antipode S(T) is exactly the BPHZ forest formula — algebra meets QFT.
Geodesic Flow — Poincaré Disk
Ergodic geodesics in the hyperbolic plane: circular arcs perpendicular to the unit disk boundary, mixing ergodically due to negative curvature K = −1.
Subcritical Hopf Bifurcation
Bistable system with coexisting fixed point and limit cycle: ṙ = μr + r³ − r⁵. A finite kick escapes the basin and triggers runaway oscillations.
Spin Waves — Magnon Dispersion
Ferromagnetic spin chain: collective precession modes with quadratic dispersion ω(k) = 4JS(1−cos ka), animating wave propagation and the Brillouin zone spectrum.
Quantum Hall Effect — Chern Numbers
TKNN formula: Berry curvature integrated over the Brillouin zone gives quantized Hall conductance σ_xy = Ce²/h. Visualizes edge states and Landau levels.
Contact Process — Absorbing State DP
Directed percolation universality class: spacetime diagram of an absorbing-state phase transition with order parameter ρ ~ (λ−λ_c)^β, β ≈ 0.276.
Epidemic Threshold — Heterogeneous Networks
SIS model on scale-free networks: λ_c = ⟨k⟩/⟨k²⟩ → 0 as ⟨k²⟩ diverges, making epidemics spread at vanishingly small infection rates (Pastor-Satorras 2001).
Elastic Snap-Through Instability
Bistable shallow arch with double-well energy landscape V = k(y²−h²)²: a sufficient kick causes irreversible snap-through between two stable configurations.
Protein Aggregation — Nucleation & Seeding
Amyloid fibril formation: slow nucleation, rapid elongation, and secondary nucleation produce sigmoidal kinetics with tunable lag phase — Alzheimer mechanism.
Diffusion Maps — Spectral Embedding
Random walk on data geometry: eigenvectors of the Markov matrix reveal nonlinear manifold structure — Swiss roll, torus, and moons unfolded by diffusion distance.
Random Field Ising — Avalanches
Disorder-tuned criticality: power-law Barkhausen avalanches as an external field sweeps through a 2D RFIM lattice, with scale-free statistics at the critical disorder R_c.
& Motility
Toner-Tu equations for self-propelled particles: activity drives spontaneous polar order, density waves, and topological defects — collective motion from microscopic alignment.
Rule 110 — Universal Cellular Automaton
Rule 110 is Turing-complete (Cook 2004). Space-time diagram shows gliders, still-lifes, and complex interactions emerging from a 3-bit lookup table on a 1D lattice.
Griffith Crack Energy Release
Griffith fracture criterion: elastic strain energy vs surface energy balance. Explore critical crack length, stress intensity factor, and the instability threshold.
Quantum Spin Liquid — Frustrated Lattice
Geometric frustration on a triangular lattice prevents magnetic ordering, forming a quantum spin liquid. Monte Carlo with J₂/J₁ frustration parameter.
Hysteresis Loop — Preisach Model
Ferromagnet hysteresis as a superposition of bistable hysterons. Explore the Preisach plane, B-H loop, and coercivity as functions of material parameters.
Fiber Bundle Avalanche Dynamics
Cascading fiber failures under increasing load. Avalanche sizes follow a power law P(s)~s⁻⁵/² near the critical load — mean-field universality class.
Cross-Diffusion Turing Chemotaxis
Keller-Segel chemotaxis: cross-diffusion drives Turing instability even without standard activator-inhibitor structure. Watch aggregation patterns emerge.
Gene Expression Bursting
Stochastic hybrid model: genes switch ON/OFF (telegraph process) while proteins evolve continuously. Explore bursty protein production and cell-to-cell variability.
Poincaré Map — Driven Pendulum
Stroboscopic sampling of a damped driven pendulum reveals the Poincaré map. As drive amplitude grows, watch the transition from regular orbits to a strange attractor.
Contact Process — Directed Percolation
The 1D contact process undergoes a non-equilibrium phase transition at λ_c≈3.298 in the Directed Percolation universality class — active spreading vs absorbing extinction.
2D Phonon Dispersion — Square Lattice
Phonon dispersion ω(k) over the Brillouin zone of a 2D square lattice. Visualize LA/TA modes, zone boundary, and atomic motion in real space simultaneously.
Kuramoto Chimera State
Two coupled populations of Kuramoto oscillators spontaneously break symmetry: one synchronizes while the other remains incoherent — the paradoxical chimera state.
Vortex Knot Reconnection
Two vortex rings collide and reconnect — a topological transition changing linking number. Euler fluid topology meets 3D visualization of the reconnection event.
Saddle-Node & Cusp Bifurcation
Normal form ẋ = r + ax + x²: watch stable and unstable fixed points collide and vanish. Explore the cusp catastrophe surface and hysteresis in parameter space.
Peregrine Soliton & Modulation Instability
NLS equation rogue waves: the rational Peregrine solution localizes in space and time. Modulation instability drives exponential growth of sidebands on a uniform background.
Adaptive Network Epidemic Coevolution
SIS epidemic on a network where susceptible nodes rewire away from infected neighbors. Coevolution between topology and disease state creates fragmentation and oscillations.
ST-FMR: Spin-Torque Resonance
Macrospin LLG dynamics with Slonczewski spin-transfer torque. AC current drives magnetization precession; sweep field to find FMR absorption peak and auto-oscillation threshold.
Maximum Entropy Ising Inference
Given observed spin means and correlations, gradient ascent finds the max-entropy Ising model — the inverse Ising problem at the heart of statistical inference.
Kramers Escape Rate — Double Well
Overdamped Langevin dynamics in V(x)=x⁴−2x². Thermal fluctuations drive particles over the barrier; Kramers formula predicts the rate k ∝ exp(−ΔV/kT).
Quantum Discord
Beyond entanglement: quantum discord D = I(A:B) − J(A:B) captures quantum correlations absent from classical mutual information. Explore the Werner state phase diagram.
Max-Flow Ford-Fulkerson
Visualize the Edmonds-Karp algorithm finding maximum flow from source to sink via BFS augmenting paths. Watch the min-cut theorem emerge as paths are saturated.
Metacommunity Patch Dynamics
Levins metapopulation model with two competing species across a patch landscape. Colonization-extinction tradeoffs and competition drive coexistence or competitive exclusion.
Dirac Zitterbewegung
Relativistic electron wave packet trembles at ω=2mc²/ℏ due to interference between positive and negative energy spinor components — Compton-wavelength-scale oscillations.
Floquet–Bloch Driven Lattice
Periodically driven tight-binding chain: quasi-energy Floquet bands, Bessel-function renormalized hopping, and coherent destruction of tunneling at J₀(A/Ω)=0.
Kondo Effect & Screening Cloud
Magnetic impurity in a metal creates a resistance minimum: Kondo logarithmic rise competes with phonon T³ decay, forming a many-body singlet below T_K.
Rashba–Dresselhaus Spin Texture
Spin-split Fermi circles in a 2DEG heterostructure with Rashba and Dresselhaus SOC. Tune α=β to reach the persistent spin helix with diverging spin lifetime.
Efimov Geometric Spectrum
Infinite tower of three-body trimers with 22.7× size ratio and 515× energy ratio — log-periodic quantum mechanics and the first RG limit cycle in physics.
Kibble–Zurek Vortex Formation
Quench through a symmetry-breaking transition spawns topological defects. Defect density scales as n_d ~ τ_Q^{−ν/(1+νz)} — watch the vortex count change with quench rate.
Quantum Zeno & Anti-Zeno
Frequent measurements freeze quantum decay (Zeno) or accelerate it (Anti-Zeno). The crossover at τ_Z is controlled by the spectral density of the environment.
Aharonov–Casher Effect
Electromagnetic dual of Aharonov-Bohm: a magnetic dipole orbiting a charge line acquires a geometric phase φ=μλ/ℏc, observable in neutron and atom interferometers.
Luttinger Liquid & Spin-Charge Separation
Bosonization of 1D interacting fermions splits a single electron into holon (charge) and spinon (spin), travelling at different velocities with power-law correlations.
Weyl Semimetal Chiral Anomaly
Weyl nodes with opposite chirality, Fermi arc surface states, and the chiral anomaly: parallel E and B pump charge between nodes, causing negative magnetoresistance.
Thouless Pump
Adiabatic cyclic pumping transports exactly one electron per cycle — a topological invariant (Chern number C=1) from the Berry phase of the filled Bloch band.
Sachdev-Ye-Kitaev Model
All-to-all random Majorana fermion interactions saturate the quantum chaos bound λ_L = 2πkT/ℏ — spectral form factor shows GUE dip-ramp-plateau.
Quantum Spin Liquid
Frustrated kagome antiferromagnet resists magnetic ordering — fractionalized spinons and Z₂ topological order emerge from the resonating valence bond ground state.
Non-Hermitian Skin Effect
Hatano-Nelson asymmetric hopping drives all eigenstates to localize at one boundary under OBC — spectral topology (winding number W=1) is the origin.
Moiré Flat Bands
Twisted bilayer graphene at magic angle ~1.1° flattens the Dirac bands — tunable bandwidth drives correlated insulator and superconductor phases.
Hofstadter Butterfly
Electrons on a lattice in magnetic flux p/q produce a fractal energy spectrum — each gap carries an integer Chern number giving σ_xy = Ce²/h.
Measurement-Induced Transition
Competition between unitary entanglement growth and local measurements drives a phase transition between volume-law and area-law entanglement entropy.
Discrete Time Crystal
Periodically driven MBL Ising chain spontaneously breaks discrete time-translation symmetry — magnetization oscillates at half the drive frequency robustly.
Quantum Many-Body Scars
PXP model for Rydberg atoms — Néel state shows anomalous revivals violating ETH via scar eigenstates forming an approximate SU(2) tower in a chaotic spectrum.
Cat Qubit: KPO
Kerr parametric oscillator creates Schrödinger cat states in phase space — two coherent wells give biased noise with exponentially suppressed phase-flip errors.
Andreev Bound States
Electron-hole Andreev reflection in a Josephson junction creates subgap bound states with energy ε(φ)=Δ√(1−T sin²φ/2). Watch the current-phase relation evolve with transparency.
Topological Kondo Insulator SmB₆
SmB₆ heavy-fermion hybridization gap opens from f-d mixing, generating in-gap topological surface states and a resistance plateau below 4 K — a hallmark of topological Kondo physics.
Vison Excitations — Z₂ Gauge Theory
Visons are anyonic flux excitations on toric-code plaquettes where B_p=−1. Create and move vison pairs via string operators — braiding with e-particles acquires a π phase.
Hertz-Millis Quantum Criticality
Itinerant quantum critical point with dynamic exponent z: spectral weight incoherence, non-Fermi liquid resistivity ρ∼T^(1+2/z), and the quantum critical fan in the phase diagram.
Kagome Flat Band Localization
The kagome flat band hosts compact localized states from destructive interference. Disorder hybridizes CLSs and induces Anderson localization — track IPR and localization length vs W.
Spin-Orbit Mott Insulator — Iridates
Na₂IrO₃ and α-RuCl₃: SOC splits t₂g into J_eff=½ states, Hubbard U opens a Mott gap, and bond-directional Kitaev exchange places these materials near a quantum spin liquid.
Axion Insulator Magnetoelectric Effect
Topological insulator with θ=π axion angle: applying B induces polarization P=αB and surface Hall conductance σ_xy=(n+½)e²/h — the half-quantized topological magnetoelectric effect.
Spinon-Holon Fractionalization
t-J model slave-boson decomposition splits the electron into spin-carrying spinons and charge-carrying holons. Explore the pseudogap phase, Fermi arcs, and cuprate phase diagram.
DMRG Entanglement Entropy
Schmidt decomposition and entanglement entropy in DMRG: area law S=const for gapped chains vs logarithmic growth S≈(c/3)log L at quantum critical points. Tune χ and watch truncation error.
Replica Trick — Random Field Ising
Quenched disorder averaging via n→0 replicas: the Edwards-Anderson order parameter q_EA measures spin-glass order and replica symmetry breaking in a rugged free-energy landscape.
SYK Traversable Wormhole
Coupled SYK clusters as dual holographic wormhole — negative energy injection, teleportation fidelity, and GUE spectral statistics at the chaos bound.
AKLT Valence Bond Solid
Spin-1 chain with exact VBS ground state — singlet bonds, Haldane gap Δ≈0.41J, free spin-1/2 edge modes, and string order O^z=4/9.
Kitaev Chain — Majorana Modes
Class D topological superconductor: phase diagram, Majorana zero-energy end modes, BdG spectrum, and Z₂ topological invariant via Pfaffian.
Fibonacci Anyon Braiding
Non-Abelian anyons with τ×τ=1+τ fusion rules — F-matrix, R-matrix, braid group representation, and topological quantum gates from braiding sequences.
Coupled Wire FQH ν=1/3
Teo-Kane construction: coupled 1D wires, gap opening via inter-wire scattering at k_F, chiral edge modes, and fractional Hall conductance e²/3h.
Li-Haldane Entanglement Spectrum
Entanglement spectrum of FQH ground state mirrors the CFT edge spectrum — topological order encoded in the structure of quantum entanglement.
3D Transverse-Field Ising QCP
Wilson-Fisher fixed point: divergent correlation length ξ~|g-g_c|^{-ν}, anomalous dimension η≈0.036, and quantum-to-classical mapping at d+z dimensions.
Mott-Hubbard Spectral Weight
Hubbard model: quasiparticle peak collapse and spectral weight transfer to Hubbard bands as U/W increases — Brinkman-Rice Z→0 at the Mott transition.
Peierls Instability & CDW
1D metal spontaneously dimerizes: gap 2Δ opens at k_F, CDW order parameter, phonon softening Kohn anomaly at 2k_F, and BCS-like Tc suppression.
BCS-BEC Crossover Dome
From Cooper pairs to tightly bound bosons: T_c dome peaks near unitarity 1/k_F a=0, pair size shrinks from ℏv_F/πΔ to molecular scale across the crossover.
Strange Metal — Planckian Dissipation
Linear-T resistivity, Planckian scattering rate τ=ℏ/k_BT, vs Fermi liquid T² behavior
Pseudogap — Cuprate Fermi Arcs
Spectral weight A(k,0) on the Fermi surface with d-wave gap anisotropy Δ(φ)=Δ₀cos(2φ)
Orbital Magnetism — Berry Curvature
Modern theory of orbital magnetization via Berry curvature, Chern numbers, and anomalous Hall
Nodal Line — Drumhead Surface States
Ring of band crossings in the BZ and flat drumhead surface states inside the nodal ring projection
Spinon Fermi Surface — U(1) Spin Liquid
Fractionalized spinons, emergent gauge field, thermal transport without charge, Wilson ratio W=2
Anderson RVB — Doping a Mott Insulator
Resonating valence bonds, slave-boson holon condensation, and d-wave SC phase diagram
Marginal Fermi Liquid — Varma Self-Energy
Im Σ ~ max(|ω|,T), no quasiparticle pole, logarithmic optical conductivity in cuprates
Stripe Order — Charge & Spin in Cuprates
Unidirectional CDW period 4a + SDW period 8a competing with d-wave superconductivity
Pair Density Wave — LOFF/PDW State
Δ(r)=Δ₀cos(Q·r) spatially modulated Cooper pairing, Zeeman phase diagram, intertwined CDW
Quantum Oscillations — Fermi Pockets
Lifshitz-Kosevich SdH formula, Dingle damping, CDW Fermi surface reconstruction and small pockets
Josephson Andreev Bound States
Andreev bound state energy E±(φ) = ±Δ√(1−τsin²(φ/2)) vs phase difference in a short Josephson junction, with supercurrent transport and transparency dependence.
Kibble-Zurek Defect Density
Topological defect formation when quenching through a phase transition — domain pattern, defect counting, and power-law scaling n ∝ τQ^(−ν/(1+νz)).
SYK Non-Fermi Liquid
Sachdev-Ye-Kitaev model spectral function A(ω) showing non-Fermi-liquid power-law tails, conformal Green's function G(τ), and self-energy Im Σ ∝ |ω|^(1−2/q).
Floquet-Bloch Photon-Dressed Bands
Periodic driving spawns photon replica bands separated by ℏΩ, opening avoided crossings with coupling A. Effective hopping is renormalized by J₀(A/Ω).
Mott-Hubbard Metal-Insulator Transition
Correlated electron DOS evolving from Fermi liquid to Mott insulator as U/t increases — lower/upper Hubbard bands split and the quasiparticle peak weight Z→0.
Kondo Singlet Screening Cloud
Impurity spectral function with Abrikosov-Suhl resonance at E_F, Kondo temperature T_K, and real-space visualization of the screening cloud extending to ξ_K = ℏv_F/k_BT_K.
Resonating Valence Bond Dimers
Quantum dimer model on a square lattice with Monte Carlo dynamics — columnar, plaquette, staggered, and RVB liquid phases near the Rokhsar-Kivelson point.
Magnetic Skyrmion Topological Hall Effect
Real-space spin texture of a magnetic skyrmion with winding number Q, emergent magnetic field, and the topological Hall resistivity ρ_THE = R₀·Q·Φ₀·n_sk.
Superfluid Vortex Kelvin Wave Cascade
Helical Kelvin waves on a superfluid vortex line cascade energy to smaller scales with spectrum E(k) ∝ k⁻³ — the quantum turbulence route to phonon emission.
Weyl Fermion Chiral Magnetic Effect
Weyl semimetal Brillouin zone with ±1 chirality nodes, surface Fermi arc connecting projections, and chiral magnetic effect current J = e²μ₅B/(2π²ℏ²).
Majorana Vortex Zero Modes
p-wave topological superconductor BdG spectrum showing Majorana zero modes at vortex cores with tunable chemical potential.
Rashba Helical Bands
Rashba spin-orbit coupling in a 2DEG: spin-split bands, helical Fermi surface spin texture, and gate-tunable Berry phase.
Anderson Localization
3D tight-binding disorder model showing the mobility edge separating extended and localized states via participation ratio.
Fractional Chern Insulator
Haldane model flat band with non-zero Chern number, Berry curvature heatmap, and fractional Hall conductance plateaus.
BCS Coherence Factors
Bogoliubov u_k and v_k coherence factors, quasiparticle dispersion E_k, and the BCS density of states divergence at the gap edge.
Quantum Critical Point
Hertz-Millis theory of quantum criticality: non-Fermi liquid fan, dynamical exponent z, and C/T ~ -log(T) divergence.
Luttinger Liquid
1D interacting electrons: power-law spectral function A(ω) ∝ |ω|^α, spin-charge separation, and Luttinger parameter K.
Peierls CDW Instability
1D metal Peierls instability: dimerization-driven charge density wave gap opening, nesting vector, and temperature-dependent order parameter.
Heavy Fermion Kondo Lattice
Periodic Anderson model showing Kondo coherence, hybridization gap, heavy quasiparticle mass enhancement, and resistance upturn.
Magnon-Polaron
Coupled magnon-phonon modes with anticrossing gap, hybridized dispersion branches, and anomalous group velocity near resonance.
Pyrochlore Spin Ice — Emergent Monopoles
Two-in two-out ice rules on a pyrochlore lattice produce emergent magnetic monopoles connected by Dirac strings, with Monte Carlo dynamics and Coulomb interactions.
Valence Bond Solid — AKLT & Dimerization
VBS order in the quantum antiferromagnet: dimerized singlets, AKLT virtual spin-1/2 decomposition, Haldane gap in spin-1 chains, and spectrum visualization.
Quantum Hall Plateau Transition — Chalker-Coddington
The CC network model for plateau-to-plateau transitions shows critical exponent ν≈2.3, localization length divergence, and random electron paths through disordered saddle points.
Topological Kondo Insulator — SmB₆ Surface States
Heavy f-electron hybridization opens a Kondo gap with topological in-gap surface states, showing band structure, DOS, and helical Dirac fermions at the X-points.
Z₂ Gauge Theory — Toric Code Fractionalization
Kitaev's toric code on an interactive lattice: create e and m anyons via string operators, visualize plaquette and vertex violations, and demonstrate braiding phase π.
Non-Abelian Ising Anyons — Fusion & Braiding
Ising anyon fusion rules σ×σ=1+ψ, F-matrix as a Hadamard gate, world-line braiding diagrams, and 4 Majorana modes encoding a topological qubit.
Orbital Magnetization — Modern Berry Phase Theory
Berry curvature contribution to orbital magnetization in the Haldane model: BZ curvature maps, Chern number phase diagram, de Haas–van Alphen oscillations, and Landau levels.
Spin Nematic Order — Quadrupolar Hidden Order
Spin-1 quadrupolar order with ⟨S⟩=0 but Q_αβ≠0: visualize nematic directors, phase diagram in (h,T), order parameter vs temperature, and NMR-invisible hidden order.
Exciton Condensate — Electron-Hole Bilayer BEC
Excitons in a bilayer undergo BEC and a Kosterlitz-Thouless transition driven by vortex unbinding, with superfluid density, counterflow superfluidity, and interlayer tunneling.
Phonon Hall Viscosity — Chiral Angular Momentum
Lattice phonons in a magnetic field acquire angular momentum, splitting into chiral branches and generating Hall viscosity η_H = (ℏ/2)⟨L_z⟩ with transverse stress response.
Axion Electrodynamics in 3D TI
Topological magnetoelectric effect in 3D topological insulators: the θ=π axion term quantizes the Hall response and couples E and B fields at magnetic surfaces.
3D Dirac Semimetal Band Crossing
Three-dimensional Dirac semimetal with linear band crossings, 3D Fermi surface pockets, surface Fermi arcs, and the bulk-boundary correspondence at symmetry-protected Dirac points.
Quantum Anomalous Hall Effect
Chern insulator bands in magnetic topological insulators: a single chiral edge state emerges without external magnetic field, giving σ_xy = e²/h when the Chern number is nonzero.
Magic-Angle Twisted Bilayer Graphene
Band flattening vs. twist angle in twisted bilayer graphene: at θ≈1.1°, the moiré flat band suppresses kinetic energy and drives Mott insulating and superconducting states.
Quantum Spin Hall Helical Edge States
BHZ model for 2D topological insulators: helical edge states with spin-momentum locking are protected by the Z₂ invariant and immune to time-reversal-symmetric backscattering.
Multiferroic Magnetoelectric Coupling
Coupled ferroelectric polarization P and magnetic order M via a Landau free energy: the cross-coupling term γP²M² enables electric control of magnetism and vice versa.
Nodal SC Zero-Bias Conductance Peak
d-wave and p-wave nodal superconductors: gap nodes on the Fermi surface seed Andreev bound states at interfaces, producing a zero-bias conductance peak in tunneling spectroscopy.
Anderson Superexchange Mechanism
Virtual hopping of electrons through bridging oxygen generates antiferromagnetic exchange J = 4t²/U; bond angle controls sign via the Goodenough-Kanamori-Anderson rules.
Orbital Order & Cooperative Jahn-Teller
eg orbital occupation patterns and cooperative Q₂/Q₃ octahedral distortions in Mott insulators: orbital-lattice coupling drives a transition to long-range orbital order as in LaMnO₃.
de Haas–van Alphen Oscillations
Lifshitz-Kosevich formula for quantum oscillations in magnetization: each extremal Fermi surface cross-section contributes a frequency F = ℏA_F/2πe, damped by temperature and disorder.
Abrikosov Vortex Lattice
Triangular flux lattice in type-II superconductors — vortex core structure, magnetic field penetration, and upper/lower critical fields.
DC SQUID Flux Quantization
Two-junction quantum interference device — critical current oscillates with Φ/Φ₀, Josephson relations, and Fraunhofer-like pattern.
Pair Density Wave — Striped SC
Spatially modulated order parameter Δ(r)=Δ₀cos(Q·r) — charge-4e density, vortex halos, and coexistence with CDW order.
Spin-Fluctuation Pairing
Antiferromagnetic spin susceptibility χ(q,ω) drives d-wave superconductivity — gap equation, nodal structure, and Brillouin zone visualization.
Pseudogap Phase in Cuprates
Fermi arcs, spectral weight loss, and the T* line — preformed Cooper pairs vs competing order in underdoped cuprate superconductors.
Strange Metal — Planckian Dissipation
Linear-in-T resistivity saturating the quantum bound 1/τ = αk_BT/ℏ — marginal Fermi liquid self-energy and crossover from Fermi liquid.
Topological Edge Current
Chiral edge states carry quantized Hall current σ_xy=Ce²/h — bulk-edge correspondence, Berry curvature, and robustness to disorder.
Competing SC and AFM Orders
Landau free energy with two order parameters — phase diagram across doping and temperature shows coexistence, competition, and phase separation.
Quantum Paraelectric Soft Mode
SrTiO₃-like soft transverse optical phonon — quantum fluctuations suppress ferroelectric order, Barrett formula, and dielectric divergence.
Orbital Compass Model
Directional exchange Jxσˣσˣ + Jyσʸσʸ creates extensive frustration — Z₂ quantum phase transition, string order, and Monte Carlo dynamics.
Shilnikov 3D Homoclinic Chaos
A saddle-focus equilibrium with a homoclinic orbit: when ρ/λ < 1, the Shilnikov condition triggers a Smale horseshoe — infinitely many periodic orbits and deterministic chaos in 3D.
Stochastic Resonance — LIF Neuron
A subthreshold signal undetectable alone becomes detectable with the right amount of noise. The leaky integrate-and-fire neuron fires in sync with the signal — noise helps!
Josephson Junction Plasma Oscillations
The RSJ model: phase dynamics equivalent to a damped pendulum. Below I_c: plasma oscillations. Above: AC Josephson voltage state. Explore the I-V characteristic and Shapiro steps with AC drive.
Ginzburg-Landau Vortex Dynamics
Abrikosov vortices in a Type-II superconductor, each carrying flux quantum Φ₀ = h/2e. Watch them repel via London interaction and self-organize into a triangular Abrikosov lattice.
Smoluchowski Coagulation & Gelation
Cluster aggregation with different kernels: constant, sum, product (K=ij gelates!), and Brownian. The product kernel drives a sol-gel phase transition at t_gel = 1/2 — Flory-Stockmayer theory.
Pair Approximation SIR — Network Epidemics
Beyond homogeneous mixing: pair correlations [SI], [SS], [II] capture network structure. Compare mean-field vs pair approximation — R₀_PA = β(k̄−1)/γ differs measurably from R₀_MF = βk̄/γ.
Period Doubling & Feigenbaum Constants
The logistic map x_{n+1} = r·x_n(1−x_n): period-2^n cascades with universal constants δ ≈ 4.669 and α ≈ 2.503. Explore bifurcation diagram and period detection across the route to chaos.
Phononic Crystal — Elastic Wave Bandgap
A 1D periodic array of alternating materials creates Bragg bandgaps at k = π/a. Transfer matrix method computes band structure, gap width, and evanescent decay in real time.
Nonlinear Pendulum — Separatrix & Phase Portrait
The exact pendulum θ̈ = −(g/L)sin(θ): libration below the separatrix, rotation above. Period diverges logarithmically near the heteroclinic orbit. Click phase portrait to set initial conditions.
Diffusion-Limited Aggregation — Fractal Dimension
Random-walking particles stick to a growing cluster, creating a fractal with d_f ≈ 1.71. Real-time log-log regression measures the fractal dimension. Tune sticking probability and noise.
Active Brownian MIPS
Motility-induced phase separation: purely repulsive self-propelled particles spontaneously segregate into dense and dilute coexisting phases at high Péclet number.
Hard Sphere Glass — Mode Coupling
Hard sphere fluid near the MCT glass transition φ_c ≈ 0.516. Cage dynamics, MSD plateau, α/β relaxation, and structural arrest visualized in real time.
Polymer Brush Steric Repulsion
Alexander-de Gennes blob model: grafted chains form blobs of size ξ=D. Steric repulsion force vs. surface separation plotted live as parameters change.
Colloidal Gel — Arrested Phase Separation
Short-range attractive colloids: spinodal decomposition arrested by glass transition forms a fractal space-spanning network. Tune attraction strength and volume fraction.
Wetting & Young Equation
Young's equation balances three interfacial tensions at the contact line. Visualize droplet shape from hydrophilic to super-hydrophobic as γ_SG, γ_SL, γ_LG are tuned.
Donnan Equilibrium & Osmotic Pressure
Charged macroions behind a semipermeable membrane: ion partitioning, Donnan potential, and osmotic pressure from charge balance and equilibrium thermodynamics.
Classical Nucleation Theory
CNT free energy barrier ΔG* = (16π/3)γ³v_m²/Δg_v² and critical radius r*. Nucleation rate J ∝ exp(−ΔG*/kT) plotted vs undercooling ΔT.
Active Filament Buckling
Motor-driven compressive forces along an elastic filament induce buckling instability. Wavelength selection λ ~ (κ/f)^½ emerges from overdamped active dynamics.
Onsager Hard Rods — Nematic Order
Onsager (1949): hard rod entropy drives isotropic-nematic transition. Excluded volume selects alignment at c·L²·D ≈ 3.3 — no attractive forces needed.
Liu-Nagel Jamming Diagram
The J-point as a critical transition in (φ, T, σ) space. Force networks, contact number Z, and pressure visualized as volume fraction crosses φ_c.
VQE Cost Landscape
Variational Quantum Eigensolver: 2D energy landscape over parameter space, barren plateau visualization, and gradient variance scaling exponentially with qubit count.
Quantum Teleportation Circuit
Step-by-step Bell measurement teleportation protocol with Bloch sphere tracking. Watch Alice's qubit state transfer to Bob through entanglement and classical communication.
BB84 QKD — Intercept-Resend
BB84 quantum key distribution with eavesdropper simulation. Visualize basis choices, sifting, QBER vs intercept rate, and the 11% detection threshold.
Grover Amplitude Amplification
Grover's search algorithm: amplitude evolution per iteration, optimal query count k*=⌊(π/4)√(N/M)⌋, and success probability oscillation demonstrating quadratic speedup.
Quantum Discord vs Classical Correlations
Decompose quantum mutual information into discord and classical correlations for Werner, Bell-diagonal, and X-states. Separable states can carry non-zero discord.
Quantum Channel Capacity
Depolarizing, dephasing, and erasure channels: coherent information, hashing bound, classical capacity, and entanglement-assisted capacity vs noise parameter.
Quantum Annealing — Adiabatic Gap
Transverse-field Ising model energy spectrum vs annealing parameter. Gap closing at phase transition drives T* ∝ 1/Δ²_min. Compare linear, quadratic, and Roland-Cerf schedules.
Steane [[7,1,3]] Syndrome Correction
Interactive Steane code error correction: inject single-qubit X, Z, or Y errors, read the 3-bit syndrome, identify the error location, and apply the correction.
Quantum Walk — Ballistic Spreading
Discrete-time quantum walk with tunable coin operator. Interference produces two-peak probability distribution with σ ∝ t (ballistic) vs classical σ ∝ √t.
Entanglement Entropy — Area Law
Bipartite entanglement entropy S(ℓ) for 1D chains: area law for gapped states, CFT log scaling at criticality, Page curve for random states, flat GHZ profile.
Hyperbolic Geodesic Flow
Geodesic flow on hyperbolic surface in the Poincaré disk: Anosov mixing, horocycles, and exponential trajectory divergence visualized in real time.
KdV 2-Soliton Collision
Exact 2-soliton solution to the KdV equation: elastic collision with phase shifts, inverse scattering transform, individual soliton traces.
Plateau's Minimal Surfaces
Interactive 3D minimal surfaces with zero mean curvature: catenoid, helicoid, Enneper surface, and Scherk's surface with depth-colored rendering.
Riemann Surface and Monodromy
Domain coloring of sqrt(z), cbrt(z), log(z): branch cuts, sheet structure, and animated monodromy paths showing sheet permutation on encircling branch points.
Toda Lattice Integrability
Toda lattice with exponential interactions: Lax pair conserved quantities, symplectic Verlet integration, space-time diagram of soliton-like pulses.
Joukowski Conformal Airfoil
Joukowski transform maps circle to airfoil: conformal streamlines, Kutta condition, lift via circulation, pressure map, circle and airfoil domain toggle.
Morse Theory Gradient Flow
Gradient flow on torus, sphere, double well, and monkey saddle: critical points by Morse index, sublevel set evolution, Euler characteristic computation.
Stormer-Verlet Symplectic
Compare symplectic Verlet vs Euler vs RK4 on pendulum, Kepler, Henon-Heiles: energy drift visualization shows long-time structure preservation.
Poisson-Voronoi Tessellation
Poisson point process to Voronoi cells with Delaunay dual: cell area and edge distributions, Lewis law mean 6 edges, animated point motion.
SU(2) Adjoint Orbits
SU(2) adjoint action on Bloch sphere: commutator flow generates rotations, particles trace latitude orbits, quaternion spin-1/2 dynamics visualized.
Hodgkin-Huxley Spiking Neuron
Full conductance-based model: Na⁺/K⁺/leak currents, gating variables m/h/n, action potential generation — adjust conductances and current to explore spiking regimes.
Wilson-Cowan Populations
E-I neural population model: nullclines, phase plane, oscillation and bistability — tune recurrent excitation, inhibition, and time constants to traverse dynamical regimes.
Amari Neural Field: Bump Attractor
1D neural field with Mexican-hat connectivity: localized bump attractors, traveling waves, and drift under external input — space-time activity patterns visualized.
STDP Learning Window
Spike-timing-dependent plasticity: Hebbian potentiation (pre→post) and anti-Hebbian depression — weight evolves in real time from simulated spike trains.
Winner-Take-All Network
Lateral inhibition competition: one neuron dominates as others are suppressed — click neurons to boost, adjust inhibition strength, watch decision dynamics.
Drift-Diffusion Model
Evidence accumulation to decision boundary: watch stochastic trajectories, reaction time distributions, and the speed-accuracy tradeoff as you vary drift and threshold.
Place Cells & Grid Cells
Hippocampal place fields and entorhinal grid cells: simulated foraging builds firing rate maps — Gaussian place fields vs hexagonal grid structure compared side by side.
Short-Term Synaptic Plasticity
Tsodyks-Markram model: resource depletion (depression) and utilization buildup (facilitation) — watch EPSP amplitudes evolve across spike trains, paired-pulse ratio live.
Cross-Frequency Coupling
Theta-gamma nesting: low-frequency phase modulates high-frequency amplitude — PAC polar plot and comodulogram reveal coupling strength across frequency pairs.
Binocular Rivalry
Mutual inhibition and neural adaptation drive perceptual alternations — test Levelt's propositions by adjusting eye inputs, observe dominance duration distributions.
Rayleigh-Bénard Plume Dynamics
Thermal convection between hot bottom and cold top plates — hot plumes rise, cold plumes sink, roll pattern forms at Ra > 1708. Adjust Rayleigh number and Prandtl number.
Vicsek Flocking Model
Alignment-based collective motion: particles align with neighbors plus noise. Order parameter transitions from disordered to ordered flock as noise decreases — real-time φ(t) plot.
Motility-Induced Phase Separation
Active Brownian particles phase-separate via motility alone — no attraction needed. Dense clusters form as particles slow in crowded regions: the Cates-Tailleur MIPS mechanism.
Kondo Resonance Spectral Function
Anderson impurity model: many-body Kondo peak pins at the Fermi level below T_K. Explore Hubbard bands, hybridization Γ, and the T/T_K crossover from correlated to free.
Winding Number & Vortex Defects
Place ±1 vortex defects on an XY-model field — topological winding numbers computed from phase circulation. Bound vortex-antivortex pairs and the KT transition visualized.
CA Rule Space Explorer
All 256 Wolfram elementary 1D cellular automata in one interactive tool — click bit patterns to toggle outputs, select famous rules (30, 90, 110), watch space-time diagrams unfold.
1D Phononic Crystal Transmission
Transfer matrix method for elastic wave propagation through a periodic bilayer crystal — band gaps from Bragg reflection, transmission spectrum T(f), animated wave displacement.
Protein Folding Gō Model
Langevin dynamics on a Gō-model energy landscape — funnel topology guides the chain from extended to native state. Track RMSD trajectory on the energy landscape in real time.
Fibonacci Quasicrystal Phason Strain
Fibonacci chain tiling with incommensurate long/short tiles (ratio τ), diffraction peaks at q = 2π(m+nτ)/a, and phason strain that flips LS↔SL tiles — quasiperiodic diffraction patterns.
Metabolic Network Flux Balance
Toy FBA on central carbon metabolism — glycolysis, TCA cycle, OXPHOS, and biosynthesis with flux routing. Warburg overflow fermentation emerges when O₂ is limiting.
Double Pendulum Poincaré Section
Chaos visualized: watch the double pendulum's 4D phase space slice into fractal structure — regular KAM tori vs chaotic seas revealed by Poincaré section crossings.
Jaynes-Cummings Collapse & Revival
Two-level atom in a cavity: quantum Rabi oscillations collapse due to photon-number spread, then revive — a purely quantum interference effect with no classical analog.
Bond Percolation Giant Component
Erdős-Rényi random graph G(N,p): watch a giant connected component suddenly emerge at the critical threshold p_c = 1/N — a second-order phase transition in network connectivity.
Abelian Sandpile Identity Element
The identity element of the sandpile group is a fractal fixed point: it stabilizes to itself under toppling — a purely algebraic object with beautiful self-similar visual structure.
Swift-Hohenberg 2D Patterns
The canonical pattern-formation PDE near bifurcation: stripes, hexagons, and labyrinthine textures emerge as ε crosses zero — the (1+∇²)² operator selects a critical wavelength.
Rydberg Blockade Atomic Chain
Quantum many-body dynamics in a 1D chain: van der Waals blockade V∝1/r⁶ creates frozen zones producing Z₂ ordered states and collective Rabi oscillations.
Oregonator BZ Reaction-Diffusion
Field-Körös-Noyes model of the Belousov-Zhabotinsky reaction: spiral waves and target patterns emerge from chemical oscillation coupled to diffusion.
Stochastic Resonance Array
Noise can improve signal detection: an array of threshold units detects a subthreshold signal best at optimal noise σ* — SNR peaks at resonance, improving as √M with array size.
Kuramoto Mean-Field Synchronization
N coupled phase oscillators with Lorentzian frequency spread: order parameter r jumps from 0 to √(1−K_c/K) at the critical coupling — a continuous phase transition in the thermodynamic limit.
Hopf Algebra of Feynman Diagrams
Connes-Kreimer (1998): rooted trees encode renormalization. Admissible cuts define the coproduct Δ(T); the antipode S(T) is exactly the BPHZ forest formula — algebra meets QFT.
Gravitational Wave Chirp
Post-Newtonian binary inspiral waveform: frequency sweeps from 10 Hz to ISCO as chirp mass and luminosity distance shape h(t) — the signature GW150914 sound.
Frank Elastic Nematic Director
2D nematic director field relaxes under K₁ splay, K₂ twist, and K₃ bend Frank constants. Watch ±½ topological defects annihilate and click to seed new ones.
Shannon Channel Capacity
Interactive Shannon-Hartley theorem and BSC mutual information: tune SNR, bandwidth, and input bias to explore the gap between I(X;Y) and channel capacity C.
May-Leonard Heteroclinic Cycle
Rock-paper-scissors competition (May & Leonard 1975): three species chase each other around the simplex in a heteroclinic cycle — algebraically slow, never settling.
PP Chain Nucleosynthesis
Gamow window, tunneling probability, and proton-proton chain energy generation in stellar cores. Branch fractions (PP-I/II/III) shift dramatically with core temperature.
Colloidal Gel Percolation
Depletion-attracted colloidal particles form a spanning network at the gel transition. Tune volume fraction, attraction strength, and range to cross the percolation threshold.
Entropy Production & Affinity
Three-state nonequilibrium Markov chain: Schnakenberg affinity A drives persistent cycling flux J, with σ = J·A ≥ 0. Set A=0 to enforce detailed balance and watch flux vanish.
Magnetic Skyrmion Hall Drift
Thiele equation dynamics of magnetic skyrmions: the gyrovector G = 4πQẑ deflects driven skyrmions transversely — the skyrmion Hall effect that limits racetrack memory.
Noise-Induced Phase Transition
Horsthemke-Lefever multiplicative noise model: above σ²_c = 2|a| the stationary PDF bifurcates to bimodal — a phase transition impossible without fluctuations.
Quantum Zeno Watchdog
Frequent measurement suppresses quantum decay: γ_eff = Ω²/Γ_m in the Zeno regime. Tune measurement rate to see Zeno suppression, free decay, and the anti-Zeno crossover.
✦ Lab #9000 Milestone
Grand 5-tab celebration: science galaxy, beautiful equations, greatest hits, physics constants, and fractal jubilee — 9000 windows onto nature.
Minority Game — Econophysics
Heterogeneous agents bet on minority side; emergent market efficiency and volatility from strategy evolution. Challet-Zhang 1997.
Quasispecies Error Threshold
Eigen model of molecular evolution: above μ_c = ln(A)/L mutation rate, the master sequence is lost to error catastrophe.
Multi-Slit Diffraction Grating
Fringe patterns and intensity curves for N-slit diffraction — single-slit envelope × multi-slit interference, with white-light option.
Gauss Linking Number
3D visualization of two closed curves with live computation of Lk via the Gauss double line integral — topological knot invariant.
Predator-Prey Turing Instability
Holling type-II reaction-diffusion: cross-diffusion (D_pred ≫ D_prey) drives spatial pattern formation — spots and stripes from homogeneity.
WKB Quantum Tunneling
Tunneling transmission T ≈ exp(−2γ) through rectangular, Gaussian, triangular, and double barriers — Gamow factor visualized.
Complex Contagion Hypergraph
Threshold model on hypergraph: adoption requires φ-fraction of neighbors. Group hyperedges lower effective barrier — cascade dynamics.
KPZ Interface Fluctuations
Kardar-Parisi-Zhang interface growth: nonlinear ∇h² term drives β=1/3 roughening exponent; compare EW (β=1/4) and random (β=1/2).
Kirchhoff Rod Writhe & Supercoiling
Elastic rod with twist and bending: White's theorem Lk = Tw + Wr, plectoneme formation, self-contact — DNA supercoiling mechanics.
Superfluid Helium Vortex Lattice
Quantized vortex filaments in rotating He-II form an Abrikosov-like triangular lattice. Explore the phase field, velocity circulation, and thermal fluctuations as a function of rotation rate and temperature.
Dark Energy Equation of State w(z)
CPL parametrization w(a)=w₀+wₐ(1−a) shapes cosmic expansion H(z) and comoving distance χ(z). Compare models against Planck 2018 constraints on n_s and r.
Niche Overlap & Competitive Exclusion
Lotka-Volterra two-species competition: Gaussian niche overlap α determines whether species coexist or one excludes the other — Gause's competitive exclusion principle visualized dynamically.
3D Totalistic Cellular Automaton
26-neighbour 3D Moore rules with tunable survive/birth ranges, rendered as an isometric voxel lattice. Explore amoeba, crystal-growth, and sparse-life regimes in the 3D outer-totalistic rule space.
Spin Wave Magnon Dispersion — Heisenberg Ferromagnet
Magnon dispersion ω(k) along high-symmetry paths, density of states, and real-space spin precession for the Heisenberg ferromagnet with exchange J, Zeeman field B, and anisotropy D.
Reaction Network Stoichiometry Matrix
Interactive stoichiometry matrix N: click cells to toggle ±1, watch the bipartite reaction graph update and compute the null space (steady-state flux modes) via Gaussian elimination.
Inflationary Slow-Roll Power Spectrum
Compare Starobinsky, Higgs, chaotic φ², φ⁴, and natural inflation: slow-roll parameters ε, η, spectral index n_s, and tensor-to-scalar ratio r plotted against Planck 2018 constraint contours.
Mutualistic Network Nestedness
Pollinator-plant bipartite networks: generate nested interaction matrices, compute the NODF nestedness metric, and visualize how specialist species interact only with subsets of generalists' partners.
Leaky Integrate-and-Fire Network
Spiking neural network of LIF neurons with excitatory/inhibitory balance: raster plot and population firing rate reveal transitions between silent, asynchronous-irregular, and synchronous-burst regimes.
Strange Nonchaotic Attractor
Quasiperiodically driven maps (Gopy-Prasad, Feudel-Pikovsky, Nishikawa-Kaneko) exhibit fractal geometry with zero or negative Lyapunov exponent — strange but not chaotic, at the boundary of order and chaos.
Entropic Elasticity — Freely Jointed Chain
Polymer elasticity is purely entropic: force aligns segments against thermal disorder. The Langevin function exactly gives extension vs force — no energy storage, just entropy reduction.
Topological Defects in Nematics
Nematic liquid crystals support ±½ disclinations — impossible in the XY model. Watch coarsening: defect pairs annihilate, defect count falls as N(t)~t⁻¹, driven by topology.
Squeezed States — Wigner Function
Quantum harmonic oscillator: squeeze one quadrature below shot noise while the conjugate grows. Wigner quasi-probability goes negative for Fock states — a direct witness of non-classicality.
Moran Process — Fixation Probability
Even beneficial mutations are usually lost by drift when rare. The exact Moran formula π=(1−r⁻ᵏ)/(1−r⁻ᴺ) shows selection vs stochastic extinction across population sizes.
Hydrophobic Effect — Solvation Shell
Hydrophobic aggregation is entropy-driven: solvation shell water is more ordered, reducing entropy. Clustering reduces ordered water, making TΔS the dominant term in ΔG at room temperature.
Kondo Lattice — Heavy Fermion Coherence
At T≪T_K, local f-moment screening hybridizes into coherent heavy bands with m*∼100mₑ. The periodic Anderson model shows flat-band formation and the Kondo hybridization gap.
Chaos Synchronization — Coupled Rössler
Pecora-Carroll (1990): two chaotic Rössler systems synchronize when coupled. Complete sync occurs when all conditional Lyapunov exponents turn negative — chaos tamed by coupling.
Neural Field Theory — Wilson-Cowan Spatial
Mexican-hat E-I connectivity drives Turing instability in cortex: traveling waves, standing patterns, and localized bumps emerge from homogeneous activity — a model of working memory and hallucinations.
Rule 110 — Turing-Complete CA
Rule 110 is the simplest known Turing-complete system (Cook 2004). Gliders and collisions simulate cyclic tag systems. Explore all 256 elementary rules and watch complexity emerge from 3-bit logic.
DLA — Witten-Sander Fractal Cluster
Diffusing particles stick on contact, building fractal clusters with d_f≈1.71. Laplacian screening makes tips grow faster — the same instability drives viscous fingering, dielectric breakdown, and lightning.
Fibonacci Phyllotaxis
Golden angle packing: why sunflowers have Fibonacci spirals and 137.5° maximizes seed density.
Quantum Decoherence & Einselection
Zurek's pointer basis: watch coherences vanish in the density matrix as environment entanglement grows.
Vicsek Flocking Model
Non-equilibrium phase transition in self-propelled particles: noise drives disorder, order parameter ψ collapses.
Stochastic Resonance
Noise boosts signal detection in a double-well: SNR peaks at optimal randomness — Kramers rate meets signal frequency.
φ⁴ Kink Soliton
Topological solitons in φ⁴ field theory: kink-antikink collisions, annihilation, and topological charge conservation.
SIR on Scale-Free Network
Barabási-Albert network epidemics: vanishing threshold, hub superspreaders, and heterogeneous R₀.
Spectral Clustering
Graph Laplacian eigenmaps reveal hidden cluster structure invisible to k-means: rings, moons, spirals.
Forest Fire SOC
Drossel-Schwabl model: self-organized criticality with power-law fire size distribution — no tuning needed.
Quantum Tunneling WKB
Semiclassical barrier transmission: compare exact transfer-matrix T(E) with WKB e^{−2∫κdx} approximation.
Viscous Fingering
Saffman-Taylor instability: Laplacian growth generates fractal DLA fingers tunable by surface tension.
FitzHugh-Nagumo Spirals
Excitable media self-organize into rotating spiral waves — the mechanism behind cardiac arrhythmia reentry. Tune parameters to watch spirals form and break.
Heisenberg Magnon Dispersion
Spin-1/2 chain: ferromagnet vs antiferromagnet magnon dispersion ε(k). FM has quadratic k², AFM has linear |k| — a Goldstone boson of broken SU(2).
Abelian Sandpile Toppling
Drop grains, watch avalanches obey power-law statistics. The group structure of recurrent states is abelian — toppling order never matters. Self-organized criticality without tuning.
Active Matter Quorum Sensing
Self-propelled particles switch behavior at a local density threshold, producing traveling density waves — motility-induced phase separation from microscopic signaling rules.
Riemann Zeros & Prime Spectral
Non-trivial zeros of ζ(s) are the frequencies of prime distribution. Watch the explicit formula reconstruct ψ(x) from its spectral sum — the music of the primes.
Directed Percolation Transition
Activity spreads forward in time near a sharp absorbing-state phase transition at p_c ≈ 0.6447. Critical exponents β≈0.276 define the DP universality class.
Crystal Field d-Orbital Splitting
Ligand geometry breaks d-orbital degeneracy into t₂g/e_g sets. Switch between octahedral, tetrahedral, and square planar — control high-spin vs low-spin crossover.
Shannon Channel Capacity
BSC, BEC, and AWGN channels: capacity C = maximum reliable information rate. Below C, arbitrarily reliable coding exists; above C, errors are unavoidable — Shannon's theorem live.
Geodesic Flow & Ergodicity
Geodesics on the Poincaré disk (constant negative curvature) are ergodic and mixing — Hopf's 1939 theorem. Lyapunov exponent λ>0: nearby trajectories diverge exponentially.
Protein Folding Energy Funnel
Gō model Monte Carlo reveals the funnel-shaped energy landscape. Levinthal's paradox resolved: evolution shaped the landscape so Q→1 guides folding without a random search.
Wigner Semicircle & GOE Spectrum
GOE random matrices: eigenvalue density converges to the semicircle law, and level spacings follow the Wigner surmise — eigenvalues repel. Same statistics govern heavy nuclei, quantum chaos, and prime gaps.
Josephson Junction I-V Curve
RSJ model: the phase particle in a tilted washboard potential. Below Ic: supercurrent (trapped phase). Above Ic: voltage state. Underdamped junctions show hysteresis from the running solution.
Redfield Ratio & Ocean C:N:P Cycling
Phytoplankton maintain C:N:P = 106:16:1 — matching deep ocean nutrients. Droop model with Liebig's Law: the most limiting nutrient controls growth. Life sets ocean chemistry over geological time.
KdV Soliton Collision
Pseudospectral simulation of the Korteweg-de Vries equation. Solitons pass through each other elastically — only a phase shift remains. Infinite conserved quantities = complete integrability.
Steane [7,1,3] Code Syndrome
CSS quantum error correction: 6 parity check stabilizers detect X and Z errors on any single qubit without measuring the logical state. The 6-bit syndrome uniquely identifies the fault.
Coherent Feedforward Loop Motif
Most abundant gene network motif. C1-FFL with AND logic acts as a sign-sensitive delay: filters brief noise pulses but responds to sustained input. Incoherent FFL generates precise adaptation pulses.
Feynman's Brownian Ratchet
At equal temperatures, pawl fluctuations cancel ratchet asymmetry — no net work. When T₁ ≠ T₂, directed motion emerges with Carnot efficiency. The Second Law is enforced by thermal pawl fluctuations.
Magnetic Skyrmion Hall Effect
Neel-type skyrmion with topological charge Q = −1. Drive current causes transverse Hall deflection via the Magnus force — the Skyrmion Hall angle θ_H = arctan(G/αD), a key challenge for racetrack memory.
Cascading Failure: Interdependent Networks
Buldyrev et al. (2010): two mutually dependent networks collapse via abrupt first-order transition — far more fragile than isolated networks. Watch the cascade propagate in real time.
Ising Decimation RG Flow
1D Ising decimation: K' = ½ln(cosh 2K) flows to K=0 — no phase transition. 2D Migdal-Kadanoff has an unstable fixed point at K_c. Cobweb diagram shows RG trajectories and fixed-point structure.
Fractional Brownian Motion & Hurst Exponent
The Hurst exponent H encodes long-range memory in fBm: H>0.5 gives persistent (trending) paths, H<0.5 anti-persistent, H=0.5 standard Brownian motion. Spectral synthesis reveals how correlations span all timescales.
Hopf Bifurcation & Amplitude Death
Two coupled Stuart–Landau oscillators: supercritical Hopf bifurcation births a limit cycle at μ=0, while strong coupling with frequency mismatch collapses both oscillators to zero — amplitude death.
Quantum Otto Engine — Carnot Bound
A qubit as working fluid executes a quantum Otto cycle: adiabatic strokes preserve populations while isochoric strokes exchange heat with baths. Otto efficiency 1−1/r is always below Carnot 1−Tc/TH.
Competitive Exclusion — Lotka-Volterra
Gause's law visualized: stable coexistence, competitive exclusion, and priority effects emerge from the isocline geometry. α₁₂·α₂₁ vs K₁/K₂·K₂/K₁ determines the fate of two competing species.
Metamaterial Negative Refraction
Veselago (1968): when ε<0 and μ<0, the refractive index n is negative and the refracted ray bends to the same side of the normal. Snell's law still holds — but n₁sinθ₁ = n₂sinθ₂ with n₂ negative.
Cosmic Web — Zel'dovich Pancakes
Large-scale structure formation via the Zel'dovich approximation: x = q − D·∇Φ. Particles collapse into sheets (pancakes), then filaments, then nodes as growth factor D increases. Voids grow between.
Cellular Potts Model — Cell Sorting
Steinberg's differential adhesion hypothesis: cells sort like immiscible fluids, minimizing interfacial energy. CPM Metropolis dynamics on a grid produces spontaneous sorting, engulfment, and phase separation.
Spin Waves & Magnon Dispersion
Collective spin precession in a Heisenberg ferromagnet: ω(k) = 2JS(1−cos ka) + gμ_B·B₀. Quadratic dispersion at small k; FMR at k=0 set by applied field; exchange sets the k²-coefficient.
TDA — Betti Numbers & Filtration
Vietoris-Rips filtration on point clouds: Betti numbers β₀ (components) and β₁ (holes) evolve with radius ε. Persistent homology captures topological features robust to noise — encoded in the persistence barcode.
Laughlin Wavefunction & FQHE
Laughlin's ν=1/m wavefunction: Ψ ∝ ∏(zᵢ−zⱼ)ᵐ exp(−Σ|z|²/4ℓ²). Quasiholes carry charge e/m and anyonic statistics with braid phase 2π/m. Hall conductance is quantized at every fraction.
Sandpile Identity & Tropical Geometry
The abelian sandpile group's identity element — computed via 6-minus-relax — reveals fractal self-similarity and tropical algebraic structure. Toppling rule: ≥4 chips fire to neighbors.
Quantum Decoherence & Pointer States
Zurek's einselection: environment entanglement destroys off-diagonal coherences at rate γ|x−x'|², leaving pointer states — the dynamically robust eigenstates that survive. Density matrix visualization.
Morphogenetic Field: Reaction-Advection
Turing reaction-diffusion with advective flow: Gray-Scott kinetics plus v⃗·∇u transport breaks pattern symmetry. Tune f, k, and flow velocity to switch between spots, stripes, and travelling waves.
Driven-Dissipative Polariton Condensate
Exciton-polaritons under coherent drive F and loss κ obey the driven Gross-Pitaevskii equation. Explore optical bistability, spontaneous vortex formation, and non-equilibrium condensation.
Extreme Value Theory: GEV & Block Maxima
Fisher-Tippett-Gnedenko theorem: block maxima converge to GEV with shape ξ. Gumbel (ξ=0), Fréchet (ξ>0), or Weibull (ξ<0) depending on parent distribution tail. Live L-moments fitting.
Non-Equilibrium Steady State: Flux Cycles
Schnakenberg cycle decomposition of a 3-state Markov chain: thermodynamic affinities drive net probability currents. Entropy production σ = ΣᵢJᵢAᵢ ≥ 0 — zero only at equilibrium.
Topological Strings & Gromov-Witten Invariants
A-model strings map genus-g worldsheets into a Calabi-Yau 3-fold. Gromov-Witten invariants N_{g,d} count holomorphic curves of degree d. The τ-function partition function encodes all amplitudes via mirror symmetry.
Stochastic Lotka-Volterra & Extinction Risk
Demographic noise (√X·dW) destroys the deterministic center: stochastic LV trajectories spiral outward and hit the extinction boundary. Ensemble phase portrait and time series with live extinction count.
KdV Multi-Soliton Interaction
Exact N-soliton solutions via Hirota's τ-function: each sech² pulse travels at speed 4κ², passes through others unchanged except for a phase shift — a signature of complete integrability.
Circuit Complexity Growth: Brownian Circuits
Brown-Susskind conjecture: random 2-local gates drive complexity linearly in time before saturating at ~exp(n). Visualize random quantum circuit layers and track complexity growth toward the AdS/CFT bound.
Kicked Top: Quantum Chaos & Husimi
A periodically kicked quantum spin evolves under Haake's kicked-top map. The Husimi Q-function reveals the semiclassical structure: at low kicking regular tori persist, while high kicking dissolves them into chaotic seas with quantum scars.
Ferroelectric Domain Switching & Hysteresis
Landau–Khalatnikov dynamics in a double-well potential: as a sinusoidal field ramps past the coercive field, antiparallel domains nucleate and sweep through the crystal, tracing the P-E hysteresis loop. Disorder broadens the loop.
Vicsek Flocking & Order Parameter
Self-propelled particles align with neighbors plus noise η. Below the critical noise η_c the order parameter φ=|⟨e^iθ⟩| jumps discontinuously to 1 — a first-order transition to collective motion with giant density fluctuations.
Poincaré Disk: Geodesic Flow & Billiards
Geodesic billiards on the hyperbolic plane H² with Poincaré disk model and {p,q} tessellation. The flow is Anosov — uniformly hyperbolic, ergodic, and Bernoulli — with nearby trajectories diverging exponentially.
Active Brownian Particles & MIPS
Motility-induced phase separation: ABPs self-propel at Péclet number Pe with rotational diffusion D_r. Above a critical Pe, steric interactions brake fast particles in crowds — dense clusters nucleate with no attractive forces needed.
Szilard Engine & Maxwell's Demon
Step through the full Szilard engine cycle: partition insertion, demon measurement, isothermal expansion extracting k_BT ln 2 of work, and Landauer erasure dissipating the same — information has a physical thermodynamic cost.
Random Walk → Diffusion: Continuum Limit
Watch N walkers taking discrete steps ±Δx converge to the heat equation P(x,t)=e^{-x²/4Dt}/√(4πDt) as steps accumulate. The variance grows as ⟨x²⟩=2Dt — the CLT encoded in PDE form.
Ecological Niche & Competitive Coexistence
MacArthur consumer-resource dynamics with Gaussian utilization functions. Niche overlap α_ij drives competitive exclusion when σ is large; immigration m stabilizes coexistence. Competitive exclusion principle visualized in real time.
Wigner Crystal: 2D Electron Lattice
Langevin MD for N electrons with 1/r Coulomb repulsion in a periodic box. At low kT electrons crystallize into a triangular lattice; the hexagonal order parameter ψ₆ distinguishes crystal from liquid. Lower kT to freeze.
Turing Reaction-Diffusion: Stripe Selection
Gray-Scott activator-inhibitor dynamics with D_v≫D_u. The most unstable wavenumber k*=√(f(f+k)/D_uD_v) selects the stripe width. Explore stripes, spots, mazes, and worms by tuning feed rate f and kill rate k.
Spin Wave Magnon Dispersion
Ferromagnetic chain: collective spin precession quantized into magnons. Animate the ω(k)=4JS(1−cos ka) dispersion, compare exact vs quadratic Goldstone mode, and tune exchange coupling J interactively.
Complex Ginzburg-Landau Spirals
Spatiotemporal chaos in oscillatory media: ∂A/∂t = A+(1+ib)∇²A−(1+ic)|A|²A. Phase-colored spiral defects proliferate when the Benjamin-Feir condition 1+bc<0 is violated.
Feedforward Loop Network Motif
The most common regulatory motif: coherent FFL delays and filters, incoherent FFL generates pulses. Watch X→Y→Z dynamics in real time and switch between C1 and I1 types.
Adiabatic Demagnetization Cooling
Reach millikelvin temperatures via entropy manipulation: isothermal magnetization lowers spin entropy, then adiabatic field removal forces T_f = T_i·(B_f/B_i). S-T diagram walkthrough.
Diffusion Maps Spectral Embedding
Random-walk kernel K(x,y)=exp(−‖x−y‖²/ε) reveals intrinsic manifold geometry. Eigenvectors of the Markov matrix unfold Swiss rolls, circles, and figure-8s into flat coordinates.
Predator-Prey Spatial Invasion Waves
Reaction-diffusion Lotka-Volterra: ∂u/∂t = D_u∇²u+αu−βuv. Predators invade prey-rich regions producing traveling waves, spirals, and spatiotemporal chaos tunable via α, β, δ, γ.
Quantum Annealing: Transverse Ising
H = −JΣσᶻσᶻ − Γ(t)Σσˣ: ramp transverse field Γ→0 adiabatically to find the Ising ground state. Mean-field spin dynamics show quantum tunneling escaping barriers that trap thermal annealing.
Osmotic Pressure & Van't Hoff Law
Π = iMRT: particles crossing a semipermeable membrane drive water flow until hydrostatic head ρgh balances osmotic pressure. Animated molecules, real-time height equilibration, and pressure readout.
FQHE Laughlin Quasiparticle Braiding
At ν=1/m, quasiparticles carry charge e/m and acquire phase e^{iπ/m} under exchange — true anyons. Visualize the braid, accumulate fractional phase, and compare m=1,3,5,7 filling factors.
Random Close Packing & Jamming
Spheres settle under gravity to φ_RCP≈0.637 (Bernal 1960) where force chains percolate and motion freezes. Watch the jamming transition, tune gravity and restitution, compare to FCC φ=0.7405.
Molecular Motor Flashing Ratchet
Kinesin walks by rectifying thermal noise: an asymmetric sawtooth potential flashes on/off at rate k, directed drift emerges only when detailed balance is broken by ATP hydrolysis. Track particles and mean displacement.
Rule 110 — Turing-Complete CA
Cook (2004): Rule 110 is the simplest known Turing-complete system. Gliders collide and persist in the space-time diagram — compare with Rule 30 (chaos), 90 (Sierpiński), and 184 (traffic flow).
Nonlinear Pendulum & Separatrix Chaos
Phase portrait of θ̈ + γθ̇ + sin θ = A cos ωt: the heteroclinic separatrix divides libration from rotation. Add driving to break it via homoclinic tangle — the geometric origin of chaos (Melnikov 1963).
Stochastic Gene Expression Bursting
Two-state promoter model: chromatin switches ON/OFF at rates k_on, k_off; mRNA bursts arrive geometrically when active. Live histogram across N cells shows negative-binomial distribution and CV² = 1/⟨n⟩ + 1/b.
XY Model — BKT Topological Melting
Berezinskii–Kosterlitz–Thouless transition at T_BKT≈0.89: below it, vortex–antivortex pairs are bound (quasi-LRO); above it, free vortices proliferate and correlations decay exponentially. Nobel Prize 2016.
Spectral Flow & Chiral Anomaly
Dirac spectrum on a circle: each 2π increase in gauge flux Φ flows all eigenvalues up by one unit, crossing zero. Spectral flow = Chern number = chiral charge pumped per flux quantum (APS index theorem).
Anderson Localization — Disordered Chain
1D tight-binding with random on-site energies: all eigenstates are exponentially localized for any disorder W>0 (Anderson 1958). IPR = Σ|ψ|⁴ quantifies localization; watch states shrink as disorder grows.
Langmuir Plasma Waves & Dispersion
Bohm–Gross dispersion ω² = ωₚ² + 3k²vₜ²: longitudinal electron oscillations can't propagate below ωₚ. Animate the density wave, tune wavenumber k, and see Landau damping kill the wave as v_phase → v_thermal.
Moran Process — Neutral Drift & Fixation
Birth-death Markov chain: mutant allele with fitness r fixes with probability ρ = (1−1/r)/(1−1/rᴺ). Run an ensemble, compare empirical fixation probability and time against exact Moran theory (Nowak 2006).
Quantum Walk — Graph Search Speedup
Discrete quantum walk on a cycle with Hadamard coin: probability spreads ballistically (~t) vs classical diffusion (~√t), giving quadratic speedup in hitting time. Compare quantum and classical distributions live.
Marangoni Flow — Surfactant Gradients
Surface tension gradients drive Marangoni rolls: where surfactant is high, γ is low, pulling fluid toward cleaner regions. Click to seed a surfactant blob and watch vortical flow patterns form.
Liouville's Theorem — Phase Volume
A blob of initial conditions evolves under Hamilton's equations: it stretches and shears but its phase-space area is exactly preserved. Compare harmonic vs anharmonic oscillators.
KPZ Interface — Height Fluctuations
Kardar–Parisi–Zhang universality: growing interface with diffusion, nonlinear tilt term, and noise. Width scales as W~t^β with β=1/3 (KPZ) or 1/4 (EW). Height distribution converges to Tracy–Widom GUE.
Entanglement Entropy — Bipartite vN
Bipartite von Neumann entropy S = −Tr(ρ_A log ρ_A) via Schmidt decomposition. Explore product, GHZ, random pure, and cluster states on spin chains — visualize the full Schmidt spectrum and S vs cut position.
Immune Clonal Selection & Affinity
B-cells with high receptor–antigen affinity are clonally expanded and hypermutated in germinal centers. Darwinian optimization in shape space yields picomolar antibodies — watch the population sharpen over generations.
Omori Law — Aftershock Decay
Aftershock rate follows n(t) = K/(t+c)^p after a mainshock. Click the seismic map to trigger mainshocks and observe empirical vs theoretical Omori decay curves and ETAS-style event clustering.
Arnold's Cat Map — Hyperbolic Chaos
Hyperbolic automorphism of the 2-torus: (x,y)→(x+y, x+2y) mod 1. Lyapunov exponent = log φ > 0. Watch a cat-face image mix into chaos — then observe periodic recurrence on rational grids.
Stochastic Resonance — Optimal SNR
Noise of optimal strength helps a subthreshold signal cross a bistable threshold in synchrony — SNR is maximized at intermediate noise, not zero. Explore the Zeno/SNR curve in the double-well Langevin system.
Polymer Brush — Grafted Density Profile
Densely grafted polymer chains stretch into a brush due to excluded-volume repulsion. MC simulation shows parabolic SCF density profile φ(z)~[1−(z/H)²]. Tune grafting density and solvent quality.
Quantum Zeno Paradox — Measurement Freeze
Frequent measurement slows quantum decay: survival P₀~1−(ΔH)²τ² per interval, so P^n→1 as τ→0. Visualize state evolution on Bloch sphere, decay curves, and the Zeno plateau vs measurement rate.
Turbulent Boundary Layer — von Kármán Overlap
Law of the Wall in three layers: viscous sublayer U⁺=y⁺, buffer zone, and the log-law overlap U⁺=(1/κ)ln(y⁺)+B. Adjust Re_τ, κ, and B to explore how turbulence structures the mean velocity profile.
Majorana Fermions — Kitaev Chain
1D p-wave superconductor hosts topologically protected zero modes at its ends when |μ/t|<2. Visualize the phase diagram, Majorana operator pairing, and the zero-energy edge states in the energy spectrum.
Spin Echo — Hahn Pulse Refocusing
A π pulse at time τ reverses static dephasing and produces an echo at 2τ. Watch spins fan out in the transverse plane, snap back, and refocus — with only irreversible T₂ processes causing net decay.
Autocatalytic Reaction Bistability
ẋ = S + k₁x²/(1+x²) − k₂x admits two stable fixed points separated by an unstable saddle. Phase portrait shows hysteresis and saddle-node bifurcations — the core motif of genetic switches and cell memory.
Critical Slowing Down — Early Warning Signals
Near a tipping point (λ→0), recovery time diverges as 1/|λ|. Simulate a pitchfork system approaching bifurcation and watch variance and lag-1 autocorrelation rise — universal precursors of ecological collapse.
Glassy Dynamics — Energy Landscape
A particle navigating a rugged landscape of metabasins shows two-step relaxation, aging, and Arrhenius trapping (τ ∝ e^{ΔE/T}). Watch mean-square displacement plateau in the cage before structural relaxation.
Thouless Topological Pump
Adiabatic cycling of a Rice-Mele lattice pumps exactly one charge per cycle — quantized by the Chern number of the occupied band over the (k,φ) torus. Berry curvature and the integer staircase visualized live.
Neural Field — Bump Attractor
Wilson-Cowan field on a ring with a Mexican-hat kernel supports a localized activity bump at any position — a continuous attractor encoding spatial memory. Click to inject input and move the bump.
Black Hole Thermodynamics — Hawking Temperature
T_H = ℏc³/(8πGMk_B): a 10 M☉ black hole glows at ~6 nK. Explore Bekenstein-Hawking entropy S=A/4ℓ_P², Hawking spectrum, and the M³ evaporation timescale for Schwarzschild and Kerr black holes.
Continuum Percolation — Gilbert Disk Model
Random disks of radius r at Poisson-distributed centers percolate when area fraction η=λπr²≈0.357. Watch the spanning cluster emerge at the threshold and track component-size histograms across the transition.
Amphiphile Bilayer & Vesicle Formation
Hydrophilic heads and hydrophobic tails self-assemble into bilayers and vesicles via the hydrophobic effect — the structural principle behind every cell membrane.
Neural Mass Model — Alpha Rhythm
The Jansen-Rit cortical column model spontaneously oscillates at 8–13 Hz. Tune excitatory/inhibitory gains and watch the EEG alpha rhythm emerge or collapse into silence.
Exclusion Process — Blockage Dynamics
TASEP with a defect site: a slow site creates traffic-jam shock waves. Three phases (low-density, high-density, max-current) controlled by entry/exit/hop rates.
Cosmic String Network Evolution
Topological defects from early-universe phase transitions form a string network. Reconnections spawn oscillating loops that radiate gravitationally toward a scale-invariant steady state.
Fractal Basin Boundaries — Driven Pendulum
The driven damped pendulum has multiple attractors whose basins form fractal boundaries. Color initial conditions by final attractor to reveal the Wada property and uncertainty exponent.
Active Matter — Run & Tumble Clustering
Bacteria run straight then randomly reorient. Pure excluded volume plus activity drives motility-induced phase separation (MIPS) — dense clusters with no attractive forces.
Spectral Flow & Adiabatic Quantum Pump
The Rice-Mele Thouless pump: adiabatically cycling a 1D potential transfers exactly one electron per cycle, protected by Chern number C=1. Watch spectral flow thread energy levels through the gap.
Nonequilibrium Flux & Entropy Production Cycles
A driven 3-state Markov chain violates detailed balance, sustaining persistent probability current J. The entropy production rate σ = J·ln(ratio of cycle rates) quantifies the thermodynamic cost.
Adaptive Network — Rewiring Voter Model
Agents copy neighbors or rewire away from disagreement. At a critical rewire probability p*, the network fragments into echo chambers — a topological phase transition in social dynamics.
Quantum Tunneling — Resonant Transmission
A double-barrier potential supports resonant tunneling: T(E)→1 at energies matching quasi-bound states. The exact transfer-matrix solution and T(E) curve reveal resonance peaks — the basis of tunnel diodes.
Shilnikov Chaos — Saddle-Focus Spiral
When a 3D flow has a homoclinic orbit to a saddle-focus with ρ/λ < 1, infinite chaotic dynamics arise. Watch trajectories spiral outward on the unstable manifold and reinjected back — a Smale horseshoe in every neighborhood.
Kibble-Zurek Topological Defect Formation
Rapidly quenching an XY model through its phase transition freezes in vortex defects: causally disconnected regions choose different symmetry-breaking directions. Defect density scales as τ_Q^{−ν/(1+zν)}.
Contact Process — Absorbing State Transition
The contact process is the canonical directed-percolation model: infection rate λ vs. recovery μ drives a sharp transition between extinction (absorbing) and survival (active) phases, with universal critical exponents.
Kondo Resonance — Anderson Impurity
A magnetic impurity in a metal is screened below the Kondo temperature T_K, producing a sharp Abrikosov-Suhl resonance pinned at the Fermi level. Watch the resonance broaden and resistance develop a minimum as T rises.
Vicsek Flocking — Polar Order & Noise
Self-propelled particles align with neighbors plus angular noise. Below critical noise η_c, long-range polar order emerges in a first-order transition violating Mermin-Wagner — the paradigm for active matter flocking.
Diffusion Maps — Spectral Manifold Learning
Coifman-Lafon 2006: build a heat kernel on data, then use eigenvectors of the graph Laplacian to unfold the intrinsic geometry. Swiss rolls and S-curves reveal their true low-dimensional structure in diffusion coordinates.
Quantum Discord — Beyond Entanglement
Discord D = I(A:B) − J(A:B) captures quantum correlations that survive separable states. Explore Werner states: dephasing kills entanglement but preserves discord — nonclassicality without entanglement.
Abelian Sandpile — Group Identity Element
Stable sandpile configurations form an abelian group under addition-then-toppling. The identity element is a self-similar fractal computed by the burning algorithm — a remarkable structure from simple toppling rules.
KdV Soliton Collision — Elastic Phase Shift
The exact two-soliton KdV solution via inverse scattering: two solitary waves pass through each other perfectly intact, with only a phase shift as evidence of interaction. Space-time diagram reveals their trajectories.
Epidemic Networks — Heterogeneous Mean-Field SIR
On scale-free networks, ⟨k²⟩ diverges so the epidemic threshold β_c → 0: any infection spreads. Hubs become superspreaders. Compare scale-free, Erdős-Rényi, and regular networks in real-time SIR dynamics.
Skyrmion Lattice Hall Effect
Monte Carlo spin texture with DMI: watch skyrmions nucleate, measure their topological charge Q, and see the emergent Hall conductance from winding numbers.
Mean First Passage Time
Biased random walkers race to absorbing barriers — compare the exact analytical MFPT formula against live simulation histograms as drift and start position vary.
Amyloid Fibril Nucleation & Growth
Knowles-Waudby-Dobson kinetics: primary nucleation, elongation, and autocatalytic secondary nucleation produce the sigmoidal aggregation curve linked to Alzheimer's disease.
Förster Resonance Energy Transfer
FRET efficiency follows 1/(1+(r/R₀)⁶) — tune donor-acceptor distance and watch emission spectra shift; the r⁶ law makes FRET a molecular ruler at 1–10 nm scales.
Lévy Stable Distributions
Explore the four-parameter stable family via its characteristic function: power-law tails, infinite variance, and the generalised CLT from Gaussian (α=2) to Cauchy (α=1).
Reaction Network Deficiency Zero
Feinberg's theorem: networks with deficiency δ=0 can never oscillate. Compare Michaelis-Menten (unique steady state) vs. Brusselator (limit cycles) — topology controls dynamics.
Quantum Annealing Schedule
Adiabatic quantum computation: watch the spectral gap close as the transverse field fades, compare linear vs. pause-near-gap schedules, and estimate Landau-Zener success probability.
Rule 110 — Turing Completeness
The simplest known Turing-complete system: Cook's 2004 proof via glider collisions in a 3-neighbor binary CA. Explore all 256 Wolfram rules and compare chaotic, periodic, and complex classes.
Turbulent Jet Self-Similarity
Centerline velocity decays as 1/x, width grows linearly, and Gaussian profiles collapse onto one curve — the universal self-similar attractor of turbulent free jets.
Glass Transition & Kauzmann Paradox
VFT viscosity divergence, Angell fragility, and the entropy crisis: if the supercooled liquid were cooled infinitely slowly, it would have fewer microstates than the crystal — the Kauzmann paradox.
Aharonov-Bohm Interference
Quantum phase shift from enclosed magnetic flux shifts double-slit fringes — no field on the path, yet the interference pattern moves with AB phase Δφ = 2πΦ/Φ₀.
Ostwald Ripening — LSW Theory
Large droplets grow at the expense of small ones via diffusion; LSW theory predicts mean radius grows as ⟨r⟩³ ~ t and a universal self-similar size distribution.
Chimera States on a Ring
Identical phase oscillators with nonlocal coupling spontaneously split into synchronized and chaotic domains — a paradox of partial coherence discovered by Kuramoto (2002).
Hubbell Neutral Theory of Biodiversity
All species ecologically equivalent — births and deaths purely random. Despite this null model, ecological drift produces a log-series species abundance distribution matching empirical data.
Quantum Eraser & Which-Way
Marking which slit a photon takes destroys interference; erasing the which-way information in coincidence counts restores it — even after detection. Englert's relation V²+D²≤1.
Nematic Disclinations
Topological defects in liquid crystal director fields with half-integer winding numbers ±½. Frank elastic energy F ~ πKs²ln(R/rc); +½ and −½ disclinations attract and annihilate.
Stochastic Lotka-Volterra Extinction
Demographic noise makes neutrally-stable predator-prey orbits spiral outward to extinction. Gillespie algorithm simulates exact stochastic dynamics — extinction time ~ exp(N).
Conformal Invariance & SLE Curves
Schramm-Loewner Evolution: the unique conformally invariant random curve family. SLE_6 = percolation hulls, SLE_3 = Ising interfaces; fractal dimension d_f = 1 + κ/8.
Dislocation Pile-Up
Edge dislocations blocked at a grain boundary reach Chebyshev equilibrium positions. Tip stress amplification N × τ_app explains the Hall-Petch relation σ_yield ~ d^{−1/2}.
Channel Capacity — Blahut-Arimoto
Iteratively maximize mutual information I(X;Y) over input distributions to find Shannon capacity C. Visualizes BSC, BEC, Z-channel, and 4-ary AWGN with convergence curves.
Akhmediev Breathers — Modulation Instability
NLS modulation instability seeds Akhmediev breathers — exact periodic solutions that grow, peak, and recur on a plane-wave background. Precursors to rogue waves in optics and deep water.
φ⁴ Kink — Topological Field Defects
Kink solitons in the double-well φ⁴ field theory carry conserved topological charge ±1 and scatter as relativistic objects. Watch kink-antikink collisions, bion formation, and radiation.
Voter Model — Interface Coarsening
Opinions coarsen via copying neighbors: interface density decays as 1/ln(t) in 2D and t^{−1/2} in 1D. Connected to coalescing random walks, SLE₂, and non-equilibrium critical phenomena.
Anderson Localization — Transfer Matrix
Any disorder localizes all 1D quantum wavefunctions exponentially. Transfer matrix product gives a positive Lyapunov exponent γ; localization length ξ = 1/γ collapses with increasing W.
Chemical Garden — Silicate Tubes
Metal salt crystals in sodium silicate grow hollow osmotic tubes driven by buoyancy and membrane rupture. A self-organizing dissipative structure mimicking hydrothermal vent chemistry.
Coupled Map Lattice — Kaneko Chaos
Spatiotemporal chaos from logistic maps coupled by diffusion. The space-time diagram reveals frozen disorder, travelling waves, and chaotic turbulence as coupling ε and nonlinearity r vary.
Price Equation — Evolutionary Selection
George Price's exact decomposition: trait change = Cov(fitness, trait) + transmission bias. Watch Fisher's fundamental theorem, mutation-selection balance, and drift in finite populations.
Kitaev Honeycomb — Spin Liquid & Visons
Exactly solvable honeycomb model with bond-directional interactions. Z₂ flux excitations (visons) live on plaquettes; the B-phase hosts gapless Majorana fermions and non-Abelian anyons.
Heteroclinic Cycle — Rock-Paper-Scissors
Non-transitive competition creates a heteroclinic cycle on the simplex — trajectories spiral through each corner with ever-increasing dwell time. Break symmetry to see coexistence or extinction.
Viscoelastic Rheology — Maxwell & Kelvin-Voigt
Spring-dashpot models of creep, stress relaxation, and oscillatory loss. Compare Maxwell, Kelvin-Voigt, Burgers, and Standard Linear Solid responses with live mechanical diagrams.
Butterfly Effect — Lorenz Divergence
Two Lorenz attractor trajectories separated by a tiny perturbation diverge exponentially — visualizing sensitive dependence and the maximal Lyapunov exponent in real time.
Wigner Semicircle Law
Eigenvalue density of random symmetric matrices converges to a perfect semicircle as N→∞ — with GOE level repulsion contrasted against Poisson statistics.
Ecological Tipping Points
Fold bifurcation creates catastrophic regime shifts in ecosystems — the S-shaped bifurcation diagram reveals hysteresis: recovery requires far less stress than collapse.
Josephson Junction Washboard
Phase particle trapped or running in a tilted washboard potential — supercurrent to resistive switching, Stewart-McCumber damping, and the AC Josephson effect.
Active Gel — Actomyosin Patterns
ATP-driven myosin contractility destabilizes uniform actin density into clusters and waves — an active-matter instability governing cell cortex dynamics.
Hofstadter Butterfly & Fibonacci Chain
Fractal energy spectrum of 2D electrons in a magnetic field, and the Cantor-set DOS of the Fibonacci quasicrystal — quasiperiodicity meets quantum mechanics.
Abelian Sandpile Toppling
Grains topple when a cell overflows — the sandpile group governs recurrent configurations, avalanche power laws, and the fractal identity element of SOC.
Stochastic Block Model Recovery
Community detection in random graphs: adjacency matrix and spectral embedding reveal planted partitions — the Kesten-Stigum threshold marks the detectability phase transition.
Basset Hydrodynamic Memory
The Basset history force gives fluid a t⁻¹/² memory of past accelerations — non-Markovian particle dynamics with algebraic velocity relaxation.
BB84 Quantum Key Distribution
Simulate the BB84 protocol: eavesdropping induces a 25% QBER detectable above the 11% security threshold — the no-cloning theorem as cryptographic shield.
Magnon Dispersion in a Heisenberg Ferromagnet
Spin waves as bosonic quasiparticles: ω(k) = 4JS sin²(ka/2) + 2DS gives a quadratic dispersion with an anisotropy gap — watch collective spin oscillations propagate and explore the full BZ.
Packing Frustration & Icosahedral Glass Defects
Icosahedral order locally minimizes energy but cannot tile 3D space — geometric frustration forces disclination defects, arrests crystallization, and births the glass state.
Coined Quantum Walk — Interference & Ballistic Spreading
Hadamard coin + shift operator yields σ ~ t vs. σ ~ √t classical diffusion — interference creates two-peaked distributions and quadratic search speedup.
Neutral Fixation — Wright-Fisher Genetic Drift
Random binomial sampling drives allele frequencies to fixation or loss; P(fix) = p₀ and mean time = 4N generations — Kimura's neutral theory rendered visible.
Umklapp Scattering & Thermal Conductivity
Peierls U-process: k₁+k₂ = k₃+G flips phonon momentum, creating thermal resistance. Explore the κ peak, 1/T high-T regime, and Callaway model with tunable Debye temperature.
Arnold's Cat Map — Deterministic Chaos on T²
The Anosov automorphism [[1,a],[b,ab+1]] scrambles a cat portrait on the torus with Lyapunov exponent λ ≈ 0.96 — yet perfectly restores it after a finite number of iterations.
Colloidal Gel — Arrested Spinodal Decomposition
Cahn-Hilliard spinodal with a gelation threshold: phase separation freezes mid-coarsening, locking in a bicontinuous network of fractal dimension ~2.4 — the structural origin of gels.
Crawford-Sobel Cheap Talk Equilibria
Sender bias b limits communication to N* partition intervals — explore how information degrades as interests diverge and the Crawford-Sobel partition equilibria collapse.
Carbon Nanotube Chirality & Band Structure
Chiral indices (n,m) determine metallic vs. semiconducting character: (n−m) mod 3 = 0 passes a Dirac K-point — zone-folding the graphene cone into quantized 1D subbands.
Second Harmonic Generation & Phase Matching
χ⁽²⁾ nonlinearity converts ω→2ω with efficiency ∝ sinc²(ΔkL/2): phase matching gives quadratic growth in crystal length; mismatch limits conversion to the coherence length.
Kitaev Chain Majorana Modes
Topological superconductor with edge Majorana zero modes: BdG spectrum, phase boundary |μ|=2t, and exponentially localized end states.
Fokker-Planck Probability Flow
Evolution of probability densities under drift and diffusion: watch Gaussian packets drift toward Boltzmann equilibrium in real time.
Topological Defects Winding Number
XY model vortices: place +1 and -1 vortices on a canvas and visualize how the vector field winds around topological defects.
Ising Frustration Triangular Antiferro
Geometric frustration in the triangular antiferromagnet: no ground state satisfies all bonds, yielding extensive entropy and no ordering.
Quantum Teleportation Bell Measurement
Step-by-step quantum teleportation: Bell pair creation, joint measurement, classical communication, and Bloch sphere correction.
Driven Cavity Lid Vortex
Lid-driven viscous cavity flow: primary vortex and corner eddies solved via stream-function/vorticity formulation at varying Reynolds number.
Rayleigh-Benard Convection Onset
Buoyancy-driven instability: watch conduction give way to convection rolls as Ra exceeds the critical value of 1708.
Levy Stable Distribution Alpha
Heavy-tailed Levy stable distributions with tunable stability index alpha and skewness beta, generated via the Chambers-Mallows-Stuck algorithm.
Kuramoto Order Parameter Phase
Synchronization transition in coupled oscillators: order parameter r rises at K_c as coupling overcomes natural frequency disorder.
Cellular Automaton Rule 110
Turing-complete elementary CA: Rule 110 produces gliders and complex collisions; compare with Rule 30 (chaos) and Rule 90 (fractals).
Fibonacci Heap Decrease-Key
Amortized O(1) decrease-key with cascading cuts: watch the heap tree restructure lazily, enabling Dijkstra's O(E + V log V) bound.
Stochastic Block Model — Planted Partition
Community detectability near the Kesten–Stigum threshold: tune p_in, p_out, and k blocks; watch when spectral methods can recover structure.
Rayleigh Surface Wave
Retrograde elliptical particle motion at a free elastic surface: amplitude decays exponentially with depth, confining seismic energy to one wavelength.
Quantum Eraser — Delayed Choice
Which-way information destroys interference; erasing the path tag restores fringes — even after the photon lands, preserving causality via post-selection.
Epidemic Pair Approximation
Beyond mean-field SIR: track SI, SS, II pair densities with Keeling's moment closure; compare to stochastic agent simulation on a random graph.
Debye Phonon Heat Capacity
From T³ law to Dulong-Petit saturation: quantized lattice phonons give C_V = 9Nk_B(T/θ_D)³ × Debye integral, with animated lattice vibrations.
Active Matter — Motility-Induced Phase Separation
Self-propelled run-and-tumble particles spontaneously cluster without attraction: crowding slows particles, creating dense islands — a purely non-equilibrium phase transition.
2D Navier-Stokes Vorticity-Stream
Click to inject vortex dipoles into a 2D fluid; watch the inverse energy cascade form large coherent structures as viscosity slowly damps vorticity.
Thouless Pump — Adiabatic Charge Transport
Rice-Mele model: as the pump parameter φ cycles by 2π, exactly one electron is transported — quantized by the Chern number of the occupied Bloch bands.
Branching Brownian Motion — Extremal
Particles diffuse and branch; the rightmost position follows Bramson's law m(t) = √(2β)t − (3/2√(2β))log t, connecting FKPP waves to extreme value theory.
Geodesic Flow — Poincaré Disk
Ergodic geodesics in the hyperbolic plane: circular arcs perpendicular to the unit disk boundary, mixing ergodically due to negative curvature K = −1.
Subcritical Hopf Bifurcation
Bistable system with coexisting fixed point and limit cycle: ṙ = μr + r³ − r⁵. A finite kick escapes the basin and triggers runaway oscillations.
Spin Waves — Magnon Dispersion
Ferromagnetic spin chain: collective precession modes with quadratic dispersion ω(k) = 4JS(1−cos ka), animating wave propagation and the Brillouin zone spectrum.
Quantum Hall Effect — Chern Numbers
TKNN formula: Berry curvature integrated over the Brillouin zone gives quantized Hall conductance σ_xy = Ce²/h. Visualizes edge states and Landau levels.
Contact Process — Absorbing State DP
Directed percolation universality class: spacetime diagram of an absorbing-state phase transition with order parameter ρ ~ (λ−λ_c)^β, β ≈ 0.276.
Epidemic Threshold — Heterogeneous Networks
SIS model on scale-free networks: λ_c = ⟨k⟩/⟨k²⟩ → 0 as ⟨k²⟩ diverges, making epidemics spread at vanishingly small infection rates (Pastor-Satorras 2001).
Elastic Snap-Through Instability
Bistable shallow arch with double-well energy landscape V = k(y²−h²)²: a sufficient kick causes irreversible snap-through between two stable configurations.
Protein Aggregation — Nucleation & Seeding
Amyloid fibril formation: slow nucleation, rapid elongation, and secondary nucleation produce sigmoidal kinetics with tunable lag phase — Alzheimer mechanism.
Diffusion Maps — Spectral Embedding
Random walk on data geometry: eigenvectors of the Markov matrix reveal nonlinear manifold structure — Swiss roll, torus, and moons unfolded by diffusion distance.
Random Field Ising — Avalanches
Disorder-tuned criticality: power-law Barkhausen avalanches as an external field sweeps through a 2D RFIM lattice, with scale-free statistics at the critical disorder R_c.
Vicsek Flocking & Polar Order
Self-propelled particles aligning with neighbors undergo a non-equilibrium phase transition from disorder to collective polar order as noise is reduced — color encodes heading direction.
Josephson Junction Chaotic Pendulum
The driven damped pendulum — exact analog of a Josephson junction — produces chaos and strange attractors in phase space for certain drive amplitudes, visualized as Poincaré sections.
Wigner Crystal — Electron Lattice
Classical electrons crystallize into a triangular lattice when Coulomb repulsion dominates kinetic energy at low temperature — Wigner's 1934 prediction visualized via molecular dynamics.
Nonlinear Wave & Shock Formation
Burgers' equation steepens a waveform until characteristics converge and a shock front forms — viscosity regularizes the discontinuity, color encodes local gradient steepness.
Minority Game & Maladaptation
Agents competing to be in the minority using history-based strategies exhibit a phase transition at critical memory/agent ratio α*: below it, over-coordination (maladaptation) inflates variance.
Quantum Annealing — Transverse-Field Ising
A large transverse field (quantum tunneling) is slowly reduced to anneal a random Ising problem to its ground state — path-integral Monte Carlo encodes the quantum superposition classically.
Stochastic Resonance — Noisy Neuron
A subthreshold periodic signal becomes detectable when optimal noise is added to a threshold neuron — too little or too much noise degrades SNR, demonstrating that noise can be beneficial.
Sandpile Burning Bijection
Abelian sandpile model with Dhar's burning algorithm: test recurrent configurations (elements of the sandpile group) via the bijection with spanning trees of the underlying graph.
Kosterlitz-Thouless Topological Transition
2D XY model: below T_KT vortex-antivortex pairs are bound and quasi-long-range order survives; above it free vortices proliferate and destroy order — Nobel Prize 2016 physics.
Epidemic Awareness on Multiplex Networks
Disease spreads on a physical contact layer while awareness spreads on a social information layer — aware individuals reduce transmission, coupling the two dynamics in a duplex SIR model.
Nematic Liquid Crystal Defect Coarsening
±½ topological defects annihilate and coarsen in a nematic director field — watch opposite-charge defect pairs attract and annihilate via XY-model dynamics.
XXZ Spin Chain Anisotropy
Exact Bethe ansatz energy spectrum of the XXZ Hamiltonian across ferromagnetic, gapless XY, and gapped Néel phases as anisotropy Δ varies.
Replicator Equation — Evolutionary Game Dynamics
Strategy frequency evolution on the simplex under Rock-Paper-Scissors, Hawk-Dove, and Coordination games — cycling orbits, mixed ESS, and bistability.
Atmospheric Blocking — Rossby Wave Resonance
Quasi-geostrophic barotropic flow on a β-plane: stationary Rossby waves form near resonant wavenumber k_s=√(β/U) and lock into persistent blocking highs.
Rice-Mele Topological Charge Pump
Adiabatic parameter cycling pumps exactly one electron per period — quantized transport equal to the Chern number of the occupied band over the (k,φ) torus.
Network Motif Triad Census
Count all 13 directed 3-node subgraph classes across Erdős-Rényi, feed-forward, and scale-free networks — compare to random expectations to reveal structural signatures.
Stochastic Theta Neuron Excitability
The SNIPER normal-form neuron near its Type I excitability bifurcation — noise drives stochastic spiking below threshold, ISI distributions reveal sub- vs supra-threshold regimes.
2D Electron Gas Plasmon Dispersion
The 2D electron gas hosts acoustic plasmons ω∝√q unlike the constant 3D plasma frequency — hover to probe dispersion and Landau damping boundaries.
Genetic Drift — Founder Effect & Bottleneck
Wright-Fisher multinomial sampling erodes allele diversity stochastically — trigger a population bottleneck to watch founder effects eliminate genetic variation.
Lattice Boltzmann Fluid — BGK Collision
D2Q9 LBM fluid simulation with BGK collision operator — flow around obstacles, vortex shedding, and vorticity visualization at tunable Reynolds number.
Turbulent Pipe Flow — Reynolds Transition
Dye streaklines break apart as Reynolds number crosses 2300 — watch laminar parabolic flow shatter into turbulent chaos with live velocity profile.
Quantum Dot — Coulomb Blockade
Single-electron charging energy blocks current through a nanoscale island — tune the gate voltage to open Coulomb peaks and watch electrons tunnel one by one.
Network Dismantling — Optimal Percolation
Find the minimal node set whose removal fragments a scale-free network — compare degree, betweenness, and random strategies for percolation threshold attack.
Active Nematics — Topological Defect Dance
Self-propelled +1/2 and -1/2 topological defects drive spontaneous flows in an active liquid crystal — tune activity to go from order to turbulent chaos.
Collective Motion — Turn Propagation
Click the flock to trigger a directional turn and watch it propagate as a wave — scale-free correlations enable near-instantaneous collective response.
Reaction Network — Chemical Master Equation
Gillespie SSA simulates stochastic gene expression — protein bursting, Fano factor noise, and toggle-switch bimodality emerge from discrete molecule counts.
Chaos Control — OGY Method
Tiny parameter perturbations applied near unstable periodic orbits tame the logistic map's chaos — watch the OGY algorithm stabilize period-1 orbits.
Interface Roughening — KPZ Universality
Growing interfaces obey KPZ scaling W~t^(1/3) — compare the nonlinear KPZ equation against Edwards-Wilkinson and ballistic deposition models.
Protein Coevolution — MI/DCA Coupling
Mutual information and direct coupling analysis extract residue contacts from simulated multiple sequence alignments — DCA's APC correction beats raw MI.
Quantum Scarring — Stadium Billiard
Classical chaos in the Bunimovich stadium vs quantum eigenstates scarred along periodic orbits — Heller's 1984 discovery visualized in real time.
Mean-Field Game: Crowd Dynamics
Nash equilibrium flows in a congested pedestrian system — rational agents minimize travel cost plus congestion penalty, solving coupled HJB and Fokker–Planck equations.
Superconductor Vortex: Abrikosov Lattice
Quantized flux vortices in a type-II superconductor self-organize into Abrikosov's triangular lattice via London-limit K₀ repulsion, with adjustable GL parameter κ and thermal disorder.
Neural Mass Model: Wilson–Cowan
Coupled excitatory–inhibitory neural populations display fixed points, limit cycles, and bistability — explore nullclines and phase portraits by tuning synaptic weights.
Parametric Resonance: Mathieu Instability
The Mathieu equation's Ince–Strutt stability chart reveals instability tongues where periodic driving pumps exponential amplitude growth — navigate (a, q) parameter space live.
Hyperbolic Space: Cayley Tree Expansion
The Poincaré disk reveals how hyperbolic geometry packs exponentially many nodes — a k-branching Cayley tree embeds isometrically in H² with geodesic arcs drawn exactly.
Oscillator Death: Amplitude Coupling
Coupled Stuart–Landau oscillators with frequency spread undergo amplitude death — strong enough mean-field coupling quenches all limit cycles to a stable fixed point.
Fracture Percolation: Fuse Network
Progressive bond breaking in a random fuse network mimics fracture mechanics — watch current redistribute after each failure until a spanning crack percolates through the lattice.
Quantum Chaos: Level Statistics Spacing
BGS conjecture made visible — random matrix ensembles GOE, GUE, and Poisson produce distinct level spacing distributions, separating quantum-chaotic from integrable systems.
Active Particle: Microswimmer Circle Trajectories
Chiral self-propelled particles trace circular orbits and accumulate at walls — a non-equilibrium effect with no passive analog, relevant to bacterial motility near surfaces.
Ecological Neutral Theory: Hubbell
Pure demographic stochasticity, speciation, and immigration assemble a community — Hubbell's neutral theory generates log-series species abundance distributions without fitness differences.
Voter Model Interface Coarsening
Opinion dynamics: domain coarsening via random neighbor copying in 2D
Floquet2013Bloch Driven Band Structure
Quasienergy bands of a periodically driven lattice; dynamical localisation at J0=0
Kauffman NK Boolean Network Attractors
Random Boolean networks: ordered, critical, and chaotic phases; attractor basins
Bose2013Einstein Condensation Ideal Gas
Condensate fraction vs temperature; macroscopic ground-state occupation below T_c
Brusselator Turing Pattern Formation
Reaction-diffusion chemical oscillator generating spots and stripes via Turing instability
Semiclassical WKB Tunneling Barrier
Quantum tunneling transmission through a Gaussian barrier via WKB approximation
Epidemic with Behavioral Feedback and Vaccination
SIRV model with voluntary vaccination driven by perceived epidemic risk
Hamiltonian Chaos Standard Map KAM Tori
Chirikov standard map: KAM tori, island chains, and chaos onset at K_c=0.9716
Magnetic Reconnection Current Sheet
Sweet-Parker reconnection: field line topology change and particle acceleration at X-point
Hard Disk Packing Crystallization and Jamming
KTHNY melting, hexatic order, and the jamming transition in 2D hard disk MD
Anomalous Diffusion CTRW Subdiffusion
Continuous-time random walk with power-law waiting times: subdiffusion where mean-squared displacement grows as t^alpha with alpha less than 1
Topological Magnon Chern Honeycomb
Ferromagnetic honeycomb with DM interaction: nonzero Chern number magnon bands and chiral edge states — bosonic quantum Hall effect
Opinion Polarization Echo Chamber Model
Bounded-confidence Deffuant dynamics with algorithmic homophily: how echo chambers drive opinion fragmentation vs. consensus
Quantum Walk Continuous Time on Graph
Wavefunction evolution under the graph Laplacian: quantum interference enables quadratically faster spreading than classical diffusion
Viscous Fingering Hele-Shaw Displacement
Saffman-Taylor instability: low-viscosity fluid invasion of high-viscosity medium generates fractal fingering patterns
Driven-Dissipative Dicke Superradiance
N atoms collectively coupled to a cavity: mean-field superradiant phase transition with Bloch sphere dynamics and cavity field
Adaptive Network Coevolution Rewiring
Holme-Newman model: opinions and network topology coevolve — fragmentation vs. consensus phase transition controlled by rewiring rate
Stochastic SIS Epidemic Gillespie Algorithm
Exact stochastic SIS epidemic trajectories via the Gillespie algorithm: extinction, endemic state, and ensemble fluctuations
Fiber Bundle Model Fracture Statistics
Equal load sharing fiber bundle with Weibull thresholds: failure cascade avalanches, spinodal instability, and fracture statistics
Many-Body Localization Entanglement Growth
MBL vs. ergodic phase: disorder-driven transition where entanglement grows logarithmically in time instead of linearly
✦ Lab #9300 Milestone ✦
The Iris Compendium — 9,300 interactive explorations of the universe
Cell Polarity Turing Cdc42
Wave-pinning model: active Cdc42 self-amplification breaks cell symmetry into a stable polar front
Wave Turbulence Four-Wave
Kolmogorov-Zakharov spectrum from resonant four-wave interactions in weakly nonlinear dispersive waves
NK Fitness Landscape
Kauffman NK model: evolving populations climb rugged epistatic fitness landscapes with K-tunable ruggedness
Granular Impact Crater Scaling
Click to drop impactors and measure power-law crater radius scaling across 26 orders of magnitude in energy
SSH Thouless Pump
Topological edge states and quantized charge transfer via winding number in a dimerized SSH chain
NESS Driven Chain
Heat flows and entropy production in a harmonic chain coupled between hot and cold reservoirs
Wisdom of Crowds
Individual log-normal errors cancel in the geometric mean — Galton's ox-weight experiment simulated live
Quantum Metrology Fisher Limit
Shot-noise vs Heisenberg limit for phase estimation — NOON states achieve quadratic precision improvement
Spin Glass Energy Landscape
Simulated annealing through a rugged frustrated energy landscape with random ±J couplings
Causal Emergence
When macro-scale descriptions have higher effective information than the micro-level Markov chain
Eden Cluster Growth
Random perimeter attachment grows KPZ-class rough clusters with interface width W~t^(1/3)
QAOA MaxCut
Quantum approximate optimizer solves MaxCut via alternating cost and mixer unitaries on a graph
Neural Population Manifold
PCA reveals low-dimensional manifold structure in tuning-curve neural population responses
Hawk-Dove ESS
Replicator dynamics drives mixed populations to the evolutionarily stable Hawk-Dove equilibrium
Rubber Elasticity
Entropy-driven neo-Hookean stress σ=nkT(λ−1/λ²) in crosslinked polymer networks under stretch
KMC Surface Growth
Kinetic Monte Carlo deposition and diffusion showing smooth vs. rough crystal growth regimes
Subcritical Pipe Turbulence
Bypass transition in Hagen-Poiseuille flow: turbulent puffs via finite-amplitude perturbations
Active Nematics
Self-propelled ±½ topological defects drive spontaneous flows and active turbulence in nematics
Network Robustness
Scale-free networks fail fast under betweenness-targeted attacks but resist random node removal
RGG Percolation
Giant component emerges in random geometric graphs as connection radius crosses the percolation threshold
Nonequilibrium Heat Flux
Harmonic chain between hot and cold baths reaches a steady state with measurable temperature gradient
Vortex Annihilation
2D XY model vortex-antivortex pairs attract and annihilate, reducing topological defect density
Stochastic Block Model
Spectral embedding reveals community structure until SNR falls below the detectability threshold
Entanglement Area Law
Ground state entanglement saturates (area law) while random states grow linearly with subsystem size
Opinion Cascade
Watts threshold model shows global cascades when heterogeneous adoption thresholds align with network degree
Helfrich Membrane
Thermal fluctuations of a fluid membrane obey the Helfrich bending-rigidity power spectrum
Adaptive Dynamics PIP
Pairwise invasibility plots reveal ESS and evolutionary branching points via invasion fitness landscapes
Directed Percolation
Contact process exhibits an absorbing-state phase transition with critical density exponent β ≈ 0.276
GUE Wigner Surmise
GUE random matrix eigenvalue spacings converge to the Wigner surmise, showing quantum level repulsion
Metabolic Allometric Scaling
West-Brown-Enquist B~M^(3/4): fractal vascular networks explain metabolism across 27 orders of mass
Transverse-Field Ising QPT
Quantum critical point at h=J: gap closes, order parameter vanishes, correlation length diverges
Epidemic Superspreaders
SIR on scale-free networks: degree heterogeneity lowers threshold and creates hub-driven outbreaks
Dislocation Forest Hardening
2D edge dislocation dynamics: Taylor hardening τ~√ρ emerges from stress-field interactions
Delay-Coupled Synchronization
Kuramoto model with transmission delay: critical coupling shifts and oscillating synchrony emerges
Fractal Coastline Dimension
Midpoint-displacement coastlines: animated box-counting measures D_f = 2−H via log-log slope
Secretary Problem
Optimal stopping: reject r*=N/e candidates, then hire the next record-breaker — success → 1/e
Stochastic Lotka-Volterra
Gillespie algorithm for predator-prey: demographic noise drives extinction from quasi-stable orbits
Chern Insulator & Berry Curvature
QWZ model: Berry curvature heatmap, Chern number, and chiral edge states across topological phases
Information Cascade Model
BHW Bayesian cascade: sequential agents ignore private signals once public history overwhelms them
Forest Fire SOC
Drossel-Schwabl fire model self-organizes to criticality with power-law cascade sizes
Toric Code Anyons
Kitaev toric code: create e/m anyons by flipping qubits, braid for topological phase
Vicsek Flocking
Self-propelled particles align with neighbors — noise drives order-disorder phase transition
Stochastic Resonance
Optimal noise helps weak signals cross double-well barrier — SNR peaks at ideal noise level
Fitness-Preferential Network
Bianconi-Barabási model: node fitness shapes attachment, Bose-Einstein condensation emerges
Wisdom of Crowds
Hong-Page diversity theorem: diverse agents outperform homogeneous experts at estimation
Phononic Crystal Bandgap
Alternating masses and springs open elastic wave bandgaps — forbidden frequencies blocked
Deficiency-Zero Theorem
Feinberg's theorem: δ=0 reaction networks always converge to unique stable equilibrium
φ⁴ Kink Scattering
Kink-antikink collisions in φ⁴ field theory: bion formation, annihilation, resonance windows
Gompertz Mortality Law
Exponentially rising hazard rate with age; frailty heterogeneity decelerates mortality at extremes
Hysteresis & Metastability
Mean-field Ising hysteresis loops, metastable states, and first-order phase transitions at varied temperature
Stadium Billiard Husimi
Quantum chaos eigenstates in the Bunimovich stadium with Husimi phase-space and wavefunction scars
Network Copying Model
Krapivsky-Redner copying: degree distribution interpolates between Poisson and power-law as copying increases
Pendulum Chaos & Poincaré
Driven damped pendulum Poincaré sections reveal transition from regular orbits to strange attractor chaos
Ant Colony Stigmergy
Pheromone reinforcement and evaporation guide ants to shortest food-to-nest paths with no central control
Spectral Graph Clustering
Fiedler vector of the graph Laplacian bipartitions nodes; algebraic connectivity measures cluster quality
Magnon Spin Wave Dispersion
Heisenberg ferromagnet spin wave dispersion, group velocity, density of states, and animated precession
Avascular Tumor Growth
Greenspan model: proliferating shell, quiescent core, necrotic center driven by oxygen diffusion limits
Bell Inequality CHSH
CHSH game with entangled qubits achieves S=2√2 violating the classical bound S≤2 — quantum nonlocality live
River Networks & Hack's Law
Simulated drainage basins on random terrain reveal Hack's law L~A^0.6 and Horton bifurcation ratios
Random Sequential Adsorption
Disks land randomly on a surface, jamming at ~54.7% coverage — the RSA limit
Langton's Self-Replicating Loop
8-state cellular automaton where a loop structure autonomously replicates itself
Josephson Junction RCSJ
Phase dynamics of superconducting junction: I-V characteristic and Shapiro steps
Temporal Activity-Driven Network
Power-law active nodes create transient edges; SIS epidemic on evolving topology
Lattice Field Theory Continuum Limit
1D scalar field dispersion ω²=m²+4κsin²(ka/2) recovering relativistic limit as a→0
Social Learning: DeGroot vs Bayesian
Networked agents update beliefs by averaging or Bayesian rule; compare consensus speed
Polar Vortex Barotropic Instability
Rotating fluid jet supports Rossby waves; barotropic instability breaks the polar vortex
Amyloid Nucleation & Aggregation
Oosawa nucleated polymerization: sigmoidal fibril growth with lag phase and seeding
Neutral Theory Species-Area Law
Hubbell neutral theory: speciation and dispersal generate S~A^z species-area scaling
Random Walk Graph Spectral Density
Occupation power spectrum on path, cycle, star, and random graphs; spectral gap and mixing
Abrikosov Vortex Lattice
Type-II superconductor: quantized flux vortices self-arrange into triangular Abrikosov lattice in the mixed state
Sandpile Toppling Recurrent
Abelian sandpile self-organized criticality: avalanche cascades with power-law size distributions
Vicsek Flocking Transition
Self-propelled particles align with neighbors; noise drives a discontinuous order-disorder phase transition
Geodesic Flow Poincaré Disk
Hyperbolic geodesics as circular arcs on the Poincaré disk; chaotic, ergodic geodesic flow on ℍ²
Stochastic Block Model
Planted community structure in random graphs; Kesten-Stigum threshold for detectability
Glauber Dynamics Ising
Single-spin-flip Markov chain for Ising model; critical slowing down at Tc = 2.269J
Nonlinear Pendulum Separatrix
Phase portrait of exact pendulum: libration, rotation, homoclinic separatrix, and driven chaos
Quantum Kicked Rotor Husimi
Quantum rotor under periodic kicks; Husimi phase-space distribution and dynamical localization
Adaptive Network Coevolution
Opinion dynamics coupled to network rewiring; fragmentation into echo chambers at critical rewiring rate
Diffusion Map Manifold Learning
Heat-kernel eigenvectors embed high-dimensional data preserving geodesic distances on the data manifold
Contact Process & Directed Percolation
Stochastic lattice infection model at the directed percolation universality class critical point λ_c
Fibonacci Quasicrystal Diffraction
Penrose tiling with golden-ratio inflation and its 10-fold symmetric Fourier diffraction pattern
Gray-Scott Mesa Replication
Autocatalytic reaction-diffusion system producing spots, stripes, labyrinths, and self-replicating patterns
Spin Wave Magnon Dispersion
Collective spin precession in a Heisenberg ferromagnet with dispersion ω(k) = 4JS sin²(ka/2)
Rule 110 — Universal CA
Wolfram's Rule 110 elementary CA proven Turing-complete; explore all 256 rules in real time
Triply Periodic Minimal Surfaces
Ray-marched Schwarz P, Schwarz D, Gyroid, and I-WP zero-mean-curvature surfaces in 3D
Entropic Elasticity — WLC
Worm-like chain polymer model with Marko-Siggia force-extension relation for DNA and biopolymers
Chimera States — Kuramoto Ring
Identical oscillators with nonlocal coupling spontaneously split into synchronized and incoherent domains
Quantum Discord Beyond Entanglement
Two-qubit X-state correlations: mutual information, classical correlations, discord, and concurrence
Laguerre–Power Diagram
Weighted Voronoi tessellation dual to circle packing; Lloyd relaxation toward centroidal diagrams
Chiral Active Matter
Self-propelled chiral particles with odd (Hall) viscosity forming rotating flocks
XY Model Defect Dynamics
Vortex–antivortex annihilation driving the BKT phase transition in 2D
Sandpile Identity Element
Fractal identity of the abelian sandpile group via Dhar's burning algorithm
Voter Model Consensus
Opinion coarsening with logarithmically slow interface decay and martingale magnetization
Wigner Crystal — 2D Electrons
Coulomb-driven crystallization of electrons at large r_s parameter
Baryogenesis & Sakharov
Matter–antimatter asymmetry from CP violation, B-number violation, and non-equilibrium
Magnetic Reconnection
Sweet–Parker current sheet dynamics with field-line topology change and energy release
Fibonacci Heap
Amortized O(1) insert and decrease-key in a lazy forest of heap-ordered trees
Joukowski Airfoil Flow
Conformal mapping of a cylinder to an airfoil with exact potential flow and Kutta lift
Ant Colony Stigmergy
Emergent shortest-path finding via pheromone diffusion and evaporation feedback
Contact Process & Directed Percolation
Absorbing-state phase transition on a lattice near critical λ_c ≈ 1.65
Kuramoto Order Parameter
N oscillators synchronize above critical coupling K_c = 2σ/π·g(0)
Thermodynamic Uncertainty Relation
Current precision is bounded by entropy production: Var/mean² · σ/2 ≥ 1
Sznajd Opinion Propagation
Agreeing neighbor pairs convert their neighbors — social consensus dynamics
Mullins-Sekerka Instability
Solidification front goes unstable when diffusion overcomes surface tension
Quantum Annealing (TFIM)
Ramp down transverse field Γ(t) to find Ising ground state via quantum tunneling
Non-Hermitian Skin Effect
Asymmetric hopping localizes all eigenstates at one boundary (Hatano-Nelson)
Granular Gas — Inelastic Collisions
Haff's cooling law T ~ t^{-2} from energy-dissipating particle collisions
Squirmer Microswimmer Flow
Pusher vs puller vs neutral swimmers — far-field stresslet flow signatures
Polymer Brush Density Profile
Alexander scaling L ~ N·σ^{1/3} and MWC parabolic density from SCF theory
Gutenberg-Richter Avalanche
Spring-block SOC model — power-law earthquake magnitude distribution
Feedforward Loop Motif
C1/I1 FFL transcription motif — delay filter vs pulse generator
Stochastic Lotka-Volterra
Gillespie exact simulation — extinction risk from demographic noise
Hyperbolic Tiling
Regular tilings of hyperbolic plane in the Poincaré disk model
XXZ Quantum Spin Chain
Spin-wave dynamics and entanglement spreading in 1D quantum chain
Nematic Active Turbulence
±½ topological defects drive chaotic flows in active nematics
Dislocation Pile-Up
Hall-Petch strengthening — edge dislocations pile up at grain boundary
Anomalous Subdiffusion CTRW
Continuous-time random walk with power-law trapping — MSD ~ t^α
Clonal Selection & Affinity
B-cell somatic hypermutation and affinity maturation in germinal centers
Spectral Clustering
Normalized Laplacian eigenvectors reveal community structure in graphs
Rayleigh-Bénard Convection
Thermal convection rolls form when buoyancy overcomes viscous damping above Ra_c ≈ 1708
Kelvin-Helmholtz Instability
Vortex roll-up at shear interfaces — seen in ocean waves, clouds, and Jupiter's bands
Cosmic String Network Scaling
Topological defects from symmetry-breaking evolve toward a scale-invariant attractor via intercommutation
Maier-Saupe Nematic Order
Mean-field theory of liquid crystals: rod-like molecules align collectively below T* with first-order transition
Protein Aggregation Kinetics
Amyloid fibril growth via primary and secondary nucleation — sigmoidal kinetics with lag phase
Quantum Critical Point
Transverse-field Ising model: quantum phase transition at h/J = 1 driven by quantum fluctuations
Random-Bond Ising Disorder
Quenched random ±J bonds create frustrated plaquettes and spin-glass dynamics on a 2D lattice
Allen-Cahn Coarsening
Phase-ordering dynamics: domains grow as interfaces move toward curvature centers, obeying L ~ t^(1/2)
Debye Screening & Plasma Oscillations
Mobile electrons screen a test charge over the Debye length λ_D and oscillate at the plasma frequency ω_p
First Passage Time
Random walkers race to an absorbing boundary — the Lévy-Smirnov distribution with heavy tails
KPZ Height Fluctuation
Kardar–Parisi–Zhang interface growth with nonlinear slope coupling and β=1/3 roughness exponent
Vicsek Flocking
Self-propelled particles align with neighbors — collective order emerges above a noise threshold
Metabolic Scaling Laws
Kleiber's 3/4-power law: metabolic rate across 27 orders of magnitude of body mass
Network Percolation
Erdős–Rényi random graph: giant component emerges at mean degree ⟨k⟩=1 phase transition
Jamming Transition
Compress bidisperse disks to φ_J ≈ 0.84 — watch force chains rigidify at Point J
QCD Flux Tube
Chromoelectric field collimates into a narrow tube, producing linear confinement V=κr and string breaking
Lorenz-96 Atmosphere
36-variable atmospheric model — chaos onset at forcing F≈5.7 limits weather prediction horizon
Cellular Potts Mitosis
Cells grow and divide via energy-minimizing Monte Carlo — cytokinesis from volume threshold
Stochastic Gene Expression
Two-state promoter noise creates bimodal protein distributions — phenotypic diversity without mutation
Carbon Cycle Tipping
CO₂, ocean, and permafrost feedbacks — saddle-node bifurcation can trigger runaway warming
Kuramoto Finite-Size Order
Synchronization transitions and finite-size fluctuations in coupled phase oscillators
Sandpile Group Identity
Fractal self-similar identity element of the abelian sandpile group via toppling
Quantum Discord
Quantum correlations beyond entanglement in bipartite mixed states — Werner state analysis
Turbulent Boundary Layer
Log-law of the wall: universal velocity profile from inner–outer layer matching
Epidemic Household Clumping
How household structure amplifies epidemic transmission beyond mean-field predictions
Pólya Urn Dynamics
Path-dependent reinforcement processes converging to Beta-distributed random limits
Wigner Crystal
Electron crystallization in jellium when Coulomb energy dominates kinetic energy
Network Motif Detection
Statistically overrepresented subgraph patterns — the functional building blocks of networks
Tropical Rainfall SOC
Self-organized criticality in atmospheric moisture: power-law storm size distributions
Chiral Symmetry Breaking
QCD quark condensate and spontaneous chiral symmetry breaking — how quarks gain mass
Active Matter: MIPS
Self-propelled particles phase-separate without attractions — motility-induced phase separation via Péclet number
Quantum Annealing: Transverse-Field Ising
Quantum tunneling finds ground states by slowly quenching a transverse magnetic field through the quantum phase transition
Glycolysis: Selkov Oscillator
Biochemical limit cycles in yeast glycolysis — ADP/ATP feedback creates autonomous Hopf oscillations
Abrikosov Vortex Lattice
Magnetic flux tubes in type-II superconductors arrange into a triangular lattice via London repulsion
Ecological Bistability & Tipping Points
Shallow lakes switch between clear-water and turbid algal states via catastrophic fold bifurcation and hysteresis
Marchenko-Pastur Law
Eigenvalue density of large sample covariance matrices — the Marchenko-Pastur distribution with BBP spike transitions
Colloidal Gel: Arrested Phase Separation
Attractive colloids undergo spinodal decomposition then vitrify into a space-spanning gel network
Nonlinear Pendulum: Exact Phase Portrait
Exact phase space with separatrix, libration vs rotation, driven chaos and Poincaré sections
Entropy Rate: Hidden Markov Process
Computing fundamental compression limits for structured sequences — LZ complexity, conditional entropy, and the HMM entropy rate
Atmospheric Blocking Anticyclone
Persistent high-pressure systems deflect the jet stream via stationary Rossby waves — associated with extreme weather
Shear Banding Flow Instability
Viscoelastic fluids split into coexisting high- and low-shear bands — a fluid phase transition driven by the non-monotonic flow curve
Kitaev Honeycomb Spin Liquid
Exactly solvable Kitaev model on the honeycomb lattice: fractionalized Majorana fermions, Z₂ flux visons, and topological phases
Stochastic Gene Expression Bursting
Two-state promoter model of transcriptional bursting: negative binomial protein distributions and cell-to-cell variability
River Meander & Oxbow Formation
Fluvial geomorphology: helical flow erodes outer banks, amplifies curvature, and eventually cuts off oxbow lakes
Spiked Random Matrix BBP Transition
Baik-Ben Arous-Péché (2005): an outlier eigenvalue detaches from the Marchenko-Pastur bulk only above a sharp signal threshold
Amari Neural Field Bump Attractor
Continuum neural field with Mexican-hat connectivity supports stable localized bumps — a continuous attractor model of working memory
Cooperative Binding & Hill Equation
Hemoglobin oxygen binding: Hill coefficient quantifies cooperativity, MWC allosteric model, and sigmoidal switch behavior
Acoustic Emission Dislocation Avalanche
Crackling noise in crystal plasticity: dislocation avalanches follow power-law size distributions — a signature of self-organized criticality
Skyrmion Nucleation & Topology
Magnetic skyrmions: topologically protected spin vortices with integer winding number, stabilized by DM interaction and external field
Information Cascade & Bayesian Herding
BHW (1992): rational agents sequentially ignore private signals and follow the crowd — wrong cascades are inevitable and fragile
Granular Compaction: Tapping
Chicago experiment: packing fraction evolves logarithmically under vertical taps — reversible vs irreversible regimes and memory effects
Quantum Kicked Top
Husimi Q-function on the Bloch sphere: kick strength drives chaos and rapid entanglement generation up to the Page value
Fisher-KPP Pulled Front
Reaction-diffusion wave of invasion: minimal speed c*=2√(Dr), Allee effects create pushed fronts with sharp threshold behaviour
Spectral Flow & Index Theorem
1D Dirac operator: eigenvalue crossings under gauge parameter sweep count the winding number — Atiyah-Singer in action
Adhesion Cluster Rebinding
Bell model stochastic simulation: cooperative bond rupture, catch-bond effect, and force-dependent cluster lifetime in cell adhesion
Magnetic Reconnection
Sweet-Parker resistive MHD: X-point topology, current sheet width δ=L/√S, and the reconnection rate that's too slow for solar flares
Lattice Boltzmann D2Q9
BGK collision on a 2D square lattice: Poiseuille flow between walls converges to the exact parabolic velocity profile
BZ Spiral Tip Dynamics
Oregonator model: spiral waves in the BZ reaction undergo rigid rotation or meandering via a Hopf bifurcation of the tip orbit
Erdős-Rényi Giant Component
G(n,p) random graph percolation transition at ⟨k⟩=1: giant component emerges via β=1−e^(−cβ), with simulated phase diagram
Microtubule Dynamic Instability
GTP-cap model: stochastic catastrophe and rescue events, bounded vs unbounded growth phase, and exponential length distributions
Vicsek Flocking Model
Self-propelled particles align with neighbors: noise-driven order-disorder phase transition, order parameter φ, and the critical noise threshold
Quantum Thermometry
Cramér–Rao bound for temperature estimation: quantum Fisher information of a two-level probe, optimal noise, and MLE distribution vs theory
Ostwald Ripening (LSW)
Gibbs-Thomson coarsening: large droplets grow at expense of small ones, mean radius ⟨R⟩³ ∝ t (Lifshitz-Slyozov-Wagner theory)
Epidemic Immunization Strategies
Targeted vs acquaintance immunization on scale-free networks: friendship paradox exploited for near-optimal vaccination without network knowledge
Stochastic Resonance
Noise enhances detection of subthreshold signals: optimal noise level maximizes SNR, with dithering applications in analog-to-digital conversion
Langmuir Plasma Waves
Bohm-Gross dispersion ω²=ωₚ²+3k²v_th²: plasma frequency cutoff, phase vs group velocity, and evanescent modes below ωₚ
Conway's Life: Glider Gun
Gosper glider gun, R-pentomino, pulsar, and random soups: emergent complexity and Turing-complete computation from B3/S23 rules
Spin Waves & Magnon Dispersion
Heisenberg ferromagnet: Holstein-Primakoff spin-wave dispersion ω(k)=2JS(1-coska)+gμH, magnon DOS, and animated spin precession chain
Hele-Shaw Viscous Fingering
Saffman-Taylor instability: low-viscosity fluid invades high-viscosity medium, linear growth rate σ(k), critical wavenumber, and fingering patterns
Cusp Bifurcation
Codimension-2 normal form ẋ=h+rx−x³: cusp catastrophe in parameter space, bistability region, potential landscape, and sudden jump transitions
Quorum Sensing & Biofilm Switch
Collective bacterial decision-making via autoinducer signaling — the density-dependent switch to biofilm
Thouless Topological Pump
Quantized adiabatic charge transport through a slowly varying periodic potential — Chern number in action
Degree-Corrected Stochastic Block Model
Community structure with heterogeneous degree distributions — force-directed network visualization
Phononic Crystal Bandgap — 1D
Elastic wave propagation in a periodic medium — Bragg scattering creates forbidden frequency bands
Mutual Information & Transfer Entropy
Information-theoretic causality detection — directional coupling revealed by T(X→Y) ≠ T(Y→X)
Prey-Predator Spatial Turing Instability
Reaction-diffusion driven pattern formation in ecological systems — spots and stripes emerge from coupling
Bose Glass & Superfluid-Insulator Transition
Disorder-driven localization in the Bose-Hubbard model — the exotic Bose glass phase between SF and Mott
Complex Social Contagion
Threshold models and social reinforcement on networks — why behaviors spread differently than diseases
Mode Coupling Glass Relaxation
α and β relaxation, the non-ergodicity plateau, and power-law divergence of viscosity near T_c
Rényi Entropy & Multifractal Spectrum
Generalized entropies, singularity spectra, and the f(α) Legendre transform of multifractal measures
Lab #9500 Milestone — Iris Compendium V
9500 interactive explorations: quantum phase space, Turing patterns, hyperbolic tiling, Langton's ant, double pendulum chaos
Kleiber's Law — Metabolic Scaling
Metabolism ∝ M^(3/4) spanning 27 orders of magnitude, derived from fractal vascular network geometry (West-Brown-Enquist)
Entropic Elasticity — FJC vs WLC
Polymer stretching: freely jointed chain (inverse Langevin) vs worm-like chain (Marko-Siggia) force-extension curves
Rice-Mele Topological Pump
Adiabatic cyclic parameter variation pumps exactly one electron per cycle — Thouless pump, quantized by Chern number
Basset Memory Force
Particle in viscous fluid with fractional-derivative Basset force: algebraic t^(-3/2) decay vs exponential Stokes drag
Nonlinear Wave Breaking — Hodograph
Inviscid Burgers equation: characteristics carrying different speeds cross at T_break = -1/min(u'(x,0)), forming shocks
Wigner Crystal — r_s Parameter
At low electron density (r_s > 37–100), Coulomb repulsion dominates kinetic energy, freezing electrons into a triangular lattice
Contact Process — Directed Percolation
Absorbing-state phase transition at λc ≈ 3.298 (DP universality class): fractal spacetime activity clusters at criticality
Stochastic Lotka-Volterra (Gillespie)
Exact predator-prey stochastic simulation: exponential waiting times, discrete populations, extinction possible despite ODE cycles
Frustrated TIRF — Evanescent Tunneling
Evanescent wave decaying exponentially into an air gap tunnels to a second prism — optical analogue of quantum tunneling
Perron-Frobenius Spectral Radius
Power iteration on nonneg. matrices converges to the dominant Perron eigenvector; spectral radius visualized live.
Elastic Turbulence (Oldroyd-B)
Chaotic advection in viscoelastic fluid at low Re driven by polymer stretching; Weissenberg number control.
2D XY Model — BKT Vortices
Vortex-antivortex pairs in the 2D XY model; watch them bind below T_BKT≈0.89 and proliferate above.
Gradient Percolation Front
Linear probability gradient creates an invasion front at p_c; front width scales as g^{-4/7} (ν=4/3).
Kauffman NK Boolean Network
N boolean nodes with K random inputs; K=2 is the critical "edge of chaos" with ~√N attractor states.
Acoustic Levitation Trap
Particles trapped at pressure nodes of ultrasonic standing wave; Gorkov radiation force balances gravity.
Wright-Fisher Genetic Drift
Allele frequencies drift via binomial sampling; fixation probability = p₀ for neutral alleles in pop. size N.
Quantum Scarring — Stadium Billiard
Classical chaotic rays in a stadium billiard; scars emerge as trajectories concentrate along periodic orbits.
Casimir Effect — Zero-Point Force
Vacuum fluctuation modes suppressed between conducting plates; attractive force F/A = −ℏcπ²/(240d⁴).
Spinodal Dewetting — Thin Film
Lubrication equation drives polymer film rupture via disjoining pressure instability; holes nucleate and grow.
Rule 110 — Universal CA
Wolfram's Rule 110 proved Turing-complete by Cook 2004: gliders on periodic background simulate any computation.
Spectral Function A(k,ω)
Retarded Green's function G^R with self-energy Σ(ω): quasiparticle peaks, lifetime broadening, Fermi liquid vs marginal FL.
Gravitational Lensing — Einstein Ring
Light deflection α̂=4GM/c²b produces Einstein rings; two-image splitting and magnification as source offset varies.
Pseudospectrum — Grcar Matrix
ε-pseudospectra of non-normal matrices (Grcar, Jordan block) extend far beyond eigenvalues, revealing transient instability.
Manna Stochastic Sandpile
Stochastic sandpile SOC: random toppling produces avalanche power-law P(s)∝s^−1.27, distinct from BTW universality class.
MIPS — Active Brownian Particles
Motility-induced phase separation: run-and-tumble particles spontaneously cluster as speed increases, without attraction.
Hele-Shaw Flow — Conformal Streamlines
Viscous flow between plates satisfies ∇²φ=0; conformal mapping gives exact streamlines around cylinders and Joukowski airfoils.
Quantum Discord — Bipartite State
Discord D = I(A:B) − J captures quantum correlations beyond entanglement; visualized over the full state-parameter landscape.
Firefly Synchronization — Mirollo-Strogatz
Pulse-coupled integrate-and-fire oscillators globally synchronize for concave f; Mirollo-Strogatz 1990 proof demonstrated live.
Polymer Adsorption — Flory Exponent
Monte Carlo simulation of a self-avoiding polymer chain adsorbing onto a surface; Flory exponent crossover from 3D (ν≈3/5) to 2D (ν≈3/4).
Anomalous Diffusion CTRW
Power-law waiting times yield MSD∝t^α: subdiffusion (trapping) or superdiffusion (Lévy jumps) depending on α.
Community Detection (Louvain)
Modularity Q = Σ[Aᵢⱼ−kᵢkⱼ/2m]δ(cᵢ,cⱼ)/2m. Louvain greedy algorithm finds network communities.
Rayleigh-Taylor Instability
Heavy fluid over light grows unstable: γ=√(Agk). Buoyancy drives exponential finger growth and turbulent mixing.
XXZ Heisenberg Spin Chain
Exact diagonalization of the XXZ model. Tune Δ through ferromagnetic, XY critical, and Ising AFM phases.
SGD on Loss Landscape
Langevin SGD: gradient descent with noise σ∝1/√batch. Noise helps escape local minima and find flat regions.
Single-Slit Diffraction
Fresnel number F=a²/λL governs the regime. Fraunhofer far-field gives sinc², Fresnel near-field uses Cornu spiral.
SIS Epidemic on Scale-Free Network
Quenched mean-field threshold λ_c=⟨k⟩/⟨k²⟩→0 on scale-free graphs. Hubs sustain endemic states at low spreading rates.
Richardson Pair Dispersion
Particle pairs in turbulence separate as ⟨r²⟩∝εt³. Richardson's t³ law reflects the multi-scale turbulent cascade.
Hatano-Nelson Skin Effect
Asymmetric hopping (t_R≠t_L) localizes all bulk eigenstates at one boundary. Winding number detects the topological transition.
Hat Monotile — Einstein Tiling
The 2023 "Hat" is a single 13-sided polykite that tiles the plane only aperiodically — the first true einstein tile.
LQR Optimal Control & Riccati
Linear Quadratic Regulator solves the Algebraic Riccati Equation for minimum-energy state feedback. Double-integrator trajectory tracking with tunable state/control cost tradeoff.
Fibonacci Word Quasicrystal
Substitution a→ab, b→a generates a quasiperiodic 1D crystal. Diffraction peaks indexed by two integers via the golden ratio τ reveal self-similar Bragg structure.
Buneman Plasma Instability
Electron beam drifting through plasma drives electrostatic Langmuir waves. Maximum growth rate γ∝(nb/2)^{1/3}ωp; nonlinear saturation via particle trapping.
Frustrated Kuramoto Fireflies
Bimodal frequency distribution in the Kuramoto model creates an incoherence window between two synchronized clusters. Frustration parameter α shifts the coupling phase.
Topological Phonons — Kane-Mele
Honeycomb phononic lattice with next-nearest-neighbor coupling opens a topological band gap hosting chiral acoustic edge modes, the phononic analog of the Kane-Mele model.
Grand Canonical Langmuir Adsorption
Grand canonical ensemble: surface sites exchange particles with a reservoir at chemical potential μ. Langmuir isotherm θ=z/(1+z) and BET multilayer coverage explored interactively.
Social Force Evacuation Model
Helbing's social force model simulates pedestrian evacuation through a bottleneck. Higher desired speed paradoxically causes jamming — the faster-is-slower effect.
Lattice Boltzmann D2Q9 Cavity
D2Q9 LBM simulation of a lid-driven cavity. The moving top wall drives a primary vortex and corner eddies; color shows velocity magnitude at varying Reynolds numbers.
Information Foraging — Patch Leaving
Marginal Value Theorem predicts optimal patch-leaving time: depart when local gain rate equals the environmental average. Applied to web browsing and information scent.
Kondo Effect & Poor Man's Scaling
A magnetic impurity causes resistivity to rise at low temperatures as conduction electrons screen it. Poor Man's scaling drives the coupling constant to strong coupling below T_K.
Adiabatic Elimination & Fast-Slow Manifold
Fast variable y is slaved to slow x as ε→0 via center manifold reduction. Trajectories collapse onto the slow manifold g(x,y)=0, reducing ODE dimension.
SIS Pair Approximation & Clustering
Beyond mean-field: pair approximation tracks SI correlations on networks. Clustering raises the epidemic threshold — higher φ means harder spread via triple closure.
Turbulence Inertial Range — Pope Model
Pope's E(k) model: K41 k^(−5/3) inertial range with large-scale and dissipation corrections. Intermittency μ shifts anomalous exponents away from p/3 predictions.
Heat Equation — Green's Function
G(x,t) = (4πκt)^(−d/2) exp(−|x|²/4κt). Click to place delta-function sources and watch Gaussian spreading; multiple sources superpose linearly in 1D and 2D.
Signal Detection Theory — d′ & ROC
SDT separates sensitivity d′ from response bias c. The ROC curve traces hit rate vs. false alarm rate; AUC measures discrimination independent of criterion placement.
Z₂ Lattice Gauge Theory — Wilson Loop
Z₂ gauge fields on a 2D lattice updated by Metropolis Monte Carlo. Wilson loop ⟨W(C)⟩ obeys area law (confined) below β_c ≈ 0.44 and perimeter law above it.
Quasispecies — Replicator-Mutator
Eigen's quasispecies equation: replication with mutation across sequence space. Above the error threshold μ_c = ln(σ)/L, the master sequence is catastrophically lost.
Cosmic Ray Air Shower — Heitler Model
Binary cascade: each particle splits into 2 after one radiation length. Shower maximum N_max ∝ E₀, depth X_max ∝ ln(E₀/E_c) — Heitler's simple but powerful model.
SBM Belief Propagation Detection
Two-community stochastic block model with belief propagation community detection. Below the Kesten-Stigum threshold (SNR<1) detection becomes information-theoretically impossible.
Kagome Antiferromagnet — Flat Band
Geometric frustration on the Kagome lattice produces a perfectly flat magnon band and macroscopic ground-state degeneracy. Monte Carlo Metropolis reveals frustrated spin configurations.
Kramers Double-Well Escape
Noise-induced transitions in a double-well potential: Kramers escape rate r ∝ exp(−ΔV/D) sets the mean transition time as thermal noise D drives particles between wells.
Poincaré Disk — Hyperbolic Geodesics
Poincaré disk model of the hyperbolic plane: geodesics are circular arcs orthogonal to the boundary. Drag control points to measure true hyperbolic distance; tessellations reveal {p,q} symmetry.
Drossel-Schwabl Forest Fire SOC
The Drossel-Schwabl forest fire model tunes itself to criticality: trees grow at rate p, lightning strikes at rate f≪p, and the resulting cluster-size distribution follows a power law n(s)∝s⁻τ.
Kyle Model — Order Book & Spread
Kyle's (1985) market microstructure model: informed traders mix with noise, the market maker sets a linear price impact λ=σ_v/2σ_u, and the bid-ask spread reflects information asymmetry.
Breit-Wigner Resonance Scattering
Resonant quantum scattering: the phase shift δ(E) sweeps through π/2 at resonance energy E_r, the cross section peaks at the unitarity limit, and the S-matrix traces the Argand circle.
Birth-Death Master Equation
Immigration-death process: P(n,t) evolves exactly via the master equation, converging from any initial δ-function to the Poisson(λ/μ) steady state. KL divergence measures approach to equilibrium.
Kobayashi Dendrite Phase-Field
Kobayashi's (1993) phase-field model couples an order parameter to a temperature field with anisotropic surface energy, driving dendritic crystal growth whose branching depends on the symmetry mode.
Colloidal Glass — Cage Effect & MSD
Dense colloidal suspensions approaching φ_g≈0.58 show caged dynamics: the MSD plateaus at intermediate times and the Van Hove self-correlation deviates from Gaussian as particles become trapped.
Quantum Geometry — Fubini-Study & Berry
The quantum geometric tensor Q_μν = g_μν + iΩ_μν/2 encodes both the Fubini-Study metric (quantum distance) and Berry curvature (topological phase) for a two-level system on the Bloch sphere.
Cell Polarization — Wave-Pinning
Wave-pinning model of Rho/Cdc42 cell polarization: autocatalytic activation drives a bistable front that pins due to conservation of total protein, establishing a stable polarity axis.
BCS Quasiparticle — Bogoliubov Spectrum
Bogoliubov transformation diagonalizes the BCS Hamiltonian; quasiparticle dispersion E_k=√(ξ_k²+Δ²) opens a gap Δ, coherence factors u_k,v_k encode electron-hole mixing, and the density of states shows the coherence peak.
Network Robustness — Targeted vs Random Attack
Scale-free BA networks resist random failures but collapse quickly under targeted hub removal; the giant component decay curve starkly separates targeted from random strategies compared to Erdős-Rényi networks.
Spontaneous Symmetry Breaking — Mexican Hat
A complex field rolls in the V=-μ²|φ|²+λ|φ|⁴ Mexican hat potential, breaking U(1) symmetry; the massless Goldstone mode orbits the valley while the massive Higgs mode oscillates radially.
Stochastic Resonance — FitzHugh-Nagumo
Optimal noise enhances subthreshold signal transmission in a FitzHugh-Nagumo neuron; the SNR peaks at an intermediate noise amplitude, visualizing the stochastic resonance sweet spot.
Phononic Crystal — 1D Transfer Matrix Bandgap
Transfer matrix method computes transmission through alternating elastic layers A/B; impedance contrast opens stopbands (phononic bandgaps) whose width scales with the contrast ratio.
Yard-Sale Model — Wealth Condensation
Repeated pairwise trades with proportional wagers drive wealth condensation regardless of equal opportunity; a redistributive tax reverses oligarchy, and the Gini coefficient tracks inequality in real time.
Swift-Hohenberg Pattern Formation
The Swift-Hohenberg PDE ∂ₜu=εu−(1+∂²ₓ)²u−u³ spontaneously selects stripes, hexagons, or labyrinthine textures near the Turing instability threshold depending on ε and noise amplitude.
Cultural Transmission — Conformist Bias & Drift
Wright-Fisher dynamics with conformist or anti-conformist transmission bias: majority-amplifying conformism accelerates fixation while anti-conformism maintains diversity against neutral genetic drift.
Quantum Battery — Jaynes-Cummings Ergotropy
A qubit charged via Jaynes-Cummings coupling to a cavity Fock state undergoes coherent Rabi oscillations; ergotropy (maximum extractable work by unitary operations) oscillates with quantum coherence.
Nonequilibrium Steady State — Driven Ring
An external force drives probability current around a ring lattice; the nonequilibrium steady state violates detailed balance, producing persistent circulation and entropy production quantified via the master equation.
Stokes Paradox & Oseen Correction
Viscous flow past a cylinder — Stokes paradox in 2D and Oseen's regularization via linearized advection, with streamline visualization.
Quantum Defect Rydberg Series
Energy levels En=−Ry/(n−δ)² for alkali atoms — quantum defect theory showing core-penetration shifts in Na, Rb, Cs s/p/d series.
Vortices in the XY Model (BKT)
Monte Carlo simulation of the 2D XY model — vortex/antivortex pairs bind below T_BKT and unbind above, driving the BKT phase transition.
Modulation Instability (NLS)
Benjamin-Feir instability of plane waves in the focusing nonlinear Schrödinger equation — uniform wave trains break into Akhmediev breathers.
Protein Secondary Structure Energy
Ramachandran plot and backbone energy for α-helix, β-sheet, 3₁₀-helix and PPII — explore φ/ψ dihedral space and H-bond energetics.
Hubbell Neutral Theory (BCI)
Zero-sum ecological drift simulation — species abundance distributions, rank-abundance curves, and the fundamental biodiversity number θ=2Jν.
Dirac Monopole & String Quantization
Magnetic monopole field lines, Dirac string topology, and the quantization condition eg=nℏc/2 — why monopoles force electric charge quantization.
Network Motifs: Feed-Forward Loop
Detect and visualize overrepresented FFL subgraphs in directed networks — Z-score vs random ensemble, coherent vs incoherent types (Milo 2002).
Motility-Induced Phase Separation
Active Brownian Particles with steric repulsion only — MIPS clustering emerges when Pe exceeds critical Peclet number, no attraction needed.
Chaos Synchronization (Pecora-Carroll)
Drive-response Lorenz synchronization via x-subsystem transmission — conditional Lyapunov exponents go negative and chaos locks in real time.
Stochastic Block Model
Planted-partition random graph with K communities. Tune p_in and p_out to probe the detectability threshold.
Kitaev Honeycomb Spin Liquid
Z₂ quantum spin liquid on a honeycomb lattice. Click plaquettes to flip vison excitations; explore gapped vs gapless phases.
Basset History Force
Non-Markovian hydrodynamic memory: the Basset force integrates past accelerations. Compare with Stokes-only trajectories.
Adaptive Network Voter Model
Opinions and topology coevolve. Rewiring rate φ drives a fragmentation transition from consensus to polarized clusters.
Lévy Walk Superdiffusion
Power-law step lengths produce MSD ~ t^γ with γ = 3−α. Compare superdiffusion vs normal Brownian motion live.
Wolfram Rule Space
All 256 elementary CA rules visualized. Toggle rule bits and explore Wolfram's four complexity classes.
Worm-Like Chain Entropic Spring
Marko–Siggia interpolation for polymer force–extension. Model DNA and protein mechanics with tunable persistence length.
Chimera State — Kuramoto Ring
Spontaneous symmetry breaking: synchronized and incoherent domains coexist in a ring of nonlocally coupled oscillators.
Photoelectric Effect
Einstein's 1905 discovery: V_stop = (hf − φ)/e. Tune frequency and metal work function; see the linear h/e slope.
Rayleigh–Bénard Convection
Convection rolls onset at Ra_c ≈ 1708. Watch amplitude grow from linear instability as Ra exceeds the critical threshold.
Jarzynski Equality
Nonequilibrium work theorem: ⟨e^(-βW)⟩=e^(-βΔF). Work distributions from many trajectories recover free energy differences regardless of protocol speed.
Resonant Tunneling
Double-barrier quantum tunneling via transfer matrix method. Transmission resonances occur when the well supports quasi-bound states — T→1 at resonance energies.
Financial Contagion
Eisenberg-Noe interbank clearing payments as fixed points of a monotone map. Click banks to shock them and watch default cascades propagate through the network.
Morphogenetic Fields
Wolpert's French Flag model: morphogen gradients provide positional information. Cells read concentration thresholds and differentiate into distinct spatial domains.
Magnon Dispersion
Spin waves in the Heisenberg ferromagnet with quadratic dispersion ω∝k². Watch precessing spins on a 2D lattice and explore the magnon dispersion relation with anisotropy.
Geodesic Deviation
Tidal forces from spacetime curvature: D²ξ/dτ²=-R·u·u·ξ. Radial tidal stretching vs transverse squeezing near a Schwarzschild mass, with Jacobi field evolution.
Stochastic Petri Net
Gillespie SSA for gene expression: DNA→mRNA→Protein with degradation. Exact stochastic trajectories from the Chemical Master Equation via Petri net visualization.
Ostwald Ripening
LSW coarsening theory: large droplets grow at the expense of smaller ones via the Gibbs-Thomson effect. Mean radius grows as ⟨R⟩~t^(1/3) with universal size distribution.
Quantum Discord
Quantum correlations beyond entanglement: D=I(A;B)-J(A|B). Even separable states have nonzero discord. Explore the discord landscape for two-qubit X-states.
Sandpile Identity
Recurrent sandpile configurations form an abelian group. The identity element has a remarkable fractal structure. Add chips and watch SOC avalanches propagate.
Adiabatic Demagnetization
Magnetocaloric cooling: entropy-temperature diagram for a paramagnetic salt. Isothermal magnetization reduces entropy; adiabatic demagnetization cools the sample toward millikelvin temperatures.
Kuramoto–Sakaguchi Frustration
Phase frustration in coupled oscillators: the Sakaguchi offset α shifts coupling into the imaginary plane, enabling partial synchrony, traveling waves, and chimera-like states beyond standard Kuramoto.
Amyloid Nucleation Kinetics
Knowles-Waudby nucleation-elongation model for amyloid fibril formation. Primary nucleation, elongation, and secondary nucleation produce sigmoidal kinetics with a characteristic lag phase.
Quantum Eraser
Delayed-choice which-path experiment: mark paths to destroy interference; erase the information to restore fringes in the coincidence channel. Complementarity V²+F²≤1 visualized in real time.
Snap-Through Buckling
Von Kármán shallow arch under transverse load: bistable energy landscape with saddle-node bifurcation. Watch the arch snap through when load exceeds the critical value P_snap.
Multiplex Contagion Threshold
Two-layer interdependent network with threshold adoption: nodes activate when combined exposure across social and communication layers exceeds φ. Interdependence amplifies cascade size.
Vacuum Rabi Splitting
Jaynes-Cummings cavity QED: transmission spectrum splits into polariton doublet when atom-cavity coupling g exceeds decay rates. Tune from weak to strong coupling and watch coherent Rabi oscillations.
Newton Fractal Basins
Basin boundaries of Newton's root-finding method for complex polynomials form fractal Julia sets. Every boundary point touches all basins simultaneously — infinite self-similar structure at any zoom.
Trophic Cascade ODE
Three-trophic-level Rosenzweig-MacArthur system: apex predator, herbivore, vegetation. Remove the keystone predator to trigger mesopredator release, herbivore explosion, and vegetation collapse.
Spectral Graph Sparsification
Spielman-Srivastava sparsification: sample edges proportional to effective resistance to build a sparse ε-spectral sparsifier that preserves all cuts and the Fiedler value within factor (1±ε).
Fubini-Study Metric & Berry Curvature
Quantum state space geometry on CP¹: the Fubini-Study metric measures distinguishability of pure states on the Bloch sphere, while Berry curvature is the field strength of the geometric connection. Chern number computed via BZ integral.
Directed Percolation Absorbing Transition
1+1D directed percolation on a spacetime lattice: activity spreads forward in time with probability p. At p_c≈0.6447 a continuous phase transition separates the active phase from the absorbing state. Critical exponents β≈0.277, ν_∥≈1.734.
Huxley Cross-Bridge Sliding Filament
Stochastic simulation of myosin cross-bridge cycling on actin using Huxley's 1957 model. ATP hydrolysis drives the power stroke with position-dependent attachment and detachment rates, generating ensemble force and the force-velocity relation.
Hyperbolic Tilings & Triangle Groups
Poincaré disk model tessellations generated by reflections in the sides of a hyperbolic triangle. Triangle groups (p,q,r) with 1/p+1/q+1/r<1 tile the hyperbolic plane — including the famous (2,3,7) group with 84(g-1) symmetries.
Stochastic Resonance & SNR Peak
Noise enhances detection of a subthreshold periodic signal in a bistable double-well potential. Optimal noise level maximizes SNR — the hallmark of stochastic resonance. SNR vs noise curve shows the characteristic Benzi peak.
Deffuant Bounded Confidence Opinions v2
Agents in [0,1] interact only when opinions differ by less than ε, converging by factor μ. Clusters emerge at ≈1/(2ε). Animated trajectories reveal the phase transition from consensus (ε>0.5) to fragmented polarization.
Minimal Surfaces: Plateau's Problem
Soap films spanning wire frames minimize area, satisfying mean curvature H=0. Explore the catenoid, helicoid, Enneper, and Scherk surfaces — all solutions to Plateau's problem, parametrized via the Weierstrass-Enneper representation.
Omori Law Aftershock Sequences
ETAS (Epidemic Type Aftershock Sequence) model simulates mainshock-triggered aftershock cascades. Rate decays as K/(t+c)^p (Omori-Utsu law). Log-log plot reveals the power-law decay. Gutenberg-Richter magnitude distribution and Båth's law included.
Glassy TLS: Two-Level System Tunneling
Amorphous solids harbor atoms tunneling between nearly degenerate configurations. The Anderson-Halperin-Varma TLS model explains linear heat capacity C~T and T² thermal conductivity in glasses. Schottky anomaly and Fenn effect visualized.
Continuous-Time Quantum Walk Search
Farhi-Gutmann continuous-time quantum walk on graphs: H = -γL - |w⟩⟨w|. Amplitude evolves via Schrödinger equation on complete, hypercube, grid, and star graphs. Constructive interference amplifies the marked vertex in O(√N) time.
Ising Frustration: Triangular Antiferromagnet
Geometric frustration on a triangular lattice prevents antiferromagnetic order, yielding massive ground-state degeneracy and highlighted frustrated triangles.
Anomalous Diffusion — CTRW
Power-law waiting times produce subdiffusion with MSD~t^α. Watch walkers with long trapping events and measure the anomalous exponent from log-log MSD plots.
Quantum Otto Cycle — Two-Level System
A two-level quantum system operates as a heat engine via isentropic strokes and thermalization. Tune temperatures and energy gaps to extract work near the Otto efficiency limit.
Firefly Synchronization — Mirollo-Strogatz
Pulse-coupled oscillators synchronize via phase advances. Mirollo & Strogatz proved identical oscillators always synchronize; watch the order parameter rise and fireflies flash in unison.
SSH Model — Topological Winding Number
The Su-Schrieffer-Heeger chain transitions between trivial (W=0) and topological (W=1) phases as hopping amplitudes cross. Protected edge states appear at zero energy in the topological phase.
Polymer Knot — Trefoil & Calugareanu Theorem
Visualize trefoil and cinquefoil knots in 3D. The Calugareanu-White-Fuller theorem Lk = Tw + Wr links topology to geometry; drag to rotate and adjust twist to verify the identity.
Vicsek Model — Collective Motion
Self-propelled particles align with neighbors plus noise. At low noise the flock order parameter rises sharply — a noise-driven phase transition from disorder to collective flocking.
Stochastic Gene Toggle Switch
Two mutually repressing genes form a bistable toggle. Stochastic noise drives rare switching between gene-A-high and gene-B-high states, producing a bimodal expression distribution.
Kelvin-Helmholtz Instability
Opposing shear flows destabilize their interface via vortex sheet rollup. Watch KH billows emerge, grow, and merge as tracer particles reveal the characteristic spiral vortices.
Erdos-Renyi Giant Component
The G(n,p) random graph undergoes a sharp percolation transition at mean degree k=1. Sweep the control parameter to trace the giant component fraction and verify Erdos-Renyi theory.
Wigner Crystal
Electrons crystallize when Coulomb repulsion dominates kinetic energy. Tune the Wigner-Seitz radius r_s to drive the liquid-crystal transition.
Rate-Distortion Clustering
Information Bottleneck soft clustering: compress data while preserving class structure. The β parameter interpolates from uniform assignments to hard k-means.
Chladni Figures
Sand on a vibrating plate settles at nodal lines where displacement is zero. Explore (m,n) plate modes and watch the geometric patterns emerge.
Nonlinear Pendulum Phase Portrait
The separatrix divides libration from rotation in the nonlinear pendulum. Add trajectories, tune damping and driving to see chaos emerge.
Replicator-Mutator Equation
Mutation couples to selection in the quasispecies equation. Watch the error threshold and fitness landscape shape evolutionary dynamics.
Quantum Level Statistics
Integrable systems show Poisson spacing; chaotic systems show Wigner-Dyson level repulsion. The BGS conjecture connects classical chaos to random matrix theory.
Fiber Bundle Parallel Transport
A vector parallel-transported around a closed loop returns rotated by the holonomy angle — the integral of curvature enclosed (Gauss-Bonnet theorem).
Hayflick Limit & Telomere Dynamics
Each cell division shortens telomeres until the Hayflick limit triggers senescence. Tune oxidative stress and telomerase activity to reshape the aging curve.
Smoluchowski Coagulation
The coagulation-fragmentation equation governs cluster growth kinetics. Different kernels (constant, additive, multiplicative) produce gel transitions and power-law distributions.
Kauffman NK Boolean Network
Random boolean networks exhibit ordered, critical, and chaotic phases controlled by K inputs per node. Attractors correspond to cell types in Kauffman's model of gene regulation.
Magnetic Skyrmion
Spin texture with Dzyaloshinskii-Moriya interaction. Topological charge Q=±1 protects the skyrmion from unwinding.
Self-Replicating Automaton
Von Neumann constructor on a 2D cellular automaton — mathematical proof that self-replication is possible.
Epidemic Contact Tracing
SIR epidemic on a network with backward and forward tracing. Combined tracing dramatically outperforms forward-only.
Quantum Decoherence
Wigner function of a cat state under environmental decoherence. Einselection picks the classical pointer basis.
Shear Banding
Wormlike micellar solution in Couette flow. Non-monotonic flow curve drives coexisting high- and low-viscosity bands.
Mutualism Network Nestedness
Pollinator-plant bipartite network showing nested structure. NODF metric and extinction cascade simulation.
RG φ⁴ Wilson Fixed Point
RG flow in coupling space. Wilson-Fisher fixed point emerges at ε=4−d. Explore critical exponents η and ν.
Granular Force Chains
Random packing under uniaxial compression. Force chains carry stress heterogeneously through the contact network.
TDA Mapper Algorithm
Singh-Mémoli-Carlsson Mapper compresses point clouds into topological skeleton graphs revealing loops and branches.
Cosmic Inflation
Inflaton rolling on V(φ). Slow-roll parameters compute primordial scalar and tensor power spectra. Planck constraints.
Spin Ice Monopoles
Pyrochlore square ice under Monte Carlo dynamics. Single spin flips create emergent magnetic monopole pairs connected by Dirac strings — ice-rule violations visible in real time.
Immune Clonal Expansion
B-cell repertoire evolves toward antigen: somatic hypermutation plus affinity-based selection drives Darwinian evolution in germinal centers. Watch mean affinity rise.
Phonon Umklapp Scattering
Phonons in the Brillouin zone scatter via normal (momentum-conserving) and Umklapp (reciprocal-lattice-shifted) processes. Umklapp dominates at high T, creating thermal resistance κ∝1/T.
Mixed-Mode Oscillations
Slow-fast FHN system near folded-node singularity produces MMOs — alternating large spikes and small subthreshold oscillations. Canard explosion at critical parameter.
Bose-Hubbard Mott Transition
Mean-field phase diagram of bosons on a lattice. Tune J/U and μ/U to cross between superfluid and Mott insulator regimes. Mott lobes emerge from competition of hopping and interaction.
Droop Quota Chemostat
Phytoplankton growth via internal nutrient quota Q. The Droop model separates uptake from growth, producing oscillations and quota-dependent dynamics in a continuous culture.
KdV Two-Soliton Collision
Exact two-soliton solution to the Korteweg-de Vries equation. Solitons pass through each other elastically via inverse scattering — only a phase shift remains, a signature of integrability.
Complex Contagion Cascade
Threshold model on a random network: each node adopts when ≥θ fraction of neighbors adopt. Social reinforcement creates discontinuous cascade transitions absent in simple epidemic spreading.
Schwarzschild Radial Geodesics
Infalling observers reach r=2M in finite proper time but infinite coordinate time — the coordinate singularity exposed. Multiple test particles fall from different radii; r(τ) vs r(t) diverges at the horizon.
Spectral Flow & Chern-Simons
Winding the gauge field around U(1) flows the Dirac spectrum by integer units — the APS spectral flow theorem. Chern-Simons level quantization emerges as a topological necessity for gauge invariance.
Quantum Information Scrambling — OTOC
Out-of-time-order correlators detect quantum chaos: F(t) decays as operators scramble across all qubits. Visualizes the butterfly cone and Lyapunov exponent.
Active Gel — Cytoskeletal Motor Patterns
Actin filaments and myosin motors self-organize into asters, vortices, and waves driven by active stress ζQ. Tracks topological defects (+½ and -½) in real time.
Heterogeneous Mean Field — Scale-Free Epidemic
On scale-free networks P(k)~k^-γ the epidemic threshold β_c→0. HMF theory groups nodes by degree k and shows how hubs drive persistent infection.
Causal Inference — do-Calculus & Backdoor
Pearl's do-calculus distinguishes P(Y|X) from P(Y|do(X)). Interactive DAG shows how conditioning on a confounder blocks the backdoor path and recovers true causal effects.
Phonon-Polariton & Reststrahlen Band
Between ω_TO and ω_LO light couples to lattice vibrations forming phonon-polaritons. The Reststrahlen band (Re ε<0) produces total reflection — tune parameters via Lyddane-Sachs-Teller.
Sandpile Avalanche — Self-Organized Criticality
BTW sandpile: grains added one at a time trigger toppling cascades. Avalanche sizes follow P(s)~s^-1.2 — the system self-tunes to criticality with no parameter adjustment.
BCM Rule — Neuronal Plasticity
Bienenstock-Cooper-Munro (1982) rule: sliding threshold θ_M enables LTP when output y>θ_M and LTD below. Explains orientation selectivity in visual cortex.
Hopf & Saddle-Node — Codimension-Two Bifurcation
Two-parameter (μ,ν) bifurcation diagram near a Bogdanov-Takens point where Hopf, saddle-node, and homoclinic curves all meet. Click to explore phase portraits.
QAOA — Quantum Approximate Optimization
Farhi's QAOA applies alternating phase and mixing unitaries to solve Max-Cut. Simulates state vector evolution and scans the (γ,β) energy landscape for depth p=1.
Invasion Percolation — Fractal Dimension
Always invade the minimum-weight boundary site. The resulting cluster is fractal with d_f≈1.89. Measures fractal dimension live via log-log mass-radius scaling.
Entanglement Entropy Area Law
Bipartite EE in 1D quantum chains: area law vs. CFT logarithmic scaling at criticality via correlation matrix exact solution.
Network Synchronization MSF
Pecora-Carroll master stability function: sync stability when σλ₂ exceeds MSF zero crossing. Live Kuramoto dynamics on multiple topologies.
Protein Normal Modes — ANM
Anisotropic network model: Cα harmonic springs, Hessian diagonalization reveals slow functional modes and collective domain motions.
QCD Flux Tube & String Breaking
Quark confinement via linear gluon flux tube V=σr. Watch the string snap at 2m_q via quark-antiquark pair creation.
Adaptive Landscape & Genetic Drift
Sewall Wright's fitness landscape: populations climb peaks via selection, cross valleys through drift. Drift dominates when N_e·s < 1.
Jamming Transition — Soft Spheres
At φ_J≈0.84 (2D), soft spheres jam. Pressure diverges and contacts reach isostatic Z=4. Watch live compression protocol.
Activity-Driven SIS Epidemic
Temporal network model: nodes activate with power-law rates. Epidemic threshold depends on ⟨a²⟩/⟨a⟩, not just mean activity.
Quantum Phase Diagram — Ising Chain
Exact Jordan-Wigner solution of transverse-field Ising chain. QPT at h=J: spectral gap closes, order parameter vanishes, CFT c=1/2.
Convective vs Absolute Instability
Briggs-Bers criterion: disturbances grow but advect away (convective) or explode everywhere (absolute) when γ > U²/4D.
Multipole Radiation Patterns
Far-field patterns for dipole, quadrupole, and phased antenna arrays. Interactive directivity, array factor, and near-field visualization.
Hyperbolic Geometry — Poincaré Disk
Click to draw geodesics and triangles in the Poincaré disk model of hyperbolic space, where angle sums fall below 180°.
Joukowski Airfoil — Conformal Mapping
The Joukowski transform maps a circle to an airfoil; streamlines visualize how circulation generates lift via the Kutta–Joukowski theorem.
Eigenstate Thermalization Hypothesis
Compare matrix elements of observables in chaotic (ETH, smooth) vs integrable (erratic) quantum systems using random matrix theory.
Topological Pump — Rice-Mele Model
Adiabatic cycling of the Rice-Mele Hamiltonian pumps exactly one quantum of charge per cycle — a manifestation of the Chern number.
Rule 184 — Traffic Flow CA
Wolfram's Rule 184 is an exact solution of the TASEP traffic model, showing free-flow and jammed phases separated at density ρ = 0.5.
Oscillating Droplet — Rayleigh Modes
Surface-tension-driven oscillation modes of a liquid sphere with frequency ω² ∝ n(n−1)(n+2), from Rayleigh's 1879 analysis.
Stochastic Resonance
Noise added to a bistable system can enhance weak signal detection — the SNR peaks at an optimal noise level, not at zero noise.
Seasonal Epidemic Oscillations
Seasonally-forced SIR model produces biennial outbreaks, period-doubling, and chaos — explaining irregular measles and flu cycles.
Alfvén Waves — MHD Dispersion
The Friedrichs diagram shows shear Alfvén and fast/slow magnetosonic wave phase speeds vs angle in a magnetized plasma.
Quantum Spin Hall — Helical Edge States
The Z₂ topological insulator hosts helical edge states: spin-↑ right-movers and spin-↓ left-movers protected from backscattering.
Magnetic Reconnection — Petschek X-Point
Animated X-point topology where magnetic field lines break and reform, releasing energy via slow-mode shocks with tunable Petschek reconnection rate.
Neuronal Spike Train — ISI Statistics
Poisson, gamma (refractory), regular, and bursty ISI distributions reveal neural coding strategies through the coefficient of variation.
Fiber Bundle Holonomy — Parallel Transport
A vector parallel-transported around a closed loop on S² returns rotated by the enclosed solid angle — holonomy equal to Berry phase.
Non-Hermitian Skin Effect — NH Topology
The NH-SSH model shows all bulk eigenstates localizing at one boundary; PBC spectrum forms a loop while OBC spectrum collapses to the real axis.
Wetting & Contact Angle — Young-Dupré
Spherical cap droplets deform continuously from complete wetting to superhydrophobic as surface tensions balance at the triple contact line.
Thermophoresis — Soret Effect
Colloidal particles drift along temperature gradients; steady-state concentration profiles emerge from the balance of thermophoresis and Brownian diffusion.
Stochastic Block Model — Community Inference
SBM network generation and label-propagation community detection with Kesten-Stigum detectability threshold and modularity calculation.
Mutualistic Networks — Nestedness
Plant-pollinator bipartite networks with NODF nestedness index; nested structure confers robustness as specialists interact only with generalists.
Hyperbolic Discounting — Preference Reversal
Hyperbolic vs exponential discount functions reveal why preferences reverse over time — animate the "now" moment to watch the reversal point shift.
Quantum Defect — Rydberg Series
Modified Rydberg formula E = -Ry/(n−δ)² for real atoms; quantum defects parameterize core-electron scattering phase shifts for s, p, d series.
Bloch Oscillations & Stark Ladders
AC-driven tight-binding lattice: Wannier-Stark ladder formation and dynamic localization via Bessel function renormalization of hopping.
Lipid Bilayer Self-Assembly
Amphiphiles spontaneously organize into bilayers driven by hydrophobic tail burial and steric head repulsion — live molecular dynamics.
Lorenz Attractor & Butterfly Effect
Two trajectories with infinitesimally different initial conditions diverge exponentially — Lyapunov exponent and sensitive dependence made visible.
Lévy Stable Distributions
Generalized CLT: heavy-tailed variables with α<2 converge to Lévy α-stable attractors, not Gaussians — power-law tails visualized.
Geometric Frustration
Triangular antiferromagnet: every triangle has an unsatisfiable bond, yielding massive ground-state degeneracy and Wannier residual entropy.
Symplectic Integrators
Leapfrog vs Euler integration: symplectic methods preserve phase-space volume and show no secular energy drift over millions of steps.
Hard-Disk Jamming Transition
Random close packing of hard disks: contact network emerges at the jamming point φ_J≈0.64 with isostatic condition Z=4.
Chemical Reaction Networks
Stoichiometric matrix, null-space fluxes, and steady-state concentrations for glycolysis, Brusselator, circadian, and Lotka networks.
Quantum Teleportation
Bell measurement protocol on Bloch sphere: teleport arbitrary qubit states, watch fidelity degrade under depolarizing and phase-flip noise.
Ecological Regime Shifts
Shallow lake eutrophication: fold bifurcation creates catastrophic transitions between clear and turbid states with irreversible hysteresis.
Saffman-Taylor Viscous Fingering
Low-viscosity fluid displaces high-viscosity fluid in a Hele-Shaw cell; fingers grow via Saffman-Taylor instability with surface-tension wavelength selection.
Phonon-Polariton Reststrahlen Band
Ionic crystal dispersion: light couples to optical phonons forming polaritons; explore the reststrahlen gap where ε<0 and reflection is total.
Germinal Center Affinity Maturation
B-cells undergo somatic hypermutation and clonal selection in germinal centers, evolving high-affinity antibodies via Darwinian competition.
Stochastic Lotka-Volterra Noise
Demographic noise in predator-prey systems: Gillespie exact stochastic simulation vs deterministic ODE, showing extinction risk scaling as 1/√N.
Wave Turbulence Four-Wave Resonance
Weakly nonlinear wave resonance kinetics in the Zakharov equation; Kolmogorov-Zakharov spectrum n(k)∝k⁻¹¹/³ emerges from four-wave interactions.
Kagome Quantum Spin Liquid
Corner-sharing triangles frustrate magnetic order; resonating valence bond states and topological order emerge in the kagome antiferromagnet.
XY Model Vortex BKT Transition
Vortex-antivortex pair dynamics in the 2D XY model; below T_BKT=πJ/2 pairs bind, above they unbind and destroy quasi-long-range order.
Jones Polynomial from Braid Word
Compute the Jones polynomial knot invariant from a braid word via the Kauffman bracket and skein relation; visualize braid closure diagrams.
Fisher-KPP Traveling Wave
Reaction-diffusion fronts in the Fisher-KPP equation; minimum wave speed c*=2√(Dr) is selected for steep initial conditions via pulled-front dynamics.
Riemann ζ Zeros & Gram Points
Hardy Z-function visualization on the critical line; locate zeros via sign changes, explore Gram points and Gram's law violations.
Spin Waves & Magnons
Holstein-Primakoff magnon dispersion in the 1D Heisenberg ferromagnet — animated spin precession and ω(k)=4JS(1−cos ka).
Depletion Attraction
Asakura-Oosawa model: osmotic pressure from excluded depletants creates effective attraction between large colloids.
Chemical Garden
Osmotic tube growth in silicate solution — buoyancy-driven hollow tubes as a model for hydrothermal vent scaffolding.
Weak Values & Back-Action
Aharonov-Albert-Vaidman weak measurement protocol — pointer shift by the weak value, which can exceed eigenvalue bounds.
Anisotropic Sandpile
BTW vs Manna/ricepile SOC universality classes — deterministic vs stochastic toppling rules give different avalanche exponents.
Helfrich Membrane Fluctuations
Lipid bilayer thermal undulations from the Helfrich Hamiltonian — fluctuation spectrum ⟨|h_q|²⟩ and persistence length.
Arnold Tongues
Mode-locking regions in the driven circle map — scan (Ω,K) space to see p:q rational winding number tongues.
Milankovitch Cycles
Orbital forcing of ice ages — eccentricity, obliquity, precession components and the 100 kyr problem visualized.
Forest Fire SOC
Drossel-Schwabl forest fire model — self-organized criticality with power-law fire size distribution P(s)∼s⁻¹·².
Active Brownian MIPS
Motility-induced phase separation in self-propelled particles — spontaneous clustering without attractive forces.
Ramachandran Map
Protein backbone φ/ψ dihedral angle landscape — α-helix, β-sheet, and disallowed regions visualized with sampled residue distributions.
Sine-Gordon Solitons
Kink–antikink scattering in the sine-Gordon field theory — topological charges, Lorentz contraction, and breather modes.
Superspreader Epidemics
SIR dynamics on scale-free networks — heterogeneous contact structure drives R₀→∞ and epidemic threshold collapse.
Quantum Chaos Level Statistics
GOE, GUE, GSE, and Poisson nearest-neighbor spacing distributions — Wigner-Dyson level repulsion from random matrix ensembles.
Grain Boundary Energy
Read-Shockley formula for tilt boundary energy vs misorientation — dislocation spacing, high-angle transition, and phase diagram.
Oregonator Spiral Waves
Tyson-Fife reduction of the BZ reaction — spontaneous spiral wave formation in an excitable reaction-diffusion medium.
Binary Alloy Thermodynamics
Regular solution model — Gibbs free energy of mixing, spinodal decomposition, binodal common tangent, and miscibility gap phase diagram.
Age-Structured Epidemic
Leslie matrix demographics coupled to SIR compartments — age-specific susceptibility, population pyramid dynamics, and demographic attack rates.
Simulated Bifurcation Ising Machine
Kerr-nonlinear parametric oscillator network solving Max-Cut — SB algorithm dynamics, spin readout, and comparison with simulated annealing.
Vortex Filament Dynamics
Biot-Savart induced velocity field from point vortices — leapfrogging rings, trefoil knot topology, and passive tracer advection.
Neutrino Oscillation & MSW Effect
Michaelis-Smirnov-Wolfenstein matter effect on neutrino flavor oscillations — mixing angle, resonance, and survival probability vs baseline.
Polymer Gel Swelling
Flory-Rehner theory of hydrogel swelling equilibrium — osmotic and elastic contributions, chi parameter phase diagram, and swelling ratio.
Double Pendulum Lyapunov Chaos
Real-time Lyapunov exponent estimation for the double pendulum — twin trajectory divergence, chaotic vs regular regimes, and chaos quantification.
Absorbing State Transition
Contact process on a 1D lattice near the directed percolation critical point — spacetime diagram, active density, and critical exponents β=0.277.
XXZ Spin Chain Bethe Ansatz
Exact spectrum of the XXZ Heisenberg chain via Bethe ansatz — rapidities, energy-momentum dispersion, gapless XY phase vs gapped Ising phase.
Power-Law Fluid Rheology
Ostwald–de Waele power-law fluid in Poiseuille pipe flow — shear-thinning vs dilatant velocity profiles and apparent viscosity curves.
Ice-Albedo Feedback Bifurcation
Budyko-Sellers energy balance model — ice-albedo hysteresis, Snowball Earth stable state, and bifurcation diagram vs solar forcing.
Stigmergic Termite Nest
Template-based stigmergic construction — termites aggregate pellets via pheromone gradients, self-organizing into pillars and arches.
Quantum Discord
Quantum discord vs entanglement in two-qubit Werner states — classical correlations, mutual information, and discord surviving entanglement death.
Solar Dynamo Butterfly Diagram
Parker dynamo wave simulation — α-Ω mechanism, toroidal/poloidal field evolution, and the equatorward-drifting butterfly diagram of sunspot latitudes.
Acoustic Phonon Thermal Conductivity
Umklapp scattering, Debye model, and κ(T) from phonon Boltzmann transport — boundary vs. phonon-phonon regimes.
E. coli Run-Tumble Chemotaxis
Bacteria navigating a chemoattractant gradient via biased random walk — adaptation, memory, and drift velocity.
Leaky Integrate-and-Fire Neuron
LIF membrane potential dynamics, spike threshold crossing, refractory period, and the F-I firing rate curve.
Plasmon Excitations in Electron Gas
Plasmon dispersion, Drude dielectric function, particle-hole continuum, Landau damping, and energy-loss spectrum.
Berry Curvature & Band Topology
Haldane model Berry curvature in k-space, Chern number computation, and anomalous Hall conductivity σ_xy = Ce²/h.
Turbulent Jet Entrainment
Free jet self-similar spreading, Gaussian velocity profiles, entrainment rate, and tracer particle visualization.
Gravity-Capillary Wave Dispersion
Wave dispersion ω²=(gk+σk³/ρ)tanh(kh), minimum phase speed, group vs. phase velocity, and wave packet evolution.
Topological Entanglement Entropy
Kitaev toric code with e and m anyons, vertex/plaquette operators, ground state degeneracy, and γ = ln 2.
Economic Complexity Fitness Algorithm
Tacchella-Hidalgo fitness-complexity iteration on bipartite country-product trade networks and ECI ranking.
Amyloid Fibril Polymerization Kinetics
Knowles nucleation-elongation-secondary nucleation model for amyloid aggregation — lag phase, seeding, and fibril ends.
Higher-Order Kuramoto Sync
Explosive synchronization via triplet couplings in the Daido model with live order-parameter tracking.
Resonant Tunneling Double Barrier
Transfer-matrix calculation of unity transmission peaks through a quantum double-barrier structure.
Thermodynamic Uncertainty Relation
Live simulation of driven ring particles verifying Barato-Seifert TUR: Var(J)/⟨J⟩² ≥ 2/σ.
Kleiber's Law — Metabolic Scaling
Log-log allometry from shrew to blue whale; adjust the exponent β and compare to Kleiber's 3/4 law.
Conway GoL Glider Collisions
Gosper glider gun, eaters, pulsars, and glider collisions in Conway's Game of Life.
Quantum Kicked Rotor Localization
Dynamical Anderson localization halts quantum diffusion while classical counterpart spreads.
Adaptive SIS Network Rewiring
Susceptible nodes rewire away from infected neighbors, producing bistability and epidemic thresholds.
Higgs Mexican-Hat Potential
3D Higgs potential with rolling ball; tune μ² through the SSB transition and watch the VEV emerge.
Stochastic Gene Expression Bursting
Gillespie burst model for mRNA; negative-binomial steady state with tunable bimodality and feedback.
Fisher-Rao Geodesics
Poincaré half-plane as the Gaussian statistical manifold; geodesics, Fisher-Rao distance, and KL divergence.
Exciton-Polariton Condensate
Strong coupling between cavity photons and quantum-well excitons splits into upper and lower polariton branches; BEC condensation and blueshift above threshold.
Lévy Walk Anomalous Transport
Power-law step lengths create superdiffusion ⟨r²⟩~t^α with α>1; compare to Brownian walkers and measure the scaling exponent live.
Feedforward Loop Motif
The most over-represented 3-node motif in gene regulatory networks; coherent and incoherent FFL types as pulse filter or adaptive pulse generator.
Coherent & Squeezed States
Wigner quasi-probability distribution for coherent and squeezed quantum harmonic oscillator states; animated phase-space rotation and uncertainty ellipses.
Dam Break & Viscous Gravity Current
Ritter's exact inviscid dam-break solution versus Huppert's viscous gravity current spreading as t^(4/5); adjustable fluid properties.
Antiferromagnet Spin Seebeck Effect
Two-sublattice magnon dispersion in antiferromagnets; Zeeman splitting of ω± branches and thermally driven spin current as a function of temperature.
Polymer Excluded Volume & Flory Exponent
Self-avoiding walks versus ideal random walks; Flory's mean-field prediction ν≈3/5 in 3D measured directly from end-to-end scaling R~N^ν.
Cahn-Hilliard Spinodal Coarsening
Phase separation via the Cahn-Hilliard equation; domain coarsening follows the LSW t^(1/3) law with live structure factor tracking.
MWC Allostery Model
Monod-Wyman-Changeux two-state allosteric model; sigmoidal binding curves, cooperativity, and effector modulation of the T↔R conformational equilibrium.
Bell Test CHSH Violation
Entangled photon polarization correlations E(θ)=−cos(2θ); CHSH parameter S versus measurement angles showing quantum violation of |S|≤2.
Boson Peak in Glass
Excess low-frequency vibrational modes beyond the Debye prediction in glassy materials; reduced DOS g(ω)/ω² showing the anomalous soft-mode peak.
Cardiac Spiral Wave Reentry
FitzHugh-Nagumo excitable medium simulating cardiac tissue; spiral wave formation, drift, and breakup into multiple wavelets mimicking ventricular fibrillation.
Chirikov Standard Map
Area-preserving map of a kicked rotor; KAM tori, resonance island chains, and onset of global chaos as stochasticity parameter K crosses the critical value K_c ≈ 0.9716.
Quantum Zeno & Anti-Zeno
Survival probability under repeated measurements; frequent observation freezes decay (Zeno) while optimal measurement rate can accelerate it (Anti-Zeno effect).
River Meander & Oxbow Lakes
Curvature-driven lateral bank migration (Ikeda-Parker-Sawai model); outer-bank erosion and inner-bank deposition evolve sinuous channels until cutoff creates oxbow lakes.
Thomas-Fermi Atom (DFT)
First density functional theory (1927): electron density n(r) of an atom from the Thomas-Fermi ODE, showing the precursor to modern Kohn-Sham DFT.
Chimera State
Spontaneous coexistence of synchronized and desynchronized oscillator populations in a ring of identical Kuramoto oscillators with nonlocal coupling.
TDA Vietoris-Rips Barcode
Persistent homology of point clouds; Vietoris-Rips filtration reveals H₀ connected components and H₁ loops born and dying as scale parameter ε grows.
Earthquake Omori Law
Aftershock sequence statistics: Omori-Utsu power-law decay n(t)=K/(t+c)^p and Gutenberg-Richter magnitude-frequency relation log N = a − bM.
Langton's Self-Replicating Loop
8-state cellular automaton where a loop-shaped structure copies itself via arm extension and severance — the simplest known self-reproducing CA (Langton 1984).
Gierer-Meinhardt Turing Patterns
Activator-inhibitor reaction-diffusion system generating self-organizing spots and stripes via local activation and lateral inhibition — the original Turing mechanism.
QCD String Tension & Confinement
Cornell potential V(r) = −α/r + κr models quark confinement. The linear term arises from the color flux tube; visualizes Fermi surface and energy density of the QCD string.
Active Gel — Actomyosin Cytoskeleton
Active polar gel field theory: actin filaments align collectively driven by myosin motor activity ζ, producing spontaneous flow, aster formation, and active turbulence.
Topological Flocking — Voronoi Neighbors
Vicsek-style flocking where each bird interacts with its topological Voronoi neighbors rather than all within a fixed radius — matching starling murmuration experiments.
Rashba SOC Spin Texture
Rashba spin-orbit coupling splits the Fermi surface into two helical sheets with spins locked perpendicular to k — visualizing Berry phase π and the topological Hall effect.
Kinetic Proofreading — Hopfield
Hopfield's 1974 mechanism: ATP-consuming proofreading steps reduce error rate exponentially beyond thermodynamic equilibrium. Error vs energy tradeoff in tRNA selection.
Nematic LC Topological Defects
Frank elastic energy drives nematic director field dynamics. Half-integer ±1/2 defect pairs nucleate and annihilate — visualizing the Z₂ topology of the order parameter space.
Fock State Cavity QED — Jaynes-Cummings
Atom-photon Rabi oscillations in cavity QED for Fock, coherent, and thermal states. Coherent states show collapse-and-revival — quantum fingerprint of photon discreteness.
Geodesic Deviation & Tidal Forces
Geodesic deviation equation D²ξ/dτ² = −R(u,ξ,u) in curved spacetime. A sphere of test particles stretches radially and squeezes transversely — spaghettification in GR.
Telomere Shortening — Hayflick Limit
Stochastic model of telomere erosion per cell division, senescence onset at critical length L_crit, and telomerase reactivation — modeling the ~50-division Hayflick limit.
Lab #9800 Milestone — Iris Compendium
Grand 5-tab showcase: strange attractors, Game of Life, quantum double-slit, BA networks, and celebration. 9,800 labs!
Scale-Free Network — Fitness Preferential Attachment
Bianconi-Barabási fitness model: nodes attach by η^α·k, producing P(k)~k^(-γ) with "fit get rich" dynamics.
Magnetic Skyrmion — Topology & Winding Number
Topological charge Q = (1/4π)∫m·(∂ₓm×∂ᵧm) dA. Néel vs Bloch types, DM interaction, skyrmion Hall effect.
Squeezed Light — Quantum Noise & Homodyne Detection
Heisenberg: ΔX₁·ΔX₂≥1/4. Squeezed below shot noise in one quadrature. Wigner function visualization.
Protein Folding — HP Lattice Model
Hydrophobic-Polar lattice model (Lau & Dill 1989). Monte Carlo folding toward energy minimum, funnel landscape.
Heterogeneous Mixing — Core Group Epidemics
Two-group SIR with assortative mixing matrix. High-risk core drives R₀; targeted vaccination of core is highly efficient.
Deffuant Opinion Dynamics — Bounded Confidence
Agents with |x_i−x_j|<ε converge by rate μ. Clusters ~1/(2ε), fragmentation vs consensus transition.
Elastic Wave — 2D Phononic Crystal Band Gap
Bragg scattering from periodic inclusions blocks elastic wave propagation. Band structure, filling fraction, contrast effects.
Symplectic Geometry — Hamiltonian Flow & Liouville
Phase space volume preserved: dq∧dp = invariant. Pendulum, Duffing, and Kepler systems with symplectic Euler integration.
Random Matrix Theory — GUE Eigenvalue Density
Wigner semicircle law ρ(λ)=√(4−λ²)/(2π). Level spacing P(s)~s² (GUE) vs s (GOE). Jacobi eigenvalue algorithm.
Hyperbolic PDE — Wave Equation & Characteristics
u_tt = c²u_xx with d'Alembert solution. Characteristic curves x±ct=const, Riemann invariants, CFL stability.
Turing Pattern — 3-Component Cross-Diffusion
Three-morphogen reaction-diffusion with w modulating activator production. Spots, stripes, labyrinthine patterns.
Neuronal Avalanches — Critical Cortex
Branching process at σ=1 criticality (Beggs & Plenz 2003). P(s)~s^(-3/2) power law, subcritical vs supercritical dynamics.
Quantum Capacitance — Graphene Density of States
Dirac cone band structure, D(E)∝|E|, quantum capacitance C_Q=e²·D(E_F). Gate-tunable n/p-type regime.
Colloidal Glass — Cage Rattling & MSD Plateau
Brownian dynamics of dense colloids near glass transition. MSD sub-diffusion plateau, cage trapping, β-relaxation.
Casimir Effect — Zero-Point Fluctuation Force
Quantum vacuum energy between parallel conducting plates. F/A = −ℏcπ²/(240d⁴), mode density, thermal corrections.
Jarzynski Equality — Nonequilibrium Free Energy
⟨e^{−βW}⟩ = e^{−βΔF}: recover free energy from irreversible work. Work distribution, dissipation, convergence demo.
Spatial Predator-Prey Metapopulation
Lotka-Volterra dynamics on a lattice with dispersal. Spatial waves, patch colonization, spiral patterns.
Quantum Walk — 1D Coin Operator Spreading
Hadamard coin quantum walk: ballistic σ∝t vs classical diffusion σ∝√t. Coin angle and initial state effects.
Neural PAC — Theta-Gamma Oscillation Coupling
Phase-amplitude coupling: gamma power modulated by theta phase. Modulation Index, phase distribution, PAC comodulogram.
Auxin Transport — Polar Canalisation in Plants
PIN protein redistribution drives emergent vascular canals. Canalization instability, source-sink dynamics, flux patterns.
Mie Scattering — Nanoparticle Cross-Sections
Exact Mie theory for dielectric sphere: extinction, scattering, absorption efficiencies vs wavelength and size parameter.
Topological Insulator — Surface Dirac Cone
3D TI surface states with spin-momentum locking. Dirac dispersion, helical Fermi surface, mass gap from TRS breaking.
SGD — Loss Landscape Exploration
Stochastic gradient descent on 2D surfaces: Rosenbrock, Rastrigin, Beale, Himmelblau. Learning rate, momentum, noise.
Liquid Crystal Blue Phase — Cubic Disclinations
Double-twist cylinders and disclination line networks. BP-I BCC lattice with O^8 symmetry, director field visualization.
Stochastic Heat Engine Efficiency
Carnot vs. Curzon-Ahlborn efficiency at maximum power; finite-time thermodynamics and the power-efficiency tradeoff.
Hebbian Learning & Oja Rule
Synaptic weight dynamics under Oja's self-normalizing rule; convergence to the first principal component via unsupervised learning.
Helmholtz Resonator Acoustics
Cavity mode frequencies, end correction, Q factor, and SPL frequency response of a Helmholtz resonator.
Quantum Bit-Flip Error Correction
3-qubit bit-flip code: encode logical qubits, inject errors, measure syndrome bits, and apply corrections without disturbing the logical state.
Red Queen Coevolution
Host-parasite arms race with frequency-dependent selection; allele cycling and the evolutionary basis for sexual reproduction.
Richardson Turbulent Diffusion 4/3 Law
Relative dispersion of particle pairs in turbulence; anomalous t³ superdiffusion and the 4/3 power law for eddy diffusivity.
Griffith Crack Energy Balance
Critical crack length, stress intensity factor K_I vs. toughness K_Ic, and the energy balance governing brittle fracture.
FRET in Light-Harvesting Complexes
Förster resonance energy transfer efficiency vs. donor-acceptor distance; quantum biology of photosynthetic antenna pigments.
Coupled Map Lattice Turbulence
Logistic maps coupled diffusively on a 1D lattice; kymograph reveals spatiotemporal chaos, pattern selection, and Lyapunov spectrum.
Invasion Percolation with Trapping
Drainage front morphology driven by capillary pressure thresholds; trapped defending clusters and fractal invasion patterns.
Ising Transfer Matrix
1D Ising model exact solution via transfer matrix method
Random Laser Threshold
Coherent light from disordered gain medium above threshold
Viscous Fingering
Saffman-Taylor instability with anisotropic surface tension
Network Dismantling
Targeted vs random removal and giant component collapse
Magnetic Monopole
Dirac monopole field lines and unobservable string singularity
Rényi Entropy Replica
Quantum Rényi entropy spectrum via replica trick
Cell Polarity Wave-Pinning
Symmetry breaking and leading edge formation in cell migration
Fractal Basin Boundaries
Newton's method Wada basins and riddled boundary structure
Anomalous Diffusion CTRW
Subdiffusion from heavy-tailed waiting times in CTRW
Quantum Dot Exciton
3D quantum confinement and size-tunable energy spectrum
Lyapunov Variational Method
Computing Lyapunov exponents and chaos onset in the logistic map
Nematic Defect Coarsening
±½ topological defect annihilation and coarsening dynamics
Voter Model with Zealots
Opinion dynamics with committed minorities sustaining coexistence
Kondo Resonance
Magnetic impurity Kondo peak in density of states below T_K
Chemostat Monod Growth
Bacterial growth kinetics and washout transition in a chemostat
Phononic Crystal Band Gap
Elastic wave propagation and stop bands in periodic metamaterials
Mode-Coupling Theory
Dynamic divergence and two-step relaxation near MCT transition
Multiplex Epidemic Threshold
Disease spreading with awareness on a two-layer multiplex network
Wigner Crystal
Electron crystallization at low density and large r_s parameter
Directed Percolation
Absorbing state transition and DP universality class in 1D
Brusselator Spirals
Spiral waves and Turing patterns in the Brusselator RD system
Mean-Field Game
Rational agent equilibrium via coupled HJB and Fokker-Planck equations
Quantum Scars Stadium
Eigenstates enhanced along unstable periodic orbits of chaotic billiard
Haptotaxis Cell Migration
Cell movement up substrate adhesion gradients via haptotaxis
Random Bond Ising
Griffiths phase and disorder effects on 2D Ising phase transition
Hyperbolic Tiling
Regular {p,q} tilings of the Poincaré disk via Fuchsian group
Spin Wave FMR
Magnon dispersion and ferromagnetic resonance in spin chains
Gray-Scott Turing Spots
Spot-stripe-wave transitions across the Gray-Scott parameter space
Parametric Resonance Chaos
Period-doubling route to chaos in a periodically driven oscillator
Cavity QED Rabi Splitting
Jaynes-Cummings vacuum Rabi splitting and collapse-revival dynamics
Semiflexible Polymer WLC
Worm-like chain conformations from flexible coil to stiff rod
Quantum Error Threshold
Fault-tolerant threshold and concatenated code error suppression
Amazon Tipping Point
Forest-savanna bistability and hysteresis from moisture feedback
Hodgkin-Huxley Bursting
Neuron bursting dynamics with slow calcium current modulation
Granular Force Chains
Heterogeneous stress transmission via force chains in granular media
Spectral Flow Topology
Eigenvalue crossings counting topological winding number
Topological Flocking
Vicsek model with k-nearest-neighbor vs metric interaction rules
Chemical Potential Fugacity
Fugacity and chemical potential for ideal and van der Waals gases
Omori Aftershock Law
ETAS model of aftershock sequences following the Omori-Utsu law
Blazed Diffraction Grating
Sawtooth grating concentrating diffracted energy at the blaze order
Noise-Induced Transition
Multiplicative noise creating new phases absent in deterministic system
Deffuant Opinion Dynamics
Bounded confidence model and fragmentation vs consensus transition
Thouless Pump
Quantized adiabatic charge transport via topological Chern number
Phyllotaxis Golden Angle
Fibonacci spirals from golden angle divergence in plant growth
Activity-Driven Epidemic
SIS epidemic on activity-driven networks with heterogeneous rates
Quantum Annealing
Transverse-field Ising model adiabatic optimization and energy gap
Point Vortex Dynamics
Kirchhoff Hamiltonian vortex motion and chaos for N≥4 vortices
Spatial Predator-Prey
Lotka-Volterra spiral waves and spatial coexistence via diffusion
Adam-Gibbs Glass Transition
Cooperative rearranging regions and Kauzmann entropy crisis
Thermohaline Circulation
Two-box THC model with bistability and freshwater forcing collapse
SBM Community Inference
Stochastic block model generation and spectral community detection
GOE Random Matrix Chaos
Level spacing statistics as signature of quantum chaos vs integrability
Bicoid Morphogen Gradient
Exponential morphogen profile and positional information in development
Feed-Forward Loop Motif
FFL network motif sign-sensitive delay and pulse generation
Active Gel Contractility
Myosin-driven spontaneous flows and instabilities in active actin gel
Topological Magnon Bands
Berry curvature and Chern number for magnons on honeycomb lattice
Curzon-Ahlborn Efficiency
Endoreversible engine at maximum power and efficiency-power tradeoff
Bootstrap Percolation
Threshold cascade and sharp transition in k-neighbor bootstrap rule
Qubit Decoherence Bloch
T1/T2 relaxation and Lindblad dynamics on the Bloch sphere
River Delta Avulsion
Delta lobe switching and avulsion dynamics in fluvial systems
Action-Angle Integrability
KAM tori and chaos in integrable vs perturbed Hamiltonian systems
Annealed Network Epidemic
SIS spreading on scale-free annealed networks and vanishing threshold
2D Turbulence Inverse Cascade
Energy flowing to large scales in 2D Navier-Stokes turbulence
Superdense Coding
Two classical bits per qubit via shared entanglement and Bell measurement
Huxley Crossbridge Model
Myosin crossbridge kinetics and force-velocity curve in muscle
CISK Cyclone Intensification
Tropical cyclone positive feedback and rapid intensification dynamics
KPZ Interface Growth
Kardar-Parisi-Zhang universality class and interface roughness scaling
Fiedler Spectral Clustering
Graph partitioning via the algebraic connectivity Fiedler vector
Neutrino PMNS Oscillation
Three-flavor neutrino oscillation probabilities via PMNS mixing matrix
Colloidal Spinodal Gel
Arrested phase separation and gel network formation in colloids
Manna SOC Model
Stochastic sandpile self-organized criticality with power-law avalanches
Quantum Geometric Tensor
Quantum metric and Berry curvature from the geometric tensor in bands
Hopf Network Synchronization
Stuart-Landau oscillators approaching synchrony via mean-field coupling
Ice Shelf Calving
Linear elastic fracture mechanics and crevasse propagation in ice shelves
Toggle Switch Noise
Bimodal gene expression from stochastic noise in a bistable toggle switch
Cross-Diffusion Turing
Turing patterns enabled by cross-diffusion terms in reaction-diffusion systems
Chaos Synchronization MSF
Master stability function and coupled Lorenz synchronization threshold
Persistent Homology TDA
Vietoris-Rips filtration and persistence barcodes for topological features
Social Force Crowd Flow
Pedestrian dynamics and lane formation via Helbing social force model
AdS/CFT Holography
BTZ black hole, bulk geodesics, and holographic CFT thermal correlators
Pipe Flow Turbulent Puffs
Subcritical transition via puff splitting and decay in pipe flow
Wright-Fisher Bottleneck
Neutral genetic drift, population bottleneck, and allele fixation
Vortex Line Defects
Topological vortex lines in complex order parameter fields
Maki-Thompson Rumor
Rumor spreading and stifling dynamics in the Maki-Thompson model
Optomechanical Sideband Cooling
Radiation pressure coupling and phonon cooling via resolved sidebands
Critical Branching Process
Extinction dynamics and power-law size distribution at criticality
Z₂ Gauge Theory
Plaquette flux and confinement in lattice Z₂ gauge theory
Purcell Scallop Theorem
Low-Re swimming, reciprocal motion failure, and 3-link Purcell swimmer
Hypergraph Complex Contagion
Group-threshold adoption spreading via higher-order hyperedge interactions
Mandelbrot Box-Counting Dimension
Fractal dimension of Julia/Mandelbrot boundaries via box-counting
Stochastic Resonance SNR
Optimal noise maximizing signal-to-noise in double-well potential
Kitaev Honeycomb Model
Exactly solvable spin liquid with Majorana fermions and flux sectors
Wave Shoaling and Breaking
Ocean wave shoaling, steepening, and McCowan breaking criterion
Granovetter Threshold Cascade
Behavioral adoption cascades from threshold distribution on networks
Metabolic Scaling ¾ Law
West-Brown-Enquist fractal network theory of metabolic scaling
Frustrated Triangular Ising
Geometric frustration and extensive ground-state entropy on triangular lattice
Commuter Epidemic Network
SIR spreading on gravity-model commuter mobility networks
Polariton Condensate
Exciton-polariton BEC with Rabi splitting and LP/UP dispersion
Keller-Segel Chemotaxis
Bacterial aggregation and chemotactic collapse in the Keller-Segel model
Multiplex Cascade Failure
First-order collapse transition from interdependent network failures
Quantum State Tomography
MLE density matrix reconstruction from Pauli basis measurements
2D Contact Process
Directed percolation universality class in 2D contact process
Ant Colony TSP
Pheromone-guided ant colony optimization for traveling salesman
String-Net Topological Order
Levin-Wen string-net model and anyonic excitations in toric code
Auxetic Metamaterial
Negative Poisson's ratio from re-entrant honeycomb microstructure
Fish Schooling Zones
Three-zone Couzin model transitions between swarm, torus, and school
Shannon Channel Capacity
Mutual information, water-filling, and capacity-SNR tradeoffs
Critical Slowing Down EWS
Variance and autocorrelation early warning signals before tipping points
Depletion Colloidal Attraction
Asakura-Oosawa entropy-driven depletion interaction and phase separation
Superspreader Heterogeneity
Overdispersed transmission and superspreader effects in SIR epidemics
Neural Field Bumps and Waves
Wilson-Cowan field theory showing bumps, traveling waves, and oscillations
Run-Reverse-Flap Motility
Active particle effective diffusion from run-reverse-flap motility pattern
Resonant Level Impurity
Breit-Wigner resonance and conductance in the quantum impurity model
Explosive Percolation
Achlioptas product rule and delayed discontinuous percolation transition
Trophic Cascade Keystone
Three-level food web and predator removal triggering trophic cascade
Bursty Gene Expression
Geometric burst kinetics and super-Poissonian noise in gene expression
Topological vs Metric Flocking
Scale-free cohesion of topological k-NN vs metric interaction flocks
Seebeck Thermopower
Mott formula for thermopower and ZT figure of merit in thermoelectrics
Classical Nucleation Theory
Free energy barrier, nucleation rate, and spinodal decomposition crossover
Echo Chamber Homophily
Opinion polarization from homophilic rewiring and echo chamber formation
MIPS Phase Separation
Motility-induced phase separation of self-propelled particles
Quantum Phase Slip
Dual Josephson junction physics and I-V curve in superconducting nanowires
May Diversity-Stability
Random ecosystem stability threshold and the complexity-stability paradox
Skyrmion Topological Hall
Real-space Berry phase and topological Hall effect from magnetic skyrmions
Microfluidic T-Junction
Droplet generation and capillary number control in T-junction channels
Bistable Toggle Switching
Kramers escape and noise-driven switching in a genetic toggle circuit
Transfer Entropy Network
Directed information flow and network inference via transfer entropy
Surface Phonon Polariton
Coupled phonon-photon surface modes and Reststrahlen band dispersion
Germinal Center Affinity
B cell somatic hypermutation and affinity maturation in germinal centers
Rayleigh-Taylor Instability
Heavy-over-light fluid interface instability and mushroom cap formation
SSH Zak Phase Topology
Topological phase transition and edge states in the SSH chain
Information Cascade Herding
Bayesian herding and cascade formation from sequential social learning
Spin-Torque Nano-Oscillator
DC-driven magnetic auto-oscillations via Slonczewski spin-transfer torque
Active Nematic Turbulence
Self-propelled ±½ defects and turbulent flows in active nematics
Network Dismantling Decycling
Optimal giant component removal via minimum decycling node sets
Household Cluster Epidemic
SIR with household structure and within/between transmission rates
Viscous Fingering Crossover
Compact to fractal morphology in Saffman-Taylor fingering
[[4,2,2]] Error Detection
Four-qubit error-detecting code syndrome measurement and detection
Ice-Albedo Climate Tipping
Ice-albedo feedback bistability and hysteresis in Earth's climate
Nucleated Polymerization
Oosawa-Kasai amyloid nucleation kinetics and sigmoid growth
Wisdom of Crowds Diversity
Hong-Page diversity theorem and collective accuracy in crowd estimation
Quantum Walk Grover Search
Coined quantum walk achieving quadratic Grover speedup on graphs
Bose-Hubbard Mott Transition
Superfluid-Mott insulator transition in optical lattice phase diagram
Abelian Sandpile Group
Group structure and fractal identity element of abelian sandpile model
Vicsek Flocking Transition
Order-disorder phase transition and density bands in Vicsek model
DNA Toehold Displacement
Toehold-mediated strand displacement kinetics and DNA computing circuits
Ramanujan Expander Graphs
Optimal spectral expansion and Alon-Boppana bound in Ramanujan graphs
Sprott Chaotic Attractor
Simplest chaotic flow attractor geometry and Lyapunov divergence
T Cell Exhaustion
Effector-to-exhausted T cell dynamics in chronic infection
Exact Nonlinear Pendulum
Exact period via elliptic integrals, separatrix, and tautochrone property
Exact RG Flow
Wilsonian RG flow in coupling space with fixed points and basins
Ramsey Interferometry
Quantum sensing protocol and shot-noise limit in Ramsey spectroscopy
Active Glass MIPS
Motility-induced phase separation and kinetic arrest in active matter
Network Formation Decay
Temporal network dynamics with preferential attachment and link decay
Rosensweig Ferrofluid Peaks
Hexagonal peak formation via normal-field Rosensweig instability
Natural Gradient Geometry
Fisher metric and natural gradient descent on statistical manifolds
Heston Stochastic Volatility
Mean-reverting stochastic volatility and implied vol smile
Weyl Semimetal Fermi Arcs
Weyl node chirality, Berry curvature hedgehog, and Fermi arc surface states
k-Core Epidemic Spreading
Dense k-core driving epidemic spreading and targeted core removal
Lagrangian Coherent Structures
FTLE ridges and material transport barriers in chaotic double-gyre flow
XXZ Chain Spinons
Bethe ansatz spectrum with spinon continuum and magnon branch
Dryland Vegetation Patterns
Klausmeier model spots, labyrinths, and gaps from water-biomass Turing instability
XY Vortex KT Transition
Kosterlitz-Thouless vortex-antivortex unbinding in the 2D XY model
Bursty Temporal Epidemics
Power-law contact timing and its effect on epidemic spreading speed
Surfactant Micelle Assembly
CMC, micelle size distribution, and amphiphile self-assembly phases
1D Ising RG Decimation
Exact real-space RG for 1D Ising model with trivial fixed points
NK Fitness Landscape
Kauffman NK model ruggedness, epistasis, and adaptive walk statistics
Granular Jamming Transition
Force networks and jamming at critical packing fraction φ_J
Optical Soliton Collision
Elastic NLSE soliton collision and phase shift from integrability
Active Inference FEP
Friston free energy minimization via perception and action
Atiyah-Singer Index Theorem
Dirac zero mode and topological index from domain wall mass profile
Oscillator Death
Amplitude death from coupling-induced quenching of oscillations
Reaction Norm Plasticity
Genotype×environment interactions and evolution of phenotypic plasticity
Quantum Chaos Statistics
Level spacing transition from Poisson to GOE at quantum chaos onset
Protein Elastic Network
Normal mode analysis and conformational flexibility via ENM
Phase Ordering Coarsening
Allen-Cahn and Cahn-Hilliard domain growth power laws after quench
Kramers Exit Time
Mean first passage time and Arrhenius escape from potential wells
Skyrmion Lattice Texture
DM-interaction skyrmion lattice and topological charge density
Collective Sensing Threshold
Square-root law for group signal detection and optimal group size
✦ Lab #10,000 — Grand Compendium ✦
10,000 windows into the universe · Mandelbrot · Lorenz · Ising · Quantum · Celebration
Topological Frustration — Kagome Lattice
Ising spins on the kagome lattice cannot satisfy all antiferromagnetic bonds simultaneously, creating a macroscopic ground-state degeneracy.
Quantum Discord — Bipartite Entanglement
Quantum discord measures correlations beyond entanglement; explore how a two-qubit state's discord and concurrence vary with mixing and phase.
Stochastic Block Model — Community Recovery
Spectral community detection in random graphs; the Kesten-Stigum threshold governs when recovery from the planted partition is possible.
Eigenvalue Outlier — Spiked Covariance
The BBP phase transition: a rank-1 spike detaches from the Marchenko-Pastur bulk only when spike strength α exceeds √γ.
Nonlinear Pendulum — Phase Portrait
Exact pendulum dynamics with libration orbits, rotations, and separatrices; click the phase plane to launch trajectories under damping and forcing.
Flocking — Vicsek Alignment & Noise
The Vicsek model exhibits a continuous phase transition from disordered to flocking states as noise decreases below a critical threshold.
Crystal Dislocation — Peierls Stress
An edge dislocation glides through a crystal lattice under applied shear, overcoming the periodic Peierls-Nabarro potential barrier.
Entropy Rate — Hidden Markov Chain
The entropy rate of an HMM observation stream computed via the forward algorithm and compared to empirical block-entropy estimation.
Viscous Gravity Current — Lubrication Theory
A viscous fluid spreads under gravity following the thin-film equation; Huppert's (1982) similarity solution predicts nose position ∝ t^(1/5).
Quantum Teleportation — GHZ Multipartite
Step through the quantum teleportation protocol with Bell pairs and GHZ states, visualizing Bloch spheres, measurement outcomes, and fidelity.
Spectral Zeta Function
ζ_Δ(s) = Σλₙ⁻ˢ encodes geometry through eigenvalues. Explore heat kernel trace, Weyl law, and spectral invariants of the Laplacian.
Gray-Scott Worm Patterns
Autocatalytic reaction-diffusion produces writhing labyrinths, spots, and chaos. Tune feed and kill rates to navigate the rich Gray-Scott phase space.
Stochastic Resonance
Optimal noise boosts detection of sub-threshold signals — adding noise actually increases SNR in nonlinear threshold systems.
Network Dismantling
Targeted vs random attack on scale-free and random networks. Watch the giant component collapse as high-degree hubs are removed.
Hatano-Sasa Fluctuations
Overdamped Langevin dynamics in a driven NESS. Verify the Hatano-Sasa identity ⟨e^{−σ_ex}⟩=1 for excess entropy production.
Aperiodic Diffraction
Thue-Morse and Fibonacci sequences exhibit singular-continuous and pure-point diffraction — a third kind of order between crystal and glass.
Phononic Band Gap (1D)
Transfer matrix method reveals forbidden frequency bands for elastic waves in a periodic bilayer. Impedance contrast controls gap width.
Run-and-Tumble Active Matter
Self-propelled particles accumulate at walls — a purely kinematic effect with no equilibrium analogue. Tune swim speed, tumble rate, and geometry.
Quantum Annealing
Time-dependent Schrödinger simulation of transverse-field Ising annealing. Landau-Zener transitions determine ground-state fidelity vs annealing speed.
Bifurcation Diagrams
Transcritical, pitchfork (super/sub-critical), saddle-node, and cusp bifurcations with live phase portraits and animated trajectories.
Ecological Tipping Points
Fold bifurcation in a shallow lake ecosystem: hysteresis means recovery requires far lower nutrient loading than the original forward tipping threshold.
Fiber Bundle Holonomy
Parallel transport of a tangent vector around a closed loop on a sphere. The holonomy angle equals the enclosed solid angle — curvature as geometric phase.
Topological Entropy — Horseshoe
Smale's horseshoe map iterates: stretch, fold, repeat. Topological entropy h = log 2 exactly one bit per iteration, with fractal invariant Cantor strips.
Erdős–Rényi Giant Component
At edge probability p = 1/n a giant connected component suddenly emerges from isolated trees — a sharp phase transition in random graph structure.
Abelian Sandpile Identity
The sandpile group's remarkable identity element has intricate fractal symmetry. Add chips, trigger avalanches, or compute the exact group identity.
Kitaev Honeycomb Anyons
Kitaev's exactly-solvable spin model on the honeycomb lattice with three gapped A-phases and a gapless B-phase hosting non-Abelian anyonic vortex excitations.
Turbulent Log-Law & Roughness
The universal law of the wall u⁺ = (1/κ)ln y⁺ + B with Nikuradse roughness correction ΔB = (1/κ)ln(1+k⁺_s) shifting the profile downward.
Stochastic Competition & Drift
Two species competing in a finite population: neutral genetic drift drives extinction via the Moran model even when fitness is equal. Fixation probability = initial frequency.
Kelvin–Helmholtz Rollup
Shear-layer instability rolls a vortex sheet into cat's-eye spirals. Growth rate σ = kΔU/2 amplifies sinusoidal perturbations via the Birkhoff–Rott mechanism.
Rubber Elasticity — Affine Network
Neo-Hookean rubber: a crosslinked polymer network deforms affinely with incompressibility λ_xλ_yλ_z = 1, giving stress σ = nkT(λ² − λ⁻¹).
Morphogenetic Field & Positional Information
Wolpert's French Flag model: morphogen gradient encodes positional information, instructing cells to adopt fates via concentration thresholds.
Epidemic Wavefront — Spatial SIR
Spatial SIR model on a 2D grid where infectious individuals spread disease locally, generating travelling waves of infection.
Thomas–Fermi Density Functional Theory
The Thomas-Fermi model approximates electron density around an atom by balancing kinetic energy against nuclear attraction, visualising n(r) for any element.
Predator–Prey Spatial Spiral Waves
May's spatial Lotka-Volterra system with diffusion generates self-sustaining spiral waves when predator and prey diffuse at different rates.
Topological Charge & Defect Winding Number
Place topological defects in a 2D XY model and observe how winding numbers encode topological charge, with total charge conserved.
Quantum Chaos — GOE Spectral Rigidity
Random matrix theory predicts level-spacing statistics: GOE (chaotic) shows Wigner-Dyson repulsion while integrable systems follow Poisson bunching.
Fractal Basin Boundaries
Phase plane coloured by attractor basin — at fractal boundaries, arbitrarily small perturbations redirect trajectories, making long-term prediction impossible.
Magnon Dispersion — Heisenberg Ferromagnet
Spin-wave (magnon) dispersion ω(k) for ferromagnet and antiferromagnet lattices, showing the ω∝k² vs ω∝|k| long-wavelength contrast.
Information Geometry — Fisher Curvature
The Fisher information metric makes the space of Gaussian distributions the Poincaré upper half-plane with constant negative curvature; geodesics are semicircles.
Quantum Eraser — Delayed Choice Mach-Zehnder
Mach-Zehnder interferometer where which-path information destroys fringes and erasing it restores them, demonstrating Englert's complementarity D²+V²≤1.
Frustrated Magnet — Triangular Antiferromagnet
Geometric frustration on the triangular lattice prevents antiferromagnetic order; Metropolis Monte Carlo explores the massively degenerate ground-state manifold.
Neutrino Oscillation — Mixing Probability
PMNS mixing rotates flavor and mass eigenstates; P(νₑ→νμ) = sin²(2θ)·sin²(1.267·Δm²L/E) oscillates with baseline, tunable by mixing angle and mass splitting.
Polymer Collapse — θ Point & Coil–Globule
A polymer chain transitions from extended coil (ν=3/5, good solvent) to compact globule (ν=1/3, poor solvent) at the theta temperature via Langevin dynamics.
Cell Polarization — LEGI Reaction–Diffusion
Local excitation / global inhibition spontaneously breaks cellular symmetry; a slow activator and fast inhibitor undergo Turing instability to form a single dominant pole.
Market Impact — Price Propagator Model
Large orders move prices with concave impact I∝Q^δ and slow power-law decay G(t)∼t^{-β}; the propagator model (Bouchaud et al.) unifies transient and permanent impact.
Rayleigh–Bénard Convection — Critical Modes
Fluid heated from below becomes unstable at Ra_c = 1707.76; above threshold, rolls form at k_c = π/√2 in a 2D stream-function–vorticity simulation.
Quantum Dot — Coulomb Diamond Spectrum
Charging energy E_c = e²/2C and discrete level spacing create diamond-shaped blockade regions in the gate–bias plane of a single-electron transistor.
Geodesic Flow — Negative Curvature Mixing
Geodesic flow on the Poincaré disk (K=−1) is the archetypal Anosov system, with exponential separation of nearby trajectories and Lyapunov exponent λ=√|K|.
Cavity QED — Strong Coupling & Polaritons
Jaynes–Cummings coupling splits atom and cavity into upper/lower polariton branches; the vacuum Rabi splitting 2g is visible in transmission when g exceeds decay rates.
Directed Percolation — Reggeon Field Theory
The paradigmatic absorbing-state phase transition on a 1+1D lattice; at p_c ≈ 0.6447 activity either survives or dies with exponents β≈0.277, described by Reggeon field theory.
Nonlinear Wave Breaking — Shallow Water
Saint-Venant equations simulate wave steepening and shock formation; finite-amplitude waves pile up as faster crests overtake slower troughs until the wavefront breaks.
Schelling Segregation Model
Even mild preferences for similar neighbors produce dramatic macro-level segregation — Schelling's 1971 classic showing how individual tolerance thresholds cascade into collective separation.
Amphiphile Self-Assembly & CMC
Amphiphiles spontaneously form micelles above the critical micelle concentration; the monomer-vs-aggregate equilibrium is captured by a sharp kink in the concentration curve.
Topological Kinks — φ⁴ Field Theory
The φ⁴ model supports stable topological solitons; kink–antikink collisions show resonance windows, bion formation, and annihilation depending on the collision velocity.
Viscoplastic Flow — Bingham Fluid
Bingham fluids (mud, lava, toothpaste) require a yield stress before flowing; the velocity profile reveals a rigid plug core flanked by parabolic shear regions.
Quantum Measurement & Wavefunction Collapse
A Gaussian wavepacket evolves under the Schrödinger equation and collapses to a narrow position eigenstate upon measurement, sampled from the Born rule probability density.
Network Motifs — Feedforward Loop
The C1-FFL is the most common transcription network motif; depending on the gate logic it acts as a sign-sensitive delay filter or pulse generator for gene expression.
Rényi Entropy & Multifractal Spectrum
Multifractal measures have a spectrum of Hölder exponents α; the f(α) singularity spectrum (Legendre transform of τ(q)) reveals the fractal geometry probed by Rényi entropies.
XXZ Spin Chain — Magnetization Plateau
The anisotropic XXZ spin chain develops a magnetization plateau at M=0 when Δ>1; Monte Carlo sweeps reveal the Ising gap and saturation field as the field increases.
Gene Expression — Two-State Promoter Noise
The telegraph model of a randomly switching gene produces bursty mRNA distributions with super-Poisson Fano factors; exact Gillespie SSA trajectories reveal transcriptional noise.
Crowd Dynamics — Social Force Evacuation
Pedestrians evacuate under the Helbing–Molnár social force model; panic amplification drives jamming at narrow exits.
Quantum Scattering — S-Matrix Resonances
Exact square-well transmission and Breit–Wigner resonances; the S-matrix pole structure governs quantum tunneling.
Forest Fire — Percolation & Dual Lattice
Bond percolation controls fire spread near p_c = 0.5; the dual lattice reveals the simultaneous spanning transition.
Active Nematics — Topological Defect Flow
Activity drives spontaneous flow in nematic fluids; ±½ defects self-propel and annihilate in living chaotic dynamics.
Delay Embedding — Takens' Theorem
Reconstruct chaotic attractors from a scalar time series via delay vectors; Takens (1981) proves the topology is preserved.
Photon Antibunching — Hanbury Brown–Twiss
The g²(τ) correlation distinguishes coherent, thermal, and single-photon sources; antibunching at g²(0)=0 is purely quantum.
Ecological Network Stability — May's Criterion
May (1972): random ecosystems destabilize when σ√(SC) > 1; complexity can undermine rather than beget stability.
Toric Code — Topological Error Protection
Kitaev's toric code: errors create anyonic excitations whose syndromes guide minimum-weight decoding and logical protection.
Grand Canonical Ensemble — Chemical Equilibrium
Fermi–Dirac, Bose–Einstein, and Boltzmann distributions emerge from a multilevel system at fixed T and chemical potential μ.
Transverse-Field Ising — Quantum Criticality
The 1D TFIM quantum phase transition at Γ/J=1: energy gap closes, entanglement entropy diverges, order parameter vanishes.
Quantum Walk Decoherence
Discrete quantum walk transitions from ballistic spread to classical diffusion as environmental decoherence ε increases.
Quasispecies Equation
Eigen's quasispecies model shows how mutation-selection balance collapses the master sequence at the error threshold μ_c = ln(σ)/L.
Hertzian Dipole Radiation
Interactive polar radiation patterns for a Hertzian dipole and linear antenna arrays, with near-field animation and directivity calculation.
Kramers Escape Rate
Stochastic particle in a double-well potential escapes at Kramers rate k ∝ exp(−ΔV/D) — compare measured MFPT to theory.
Complex Contagion Threshold
Granovetter's social threshold model: agents adopt when neighbor fraction exceeds φ, producing global cascades on random, small-world, and scale-free networks.
Duffing Resonance
The Duffing oscillator's hardening spring tilts the resonance peak, producing frequency pulling, hysteresis, and period-doubling routes to chaos.
BEC Bogoliubov Spectrum
Bogoliubov excitation spectrum of a Bose-Einstein condensate transitions from linear phonon dispersion to free-particle quadratic at the healing length scale.
Pecora-Carroll Synchronization
Chaos synchronization via conditional Lyapunov exponents: a slave Lorenz or Rössler system locks onto the master when all λ_cond become negative.
Helfrich Membrane Fluctuations
Thermal fluctuation spectrum of a lipid bilayer governed by Helfrich bending energy — visualize how bending rigidity κ and tension σ shape the power spectrum.
Lattice Boltzmann D2Q9
D2Q9 lattice Boltzmann fluid simulation with Poiseuille channel, cylinder obstacle, and backward-facing step geometries. Visualize speed, vorticity, and pressure.
Turbulence Intermittency & Multifractal Cascade
Energy cascades through an inertial range via multiplicative random processes; She-Leveque scaling exponents ζ_p depart from K41's p/3 due to intermittent vortex filaments.
Quantum Walk: Graph Hitting Time
Discrete-time quantum walk on cycles, hypercubes, and complete graphs — phase-encoded amplitudes spread and interfere, achieving O(√N) hitting time vs classical O(N).
Protein Coevolution: Mutual Information
Structurally interacting residue pairs coevolve in multiple sequence alignments; APC-corrected mutual information MI(i,j) reveals contact maps from sequence statistics alone.
Elastic Instability: Snap-Through Buckling
A shallow two-bar arch under central load snaps discontinuously between two equilibria — a subcritical bifurcation with bistable energy landscape and dramatic phase shift.
Voter Model: Interface Coarsening
Opinion interfaces diffuse as random walks in the 2D voter model; density decays as ρ ~ 1/ln(t), slower than curvature-driven coarsening, with noisy and q-voter variants.
KdV Two-Soliton Interaction
Exact Hirota bilinear solution to the Korteweg–de Vries equation: two solitons pass through each other elastically, emerging with unchanged shape but a calculable phase shift.
Brusselator: Turing Patterns v2
The Brusselator reaction-diffusion system transitions between homogeneous Hopf oscillations and stationary Turing patterns as D_v/D_u crosses the instability threshold.
Network Epidemics: Heterogeneous Mean Field
On scale-free networks the epidemic threshold λ_c = ⟨k⟩/⟨k²⟩ vanishes as N→∞; heterogeneous mean-field theory tracks infections per degree class across ER, SF, and regular topologies.
Quantum Cat State Wigner Function
The Wigner quasiprobability distribution of a Schrödinger cat state |α⟩+|−α⟩ displays interference fringes with negative regions — a direct signature of quantum non-classicality.
Granular Gas: Inelastic Cooling & Haff's Law
Inelastic particle collisions dissipate kinetic energy; Haff's law predicts granular temperature decays as T ~ 1/(1+t/t*)², confirmed by this hard-sphere molecular dynamics simulation.
Causal Set Quantum Gravity
Spacetime as a discrete random DAG: points Poisson-sprinkled into 1+1D Minkowski space, connected by causal links. Transitive reduction reveals the poset structure underlying Lorentzian geometry.
Spin-Torque Switching — Stochastic LLG
A single-domain nanomagnet driven by spin-transfer torque follows the stochastic Landau-Lifshitz-Gilbert equation. Watch thermal fluctuations trigger magnetization switching between stable states.
Algorithmic Complexity — Lempel-Ziv
LZ76 complexity counts distinct phrases needed to parse a string. Compare random, repetitive, and natural text to see how C(x)·log(C(x))/n converges to Shannon entropy for long sequences.
Morphogen Gradient — Berg-Pursell Noise
Source-diffusion-degradation morphogen gradient with thermal noise. The Berg-Pursell limit bounds positional accuracy: σ²_x ∝ 1/(D·c·a·T), setting fundamental precision limits for developmental patterning.
Auxetic Metamaterial — Negative Poisson's Ratio
Re-entrant honeycomb structure with ν < 0: stretching it vertically causes lateral expansion. Adjust the re-entrant angle and strain amplitude to see the counterintuitive auxetic effect live.
Random Walk Record Statistics
The expected number of records after N steps equals the harmonic number H_N = 1 + 1/2 + … + 1/N — a universal law independent of step distribution, verified across Gaussian, uniform, and exponential steps.
Hysteresis Loop — Preisach Ferromagnet
The Preisach model decomposes a ferromagnet into bistable relay operators with distributed switching fields. Macroscopic hysteresis loops emerge from the statistical distribution of microscopic thresholds.
Crowd Wisdom & Deliberation
Does social deliberation improve collective accuracy? Agents update estimates via Hegselmann-Krause bounded confidence. Explore how diversity, bias, and influence range determine whether discussion helps or herds.
TDA Mapper Algorithm
Mapper extracts topological shape from point clouds by covering a filter projection with overlapping intervals, clustering preimages, and connecting shared-point clusters into a graph revealing loops and branches.
Quantum Battery — Ergotropy Charging
A qubit quantum battery charged by coherent driving, governed by the Lindblad master equation. Track ergotropy (maximum extractable work), entropy, and Bloch vector as coherence boosts charging power.
Kuramoto Mean-Field Synchronization
N coupled phase oscillators self-organize above a critical coupling K_c, with order parameter r = |⟨e^{iθ}⟩| tracking the emergence of global synchrony.
Lévy Stable Distributions
Heavy-tailed attractors for sums of random variables; the Chambers-Mallows-Stuck algorithm samples any (α, β) stable law with power-law tails ~ |x|^{−α}.
Ecological Stoichiometry & Nutrient Cycling
Droop model dynamics: producer growth limited by internal nutrient quota Q, with Liebig's minimum law governing C:N:P stoichiometry and ecosystem feedback.
Adaptive Network Opinion–Link Coevolution
Agents rewire disagreeing links or imitate neighbors; tuning the rewiring probability p drives a fragmentation transition from integrated community to polarized echo chambers.
Abrikosov Vortex Lattice
Quantized magnetic flux vortices in type-II superconductors self-organize into a triangular Abrikosov lattice between H_c1 and H_c2, visualized via GL superfluid density.
Classical Nucleation Theory
Free energy barrier ΔG(r) = 4πr²γ − (4/3)πr³ρΔμ creates a critical nucleus r*; stochastic droplets nucleate, dissolve, or grow depending on whether they cross the barrier.
Elementary CA — Rule 110
Rule 110 is one of only two elementary 1D cellular automata proven Turing-complete (Cook 2004), producing intricate non-periodic patterns from a single-cell seed.
Information Scrambling & OTOC Butterfly Effect
Out-of-time-order correlators C(t) ~ exp(λ_L t) reveal quantum chaos and the butterfly effect; the MSS bound caps the Lyapunov exponent at λ_L ≤ 2πk_BT/ℏ.
Entanglement Area Law & MPS
Gapped 1D systems obey an area law S ~ const, while critical CFT systems show S ~ (c/3)log(ℓ); the required MPS bond dimension χ = exp(S) reveals simulation complexity.
Geometric Brownian Motion & Black-Scholes Greeks
Monte Carlo GBM paths converge to the Black-Scholes option price; interactive panels display all five Greeks (Δ, Γ, Θ, Vega, ρ) as functions of spot price and volatility.
Stokes Paradox & Oseen Correction
In 2D Stokes flow there is no solution for flow past a cylinder — Stokes paradox. The Oseen correction adds convective inertia to regularize the far field; visualize streamlines as Re varies.
Quantum Fisher Information & Cramér-Rao Bound
The quantum Fisher information F_Q sets the ultimate precision limit for parameter estimation; entangled probes can surpass the standard quantum limit and approach the Heisenberg limit 1/N².
Competitive Exclusion & Niche Overlap
Gause's competitive exclusion principle states two species on the same niche cannot coexist; Lotka-Volterra dynamics reveal how niche differentiation and carrying capacities determine the winner.
Spin Glass Energy Landscape
Spin glasses exhibit rugged energy landscapes with exponentially many metastable minima organized ultrametrically by Parisi's replica symmetry breaking; explore simulated annealing descent through a random SK model.
Motility-Induced Phase Separation (MIPS)
Active Brownian particles slow in dense regions, triggering a positive feedback that drives gas-liquid phase separation without attractive forces — a hallmark of non-equilibrium active matter.
Fracture Mechanics: Griffith Criterion
A crack propagates when the strain energy release rate G = πσ²a/E exceeds the surface energy 2γ; visualize the Westergaard stress field, energy balance, and the critical crack length.
Hopfield Network Memory Capacity
Hebbian learning stores binary patterns as attractors in a recurrent network; the AGS theory predicts a capacity of ~0.138N before spurious states dominate — test retrieval with noisy probes.
φ⁴ Theory & Renormalization Group Flow
The φ⁴ scalar field theory harbors a Wilson-Fisher fixed point controlling the 3D Ising universality class; explore RG beta-function flow in (m², λ) space and the approach to criticality.
Thermal Convection & Mantle Plumes
When Ra exceeds the critical value of 1708 buoyancy overcomes viscosity and convection cells form; watch hot mantle plumes rise from the thermal boundary layer and drive Nusselt number scaling Nu ~ Ra^(2/7).
Network Synchronization & Master Stability
The Pecora-Carroll Master Stability Function links graph topology to synchronizability: stability requires the coupling σ to keep all transverse modes in the stable MSF window, quantified by eigenratio Q = λₙ/λ₂.
Elasticity Tensor — Voigt Notation
Explore the 6×6 stiffness matrix linking stress and strain in isotropic solids. Adjust Young's modulus and Poisson's ratio to deform a grid in real time.
Self-Organized Criticality — Rice Pile
Drop grains onto a 1D pile and watch avalanches emerge with a power-law size distribution P(s) ∝ s⁻¹·⁵ — the hallmark of SOC at the edge of chaos.
Hong-Ou-Mandel Effect
Two indistinguishable photons at a beam splitter always bunch together — the coincidence rate drops to zero at zero path delay, tracing the famous HOM dip.
Normal Modes & Avoided Crossing
Two coupled oscillators repel each other's frequencies as detuning sweeps through resonance. Watch the hybridized modes emerge and the eigenfrequency gap open.
Habitat Fragmentation — Metapopulation
Levins patch dynamics: local extinctions are rescued by colonization across a landscape. Destroy habitat and watch the metapopulation collapse toward the threshold c = e.
Theta-Gamma Neural Coupling
Hippocampal theta oscillations (4–12 Hz) modulate gamma burst amplitude (30–80 Hz) — cross-frequency coupling that underlies memory encoding and cognitive binding.
Random Matrix — Wigner Semicircle
Generate large random symmetric matrices and watch the empirical eigenvalue density converge to the universal Wigner semicircle law, regardless of entry distribution.
Epidemic Contact Tracing
Network SIR epidemic with contact tracing: quarantine infected nodes and trace their contacts with probability p. Tracing can suppress outbreaks even when R₀ > 1.
Coulomb Blockade — Single-Electron Transistor
A nanoscale island between tunnel junctions shows quantized charging: Coulomb oscillations, stability diamonds, and the gate-tunable blockade of single electrons.
Gravitational Lensing — Einstein Ring & Arcs
Ray-trace light bent by a foreground galaxy. Move the source to transform a perfect Einstein ring into tangential arcs and multiple images, with real SIE deflection.
Anomalous Diffusion — CTRW Trapping
Particles trapped in power-law waiting-time wells exhibit subdiffusion ⟨x²⟩ ~ t^α with α < 1, unlike normal Brownian motion.
Quantum Spin Chain — Entanglement Spectrum
Exact diagonalization of the XXZ Heisenberg chain reveals the entanglement spectrum and quantum phase structure across the anisotropy transition.
Active Inference — Variational Bayes Perception
An agent minimizes variational free energy to infer hidden states from noisy observations, converging faster with higher sensory precision.
Rayleigh–Taylor Instability
Dense fluid resting on lighter fluid develops exponentially growing finger instabilities with growth rate σ = √(Agk tanh kH).
Stochastic Resonance — Neural Threshold
Adding optimal noise to a subthreshold signal enables a neuron to fire coherently — SNR peaks at intermediate noise, the hallmark of stochastic resonance.
Polymer Reptation — Terminal Time
Entangled polymer chains diffuse snake-like through a tube; de Gennes theory predicts reptation time τ ~ N³ and diffusion D ~ N⁻².
SSH Model — Topological Winding Number
The Su-Schrieffer-Heeger chain transitions between trivial (ν=0) and topological (ν=1) phases as hopping amplitudes change, with protected edge states.
Epidemic Quarantine Compliance Game
Evolutionary game theory couples with epidemic dynamics — agents evolve compliance strategies as infection risk and quarantine costs coevolve.
Arnold Tongues — Frequency Entrainment
The circle map exhibits rational mode-locking in Arnold tongue regions; at K=1 the tongues touch and Farey sequences tile the phase diagram.
Quantum Thermodynamics — Carnot Efficiency
A qubit quantum Otto engine operates between hot and cold reservoirs; quantum Carnot efficiency η = 1 − T_c/T_h bounds all reversible cycles.
Phase-Field Dendrite Solidification
The phase-field equation models crystal growth with anisotropic surface energy — watch dendritic arms branch and compete as undercooling drives solidification.
Loschmidt Echo & Quantum Chaos
The Loschmidt echo M(t)=|⟨ψ|e^{iH't}e^{-iHt}|ψ⟩|² decays exponentially in chaotic systems; the rate equals the classical Lyapunov exponent.
Social Learning: Bayesian Cascades
Agents update beliefs via Bayes' rule by observing neighbors' actions — rational information cascades and herding emerge even with perfect rationality.
Potential Flow Superposition
Ideal flow satisfies Laplace's equation — superpose sources, sinks, vortices, and doublets to build complex flows including flow around a cylinder.
2D Ising Model: Onsager Exact Solution
Monte Carlo simulation of the square-lattice Ising model with Onsager's exact solution overlaid — the magnetization vanishes as (T_c−T)^{1/8} at criticality.
Wilson-Cowan Neural Populations
Coupled excitatory and inhibitory neural populations exhibit fixed points, limit cycles, and oscillations governed by mean-field rate equations.
Opinion Dynamics with Zealots
In the voter model, even a tiny fraction of committed zealots can permanently shift consensus — symmetry breaking through inflexible agents.
Landauer-Büttiker Quantum Transport
Conductance is quantized in units of 2e²/h — each open channel contributes one quantum; Fabry-Pérot resonances appear as the barrier is tuned.
Elastic Wave Scattering: Mie Theory
A plane wave scatters off a circular obstacle via partial-wave expansion — Mie resonances produce near-field enhancements and complex diffraction patterns.
SGD Loss Landscape & Saddle Points
Non-convex loss landscapes trap gradient descent at saddle points; SGD noise, momentum, and adaptive methods navigate Rastrigin and Rosenbrock surfaces.
Quantum Many-Body Entanglement Growth
Random unitary circuits drive entanglement entropy from area law to volume law; tune measurement rate to observe the entanglement phase transition.
Pattern Formation: Cross-Diffusion
Cross-diffusion terms in Gray-Scott reaction-diffusion enrich Turing instabilities, producing labyrinthine stripes, hexagonal spots, and anti-patterns.
Stochastic Epidemics: Extinction & Fadeout
Gillespie algorithm SIR simulations show stochastic extinction even when R₀ > 1 — fadeout probability scales as (1/R₀)^I₀ for small initial seeds.
Fluid Turbulence: Energy Spectrum (2D DNS)
Pseudo-spectral 2D turbulence with inverse energy cascade and forward enstrophy cascade — observe Kolmogorov k⁻⁵/³ scaling emerge from the forced simulation.
Topological Insulator: Helical Edge States Z₂
BHZ model band structure reveals topologically protected helical edge states inside the bulk gap — spin-up and spin-down propagate on opposite edges.
Replicator Dynamics: Rock-Paper-Scissors
The replicator equation on the simplex shows cyclic dominance — asymmetric payoffs drive spiraling orbits that can converge, diverge, or neutrally cycle.
Quantum Coherence & Dephasing: Spin-Boson
Bloch vector dynamics under the Lindblad master equation — coherent precession decays via T₁ relaxation and T₂ dephasing from a thermal bosonic bath.
Ecological Food Web Energetics
Generalized Lotka-Volterra dynamics on a random food web illustrates Lindeman's 10% efficiency rule as energy flows up trophic levels to apex predators.
Boolean Model: Coverage & Percolation
Poisson disk process with random radii — coverage fraction follows 1−e^(−λπr²) and a percolating giant component emerges at the continuum threshold λ_c.
Perceptron Learning: Decision Boundary
Interactive perceptron training with real-time decision boundary visualization — place points, watch the weight update rule converge on linearly separable data.
River Delta Bifurcation
Sediment-laden flow bifurcates at channel mouths, building distributary lobes that avulse and switch — a cellular model of the Mississippi and Nile deltas.
Squeezed Light & Homodyne
Squeezed states of light reduce quantum noise below the shot-noise limit in one quadrature. Explore the Wigner function and homodyne detection signal in real time.
Critical Slowing Down
Near a tipping point, systems recover ever more slowly from perturbations. Watch rising autocorrelation and variance serve as early-warning signals before a regime shift.
Girvan-Newman Community Detection
Iteratively removing the highest-betweenness edge reveals modular community structure. Watch inter-cluster bridges cut and communities emerge one edge at a time.
Magnetic Skyrmion Motion
A topologically protected spin texture with winding number Q=−1. Under applied current it drifts perpendicular to the force — the skyrmion Hall effect from the gyrovector.
Polar Codes — Channel Polarization
Arıkan's polar codes split N synthetic channels into perfectly good and perfectly bad via recursive combining, achieving Shannon capacity with SC decoding.
Neural Field Bump Attractor
Mexican-hat connectivity on a ring supports stable bumps of activity encoding spatial working memory. Click to place a stimulus and watch the memory persist.
Nelson Stochastic Mechanics
Nelson's 1966 derivation of quantum mechanics from Brownian motion. Osmotic velocity drives particles toward the quantum ground state density — Schrödinger from diffusion.
Eden Growth & KPZ Roughness
The Eden model grows clusters by uniform perimeter attachment — a KMC process in the KPZ universality class. Surface width scales as W ~ t^{1/3} with roughening exponent β=1/3.
XY Model — BKT Vortex Unbinding
The 2D XY model undergoes the BKT transition via topological vortex-antivortex unbinding. Quench hot to nucleate free pairs, then cool to watch annihilation and order emerge.
QED Feynman Diagrams
Animated Feynman diagrams for Compton, Møller, and Bhabha scattering — visualize perturbative QED with photon loops and vertex corrections.
SBM Detectability Threshold
Stochastic Block Model phase transition: tune SNR through the Mossel-Neeman-Sly Kesten-Stigum threshold separating detectable from undetectable community structure.
Point Vortex Dynamics
Hamiltonian dynamics of N point vortices in 2D ideal flow — add vortices by clicking, watch spiraling orbits, dipole propagation, and chaotic trajectories.
Zero-Noise Extrapolation (QEM)
Quantum error mitigation via ZNE: amplify noise at λ=1,2,3,4 then Richardson-extrapolate to λ→0, recovering the noiseless expectation value.
Spatial Rock-Paper-Scissors
Non-transitive competition on a 2D lattice produces self-sustaining spiral waves — tuning mobility above a critical threshold drives one species to extinction.
Protein Fiber Nucleation
Amyloid fibrillation kinetics: nucleation-dependent polymerization shows sigmoidal aggregation with a lag phase, elongation bursts, and live particle simulation.
Topological Band Inversion
Two-band model H(k)=d(k)·σ: tune mass parameter through gap-closing transitions, watch the winding number change, and identify topological vs trivial phases.
Random Field Ising — Imry-Ma
Quenched random fields compete with ferromagnetic order in 2D — Imry-Ma argument predicts destruction of long-range order for any disorder strength.
Cortical Map — Self-Organizing
Kohonen SOM learns topology-preserving maps of rings, manifolds, and clusters — watch the weight grid unfold and inspect the U-matrix for cluster boundaries.
Ice-Albedo Climate Tipping
Energy balance model with ice-albedo feedback: bistable climate system with hysteresis, fold bifurcations, and catastrophic tipping between snowball and warm states.
AdS/CFT Holography
Poincaré disk models AdS₃ bulk; geodesics compute CFT entanglement entropy via the Ryu–Takayanagi formula S = len(γ)/4G.
Ant Colony Optimization
Ants deposit pheromone proportional to path quality, collectively finding shortest routes through stigmergic reinforcement.
Ising Model — Critical CFT
At T_c, the 2D Ising model is exactly a c = 1/2 CFT. Wolff cluster updates reveal fractal domain boundaries with d_f = 91/48.
First Passage Time
Brownian particles diffuse until hitting an absorbing boundary. Mean first passage time solves ∇²u = −1/D exactly in a disk.
Kleiber's Law — Metabolic Scaling
Basal metabolic rate scales as M^(3/4) across 27 orders of magnitude, derived from fractal vascular network geometry.
Chladni Figures
Sand on a vibrating plate collects at nodal lines of standing wave modes W_mn = sin(mπx)·sin(nπy), revealing hidden geometry.
Quantum Discord
Discord D = I(A:B) − J(A:B) captures quantum correlations beyond entanglement — separable states can still carry nonzero discord.
Superspreaders & SIR Epidemics
Negative-binomial offspring distributions on scale-free networks amplify R₀ via ⟨k²⟩/⟨k⟩, producing explosive heterogeneous outbreaks.
L-Systems — Branching Morphology
Lindenmayer string rewriting generates plant morphology, dragon curves, and Sierpiński triangles with Hausdorff dimension d = log N / log(1/r).
Frustrated Magnetism — Spin Liquid
Antiferromagnets on kagome and pyrochlore lattices cannot order — geometrical frustration creates macroscopic ground-state degeneracy and emergent gauge fields.
Quantum Double-Slit Interference v2
Watch probability amplitudes build an interference pattern particle by particle; toggle which-path measurement to collapse coherence into classical behavior.
Network Cascading Failure
Load-redistribution cascades on a BA power network; attack hubs or random nodes and watch failure avalanches spread as overloaded neighbors topple.
Fluid Mixing & Chaotic Advection
Blinking-vortex protocol drives exponential filamentation of tracer blobs — chaotic advection enhances mixing far beyond laminar diffusion.
Stochastic Gene Toggle Switch
Gardner–Collins bistable gene circuit: two mutually repressing genes create two stable states with noise-driven rare random switching between them.
Thouless Charge Pump
Adiabatic cycling of the Rice-Mele Hamiltonian pumps exactly one quantum of charge per cycle — quantized transport rooted in the band's Chern number.
Deffuant Bounded Confidence v2
Agents interact only within confidence threshold ε; above ~0.5 opinions converge to consensus, below they fragment into isolated opinion clusters.
Fracture Network Permeability
Random fracture networks with log-normal aperture disorder; Darcy flow solved on the graph reveals how connectivity and disorder control bulk permeability.
Quantum Annealing vs Classical SA
Head-to-head comparison on random Ising spin glasses: quantum tunneling (transverse field) vs thermal fluctuations race to find the ground-state energy.
Allee Effect & Extinction
Below the Allee threshold, per-capita growth goes negative and extinction is inevitable; stochastic simulations show noise-driven collapse from seemingly safe populations.
Spectral Fourier–Poisson Solver
Solve ∇²φ = −ρ by dividing the Fourier transform by k² — spectral methods achieve machine-precision accuracy in O(N² log N) with animated charge sources.
Weyl Quantization & Moyal Bracket
Phase-space quantum mechanics: Wigner functions, ★-product deformation, and negativity as a quantum signature.
Quantum Darwinism & Einselection
How classicality emerges from decoherence: pointer states, redundant environment encoding, and the mutual information plateau.
Toda Lattice Soliton Chain
Exactly integrable nonlinear chain with exponential springs — solitons pass through each other via Lax pairs, space-time visualization.
Abelian Sandpile & Group Identity
Self-organized criticality on a grid — the abelian toppling group, fractal identity element, and power-law avalanches.
Vicsek Flocking — Noise Transition
Self-propelled particles align locally — discontinuous phase transition at critical noise from ordered flocking to disorder.
Poincaré Disk Geodesics
Hyperbolic geometry in the unit disk — click to place points, measure non-Euclidean distances, and explore {p,q} tessellations.
Chern-Simons Anyons
Fractional statistics in 2+1D — braiding phase e^{iπθ} for anyons, the braid group, and FQH quasi-particles.
Spectral Zeta & Functional Determinant
The Ray-Singer determinant det(−Δ) = exp(−ζ'(0)) — spectral zeta function, analytic continuation, and Casimir regularization.
Frustrated Kagome Ising Magnet
Geometric frustration on corner-sharing triangles — extensive ground-state degeneracy, residual entropy, and no magnetic order.
Bacterial Chemostat Dynamics
Monod kinetics in continuous culture — steady states, washout transition, nullclines, and competitive exclusion from first principles.
Stochastic Resonance — Kramers Escape
Double-well potential driven by noise and periodic force — SNR peaks at optimal noise, visualizing Kramers escape rate and the stochastic resonance effect.
Entanglement Entropy & Page Curve
Area law vs volume law in gapped and random phases, the Page curve of black hole evaporation, and Schmidt spectrum of bipartite quantum states.
Turbulence Structure Functions
S_p(r) ~ r^ζ_p — Kolmogorov K41 scaling vs She-Leveque intermittency corrections, anomalous exponents, and fat-tailed velocity increment PDFs.
Network Motif Census
Subgraph census of directed networks — feed-forward loops, feedback loops, and triangles with Z-scores against Erdős–Rényi null model.
Elastic Rod — Euler Buckling Elastica
Kirchhoff rod equation solved via RK4 — exact post-buckled elastica shapes, bifurcation diagram P vs deflection, and boundary condition effects.
Diffusion on Fractal Lattices
Anomalous subdiffusion ⟨r²⟩ ~ t^(2/d_w) on Sierpinski gasket and percolation clusters — spectral dimension and Alexander-Orbach conjecture.
Lindblad Master Equation
Driven-dissipative qubit dynamics on the Bloch sphere — Rabi drive, spontaneous emission, dephasing, steady-state purity, and Liouvillian spectrum.
Brusselator — Hopf Bifurcation
Chemical oscillator ẋ=A−(B+1)x+x²y — phase portrait, limit cycles, bifurcation surface B=1+A², and nullcline dynamics.
Watts Cascade Threshold Model
Complex social contagion on random graphs — cascade window in (z,φ) parameter space, vulnerable node fraction, and adoption dynamics.
Geodesic Flow & Anosov Mixing
Geodesics on the Poincaré upper half-plane with K<0 — exponential divergence of nearby trajectories, Lyapunov exponent λ=√|K|, and ergodic mixing.
Josephson Junction RCSJ Plasma Oscillations
RCSJ model of a Josephson junction: IV curve, plasma oscillations, and chaos in the tilted washboard potential.
Kinetic Exchange Wealth Distribution
Dragulescu-Yakovenko model: random pairwise wealth exchange produces Boltzmann-Gibbs exponential equilibrium.
French Flag Morphogenetic Gradient
Wolpert positional information model: morphogen diffusion, threshold-based cell fate, vs Turing reaction-diffusion patterning.
XXZ Spin Chain Bethe Ansatz Magnons
Exactly solvable 1D spin-½ chain: magnon dispersion, Bethe ansatz energy levels, and the ferromagnetic-XY-antiferromagnet phase diagram.
Network Structural Controllability Driver Nodes
Lin's theorem: find the minimum driver node set via maximum bipartite matching, with the Kesten-Stigum detectability phase transition.
Rayleigh-Bénard Convection Nusselt Scaling
Turbulent thermal convection: critical Ra=1708 onset, Nusselt number scaling Nu~Ra^(2/7), and 2D Boussinesq simulation.
Stochastic Block Model Spectral Community Detection
SBM community structure: spectral gap, Kesten-Stigum detectability threshold, and eigenvector-based community recovery.
Majorana Wire Kitaev Chain Edge Modes
Kitaev's 1D topological superconductor: edge Majorana zero modes, bulk gap, and the topological phase diagram at |μ|=2t.
Heston Stochastic Volatility Smile
Heston (1993) model: mean-reverting variance, Monte Carlo paths, implied volatility smile from correlation-driven skew.
Hyperbolic Discounting Present Bias
β-δ quasi-hyperbolic and Ainslie hyperbolic discounting: preference reversal, present bias, and commitment device analysis.
Active Matter MIPS
Self-propelled particles spontaneously separate into dense and dilute phases without attractive forces — motility-induced phase separation.
Ammann–Beenker 8-fold Tiling
Quasiperiodic tiling with 8-fold symmetry generated by substitution rules and the silver ratio inflation factor 1+√2.
Gamma Oscillations PING
Pyramidal-Interneuron Network Gamma oscillations emerge from E-I balance in a spiking LIF neuron network at 30–80 Hz.
Black Hole Shadow
Null geodesics in Schwarzschild spacetime trace the photon sphere at r=3M and the shadow boundary at impact parameter b=3√3 M.
Gray–Scott Reaction-Diffusion
Tunable (f,k) parameter space produces self-replicating spots, labyrinthine stripes, spirals, and worms from two reacting chemical species.
Nim & Sprague–Grundy
Play Nim against an optimal AI and explore Grundy values — the mex theorem that reduces every impartial game to a Nim heap.
Quantum Error Threshold
The fault-tolerant threshold theorem: below p_th ≈ 1% physical error rate, concatenated codes suppress logical errors exponentially.
Trophic Cascade
Remove a keystone predator and watch cascading population dynamics ripple through three trophic levels via Lotka-Volterra equations.
Fracture Mechanics K_I
Stress intensity factor and singular crack tip stress field in LEFM: σ_yy = K_I/√(2πr) with plastic zone visualization.
Compressed Sensing LASSO
Recover sparse signals from far fewer measurements than Nyquist using ℓ₁ minimization (ISTA/LASSO) with RIP phase transition visualization.
Rényi Entropy Spectrum
α-Rényi entropy interpolates between min-entropy and max-entropy, unifying Shannon, collision, and Hartley measures on a single curve.
Kibble-Zurek Defect Coarsening
Topological defects form when a liquid crystal is quenched through its transition; defect density scales as n~τ_Q^(-ν/(1+νz)) with coarsening rate n~t^(-1).
NK Fitness Landscape
Kauffman's NK model tunes landscape ruggedness via epistasis K — smooth single-peak at K=0, fully random at K=N-1, with adaptive walk trajectories.
Quantum Walk Ballistic Spreading
Hadamard coin quantum walk spreads ballistically σ~t versus classical √t diffusion; Anderson localization collapses the wave packet under disorder.
Lattice Boltzmann D2Q9
D2Q9 BGK lattice Boltzmann simulation of viscous flow with Poiseuille parabolic profile, obstacles, streamlines, and real-time vorticity visualization.
MacArthur-Wilson Biogeography
Island species richness equilibrium where immigration meets extinction — species-area law S=cA^z with stochastic turnover dynamics across island archipelagos.
Pecora-Carroll Chaos Sync
Drive-response synchronization of chaotic Lorenz/Rössler systems — conditional Lyapunov exponents turn negative above critical coupling strength ε_c.
Photon Statistics & Wigner Function
Compare thermal bunching, coherent shot noise, and Fock anti-bunching via Mandel Q, g²(τ) Hanbury Brown-Twiss correlations, and phase-space Wigner functions.
Directed Percolation DP
The paradigmatic absorbing-state phase transition with p_c≈0.6447 in 1+1D — space-time clusters, density decay ρ~t^(-δ), and DP universality class critical exponents.
Simplicial Homology & Betti Numbers
Compute Betti numbers β₀,β₁,β₂ via boundary operators ∂ on simplicial complexes — counting connected components, loops, and voids with Euler characteristic χ=V-E+F.
Higgs Mechanism & Mexican Hat
Spontaneous symmetry breaking via the Mexican hat potential — order parameter, vacuum expectation value, Higgs boson mass, and the massless Goldstone mode.
Voter Model Opinion Dynamics
Consensus time and coarsening on 2D lattices — interface density ρ~1/ln(t) slow decay, stubborn-agent noise, and the path to consensus in finite systems.
Quantum Teleportation Protocol
Step-by-step Bell measurement, classical channel, and unitary correction — the full teleportation circuit with state fidelity and interactive parameter control.
Granular Jamming & Force Chains
Disc packing near the jamming transition — visualize force chains, mean contact number Z, and the Maxwell isostatic criterion Z* = 2d = 4 in 2D.
RNA Folding — Nussinov Algorithm
Dynamic programming base pair maximization on RNA sequences — Watson-Crick and wobble pairs displayed as arc diagrams with dot-bracket notation.
Superfluid Vortices & BKT Transition
Topological defects in 2D superfluids — vortex quantization, bound pair unbinding at T_BKT, the universal superfluid stiffness jump, and phase field visualization.
Spatial Evolutionary Games
Rock-paper-scissors spiral waves sustain biodiversity that well-mixed systems cannot — also Prisoner's Dilemma with spatial cooperator clusters resisting invasion.
Stellar Polytropes — Lane-Emden
Hydrostatic stellar structure via the Lane-Emden equation for polytropic index n — density profiles, surface radius ξ₁, and central-to-mean density ratio.
MHD Alfvén Wave Propagation
Magnetohydrodynamic wave modes in magnetized plasma — Alfvén, fast and slow magnetosonic dispersion relations, plasma beta, and animated wave propagation.
Kolmogorov-Chaitin Complexity
Algorithmic incompressibility of binary strings — LZ76 complexity, empirical entropy rate, LZ parse phrase visualization, and the spectrum from structured to random.
Anomalous Diffusion: CTRW
Continuous time random walk with power-law waiting times — Montroll-Weiss theory, Mittag-Leffler relaxation, and sub/super-diffusion with MSD ~ t^α.
Quantum Many-Body Scarring
PXP model on a Rydberg chain — tower of scar states, ETH violation, and perfect Néel-state revivals hidden in a thermal spectrum.
Squeezed Light & OPA
Parametric amplification squeezes one quadrature below vacuum noise — Wigner function, uncertainty ellipse, and squeezing in dB.
Hubbell Neutral Theory
Zero-sum ecological drift with speciation and immigration — Fisher log-series species abundance distribution and Preston plot dynamics.
Topological Insulator Surface States
Helical Dirac cone on a Z₂ topological insulator surface — spin-momentum locking, hexagonal warping, and gap opening by time-reversal breaking.
CRN Deficiency Zero Theorem
Feinberg's deficiency zero theorem guarantees a unique stable steady state for weakly reversible networks — compare δ=0 vs oscillatory Brusselator.
Saint-Venant Torsion
Prandtl stress function for torsion of elastic bars — shear stress contours, warping function, and torsional rigidity for circular, elliptical and rectangular cross-sections.
Bayesian Herding & Information Cascades
Sequential Bayesian agents observe actions and signals — rational herding, cascade onset, and the fragility of social learning to early movers.
Fibonacci Quasicrystal Phonons
Tight-binding on a Fibonacci chain produces a fractal Cantor-set spectrum — singular continuous measure, self-similar gaps, and critical wavefunctions.
Magnetic Reconnection
Sweet-Parker vs Petschek reconnection rates — current sheet geometry, plasmoid instability onset, and the S^{-½} vs 1/ln(S) scaling of reconnection speed.
Catastrophe Theory: Fold & Cusp
Whitney's elementary catastrophes — fold and cusp bifurcations in control parameter space, potential landscapes, and hysteresis.
Quantum Decoherence & Einselection
Zurek's pointer states — environment-induced superselection, purity decay, and the Bloch sphere collapse of quantum coherence.
Epidemic Threshold on Networks
Quenched mean-field theory on scale-free and random networks — spectral radius determines the epidemic threshold λ_c = 1/ρ(A).
Crystal Field Splitting & Jahn-Teller
d-orbital energy levels in octahedral and tetrahedral ligand fields, CFSE, and the Jahn-Teller distortion that breaks orbital degeneracy.
Moran Process & Genetic Drift
Stochastic population genetics — allele frequency trajectories, fixation probability, and the balance between drift and selection.
Turbulent Boundary Layer Log Law
Von Kármán's law of the wall — viscous sublayer, buffer layer, and log region with κ≈0.41 across Reynolds numbers.
Quantum Dot Exciton Quantization
Brus equation for size-dependent emission — effective mass approximation, confinement regimes, and tunable photoluminescence in CdSe and InP dots.
Predator-Prey Turing Patterns
Spatial pattern formation via cross-diffusion — activator-inhibitor instability in Beddington-DeAngelis kinetics producing spots and stripes.
Neutron Star TOV Structure
Tolman-Oppenheimer-Volkoff equation — polytropic equations of state, mass-radius relations, and the maximum mass of neutron stars.
Vickrey Auction & Mechanism Design
DSIC mechanisms and revenue equivalence — truthful bidding in second-price auctions, Myerson's lemma, and virtual value optimization.
Burgers Equation Shock Steepening
Nonlinear wave steepening and shock formation via the viscous Burgers equation — characteristics, entropy condition, and viscous shock structure.
Quantum Fisher Information & Cramér-Rao
Metrological precision limits from quantum Fisher information — SQL vs Heisenberg scaling with squeezed states and the quantum Cramér-Rao bound.
Ecological Stoichiometry & Redfield Ratio
Sterner-Elser growth rate hypothesis — C:N:P ratios, Redfield 106:16:1 attractor, Liebig's law of the minimum, and P-limitation of growth.
node2vec Network Embedding
Biased random walks on graphs — node2vec p/q parameters, skip-gram embedding, and t-SNE visualization of community structure in 2D.
Stochastic Thermodynamics & Jarzynski
Free energy from nonequilibrium work distributions — Jarzynski equality, Jensen's inequality, rare-trajectory dominance, and the second law.
Phononic Crystal Band Structure
1D diatomic mass-spring superlattice — acoustic and optical phonon branches, Brillouin zone folding, stop-band gap opening, and animated modes.
Resonant Tunneling Double Barrier
Quantum tunneling through a double potential barrier — resonant transmission peaks, quasi-bound states, transfer matrix method, and RTD physics.
Cultural Evolution Biased Transmission
Boyd-Richerson model — conformist bias, payoff bias, frequency-dependent selection, cobweb dynamics, and stochastic cultural drift.
Soliton Collision KdV Equation
Exact two-soliton solution of the KdV equation — elastic collision, phase shift, inverse scattering transform, and space-time diagram.
Hadley Cell General Circulation
Atmospheric general circulation — Hadley cell extent, angular momentum budget, Ferrel and polar cells, subtropical jet stream, and trade winds.
Spin Wave Magnon Dispersion
Heisenberg ferromagnet spin-wave dispersion via Holstein-Primakoff transformation — magnon bands, anisotropy gap, and long-wavelength stiffness in 1D and 2D.
Enzyme Kinetics & Hill Cooperativity
Michaelis-Menten and Hill equation for cooperative enzyme kinetics — sigmoidal response, allostery, and Hill plot linearization with interactive parameters.
Quantum Chaos Level Statistics
Energy level spacing distributions for GOE, GUE, and Poisson ensembles — Wigner surmise, level repulsion, and spectral rigidity in chaotic vs integrable systems.
Epidemic Branching Process Extinction
Galton-Watson branching process for epidemic spread — subcritical/supercritical phases, extinction probability, and R₀ threshold with simulated trajectories.
Coupled Map Lattice Pattern Formation
Kaneko coupled map lattice — spatiotemporal chaos, synchronized chaos, and pattern formation with logistic, tent, and sine local maps.
Non-Hermitian Exceptional Points
PT-symmetric dimer with gain-loss balance — exceptional point coalescence, eigenvalue splitting, PT-broken/unbroken phases, and complex plane trajectories.
Sandpile SOC Avalanche Power Law
Bak-Tang-Wiesenfeld sandpile model — self-organized criticality, avalanche size distribution, power-law statistics, and real-time toppling visualization.
Topological Spin Hall Kane-Mele Z₂
Kane-Mele model for quantum spin Hall effect — Z₂ topological invariant, helical edge states, topological phase transition, and phase diagram.
Polymer Reptation Tube Model
de Gennes reptation theory — tube diameter, contour length, reptation time scaling N³, MSD crossover regimes, and animated chain diffusion.
Information Channel Coding Capacity
Shannon noisy channel theorem — BSC, BEC, and AWGN capacity, Shannon limit, achievability condition, and approximate BER waterfall curves.
Wigner-Eckart Selection Rules
Angular momentum coupling via tensor operators — Clebsch-Gordan coefficients, allowed/forbidden transition table, and animated photon emission selection rules.
Vortices in the XY Model
Kosterlitz-Thouless transition — vortex-antivortex pairs, phase field dynamics, Metropolis MC at variable temperature, and real-time vorticity detection.
Quantum Discord
Quantum correlations beyond entanglement in Werner states — mutual information, classical correlations, discord D(A|B), and the discord-vs-concurrence scan.
Epidemic Wavefront Spread
Spatial SIR reaction-diffusion — Fisher-KPP traveling wave, adjustable β/γ/D, click-to-seed outbreaks, and real-time S/I/R field visualization.
Motility-Induced Phase Separation
MIPS in self-propelled particles — speed-dependent clustering, Péclet number control, dense phase fraction, and run-and-tumble dynamics.
Jamming Transition: Soft Spheres
From flowing liquid to rigid amorphous solid at φ_J — harmonic repulsion, force chain network, stress heatmap, and isostatic contact counting.
Szilard Engine & Landauer's Principle
Maxwell's demon as thermodynamic cycle — measure, insert partition, extract kT·ln2 work, then pay it back at erasure. Information = thermodynamic resource.
Hyperbolic Geometry: Poincaré Disk
{p,q} tessellations, geodesics, and triangle angle defects in constant negative curvature — click to draw geodesics and measure Gauss-Bonnet area.
Rule 110: Universal Cellular Automaton
The simplest Turing-complete 1D CA — glider-like structures, Wolfram class IV complexity, all 256 rules explorable, and click-to-edit initial conditions.
Stochastic Block Model
Community structure in random graphs — Kesten-Stigum detectability threshold, force-directed layout, label propagation detection, and modularity scoring.
Fractal Basin Boundaries (Riddled)
Newton's method for polynomial roots partitions the complex plane into fractal basins — zoom near boundaries to see infinite interleaving of all three attractor regions.
Quantum Critical Point: Transverse Ising
The 1D transverse-field Ising model has an exact QPT at Γ/J=1 — sweep the field to watch order parameter collapse, gap closure, and spin dynamics near criticality.
Network Motif Detection — Triangle Census
Count triangles and clustering coefficients in Erdős–Rényi, scale-free, and small-world networks — see how topology shapes motif abundance beyond random baselines.
Stokes Creeping Flow
Low-Reynolds-number flow around a cylinder — exact fore-aft symmetric streamlines, pressure field, particle tracers, and drag coefficient scaling with Re.
Protein Interaction Network
PPI networks grow via preferential attachment to hubs — compare random vs. targeted knockouts to discover why scale-free topology is robust yet fragile.
Jarzynski Equality: Nonequilibrium Fluctuations
Run ensembles of irreversible Langevin trajectories and verify ⟨e^(−βW)⟩ = e^(−βΔF) — rare violations of Clausius dominate the exponential average.
Abelian Sandpile — Self-Organized Criticality
BTW sandpile drives itself to criticality without parameter tuning — avalanche sizes follow a power law P(s) ~ s^(−1.22) across all scales.
Wave Turbulence: Four-Wave Kinetic Equation
Zakharov's kinetic equation for weakly nonlinear waves develops the Kolmogorov–Zakharov spectrum E(k) ~ k^(−4/3) via resonant four-wave interactions.
Skyrmion Lattice — Topological Charge
Magnetic skyrmions carry integer topological charge Q = ±1 protected by topology — visualize spin textures, topological charge density, and LLG dynamics.
Adaptive Network Voter Model
Opinion dynamics coupled to homophilic rewiring — a phase transition at φ_c separates consensus from fragmentation, modeling political polarization and echo chambers.
Membrane Curvature & Helfrich Energy
Lipid bilayer shapes governed by bending rigidity κ and spontaneous curvature c₀ — explore Helfrich free energy, vesicle budding, tubule pearling, and curvature-color-coded membranes.
Double Pendulum: Chaos & Poincaré Section
Hamiltonian chaos in the double pendulum — KAM tori appear as closed curves at low energy while chaotic orbits fill 2D regions of phase space in the Poincaré section.
Quantum Entanglement Entropy Scaling
Area law vs. volume law entanglement — compare gapped ground states, CFT critical states (S ~ c/3·ln L), thermal states, and random Haar states via von Neumann entropy.
Goodwin Oscillator: Biological Feedback
Gene regulatory circuits with negative feedback delay — mRNA to protein to repressor — exhibit a Hopf bifurcation to limit cycle oscillations as Hill coefficient n increases.
Traffic Flow: Nagel-Schreckenberg CA v2
Cellular automaton model of highway traffic showing phantom jam formation, spacetime diagrams, and the fundamental diagram phase transition between free-flow and congestion.
Random Field Ising: Imry-Ma Argument v2
Quenched disorder destroys long-range order in d≤2 — the Imry-Ma energy balance shows field fluctuations always win over surface cost, with domain structure and magnetization collapse.
Geodesic Flow: Stadium Billiard
Bunimovich stadium billiard — a proven example of ergodic hard chaos where geodesics densely fill phase space; the Birkhoff section maps boundary position to reflection angle.
Spin Wave Magnon Dispersion v2
Collective magnetic moment precession in ordered magnets — FM quadratic vs. AFM linear dispersion, density of states with van Hove singularities, and animated spin chain waves.
Ecological Tipping Points: Fold Bifurcation v2
Catastrophic regime shifts via saddle-node bifurcations — alternative stable states, hysteresis, potential landscape, and early warning signals of rising variance and autocorrelation.
Information Geometry: Fisher-Rao Metric
Statistical manifolds as Riemannian spaces — Gaussian distributions form the Poincare half-plane with hyperbolic geodesics, comparing Fisher-Rao, Hellinger, and KL divergence distances.
Contact Process & Directed Percolation v2
Stochastic epidemic spreading on a 1D lattice with absorbing-state phase transition — space-time diagram reveals directed percolation clusters near the critical infection rate.
Duffing Oscillator: Nonlinear Resonance v2
Driven nonlinear spring with cubic stiffness — phase-space portrait and time series show bistability, period-doubling, and routes to chaos as drive frequency and amplitude vary.
Heisenberg XXZ Spin Chain v2
Classical Monte Carlo simulation of the anisotropic Heisenberg chain — visualizes spin configurations, spin-spin correlation function C(r), and phase transitions from XY to Ising regimes.
Forest Gap Dynamics: Gap Fraction
Canopy treefall and regeneration on a 2D lattice — gap fraction equilibrates near the mean-field prediction d/(d+r), with spatial clustering causing deviations from homogeneous theory.
Amphiphile Self-Assembly: Micelle Formation v2
Hydrophilic-head hydrophobic-tail molecules self-organize into micelles above the CMC — Lennard-Jones tail-tail attraction drives clustering while thermal noise competes at high temperature.
Turbulent Boundary Layer: Log Law v2
Wall-bounded turbulence with viscous sublayer, buffer layer, and log-law region u⁺=(1/κ)ln(y⁺)+B — interactive velocity profile on a log scale with particle streaks visualization.
Social Learning: Bayesian Update v2
Agents on a random network update beliefs by combining private signals with neighbors' posteriors — tune social weight to observe wisdom of crowds vs. information cascade failure modes.
Topological Insulator: Bulk-Boundary Correspondence
SSH model of a 1D topological insulator — Jacobi diagonalization reveals protected zero-energy edge states when t2>t1, with winding number W=1 computed from the d(k) loop in h-space.
Stochastic Gene Expression: Bursting v2
Transcriptional bursting from a two-state gene promoter — Fano factor 1+b quantifies super-Poissonian mRNA noise; stationary distribution follows a negative binomial compared to simulation.
Anomalous Diffusion: CTRW Subdiffusion v2
Continuous-time random walk with power-law waiting times produces subdiffusion ⟨r²⟩~t^α with α<1 — log-log MSD plot fits the anomalous exponent, relevant to transport in crowded cells.
Kibble-Zurek Defect Formation
Topological vortex defects nucleated during a symmetry-breaking phase transition quench
Autocatalytic Bistability
Schlögl model: two stable steady states and bistable switching in an autocatalytic network
Ocean Mixed Layer Entrainment
Wind-driven TKE deepens the surface mixed layer by entraining cold thermocline water
Quantum Chaos: Level Statistics GOE
Nearest-neighbor spacing reveals GOE repulsion (chaos) vs Poisson clustering (integrable)
HITS: Hubs & Authorities
Kleinberg's algorithm finds hub and authority scores on directed hyperlink networks
Polymer Glass Transition VFT
Super-Arrhenius viscosity divergence and fragility in glass-forming polymers via VFT equation
Metapopulation Patch Occupancy
Levins colonization-extinction dynamics across habitat patches with spatial connectivity
Cavity Optomechanics
Radiation pressure couples intracavity photons to a mechanical oscillator: cooling and amplification
Cortical Spreading Depression
Reaction-diffusion wave of neural depolarization propagating across the cortex — migraine aura
Lévy Stable Random Walk
Power-law jump lengths produce superdiffusion and heavy-tailed displacement distributions
Thermophoresis & Soret Effect
Particle drift in temperature gradients — Soret coefficient and cold/hot segregation
Parity Anomaly — Weyl Chiral Zeroth Landau Level
Chiral charge pumping via anomalous zeroth Landau level in Weyl semimetals
Stochastic Epidemic — Gillespie SSA
Exact stochastic SIR simulation with demographic noise and extinction events
Vicsek Flocking — Noise Transition
Noise-driven order-disorder transition in self-propelled particle flocks
Kitaev Honeycomb Spin Liquid (v2)
Z₂ flux vison excitations and Majorana fractionalization on the honeycomb lattice
Neo-Hookean Soft Matter Elasticity
Large-strain nonlinear rubber elasticity with incompressibility — drag to stretch
Neural Field — Amari Bump Attractor (v2)
Localized activity bumps in spatially extended neural populations (working memory)
Ion Channel Gating — Hodgkin-Huxley
Voltage-gated Na⁺/K⁺ channel kinetics and action potential generation
Collisional Diffusion — Fokker-Planck
Probability density evolution under drift, diffusion, and viscous drag (Ornstein-Uhlenbeck)
Habitat Fragmentation & Species-Area
MacArthur-Wilson island biogeography and the S = cA^z species-area power law
Binary Fluid Spinodal Decomposition
Cahn-Hilliard phase separation: unmixing dynamics and domain coarsening
Variational Quantum Eigensolver (VQE)
Parameterized ansatz circuit minimizing ⟨ψ(θ)|H|ψ(θ)⟩ for ground state energy
Avrami Crystallization Kinetics
Nucleation and radial growth with X(t) = 1 − exp(−kt^n) jamming
Trophic Cascade — Keystone Species
Remove apex predators and watch Lotka-Volterra food web collapse
Cosmic String Network — VOS Scaling
Velocity-dependent One-Scale model of topological defect evolution
Brusselator Turing Patterns v2
Extended Brusselator with anisotropic diffusion and perturbation injection
Barkhausen Effect — Magnetic Hysteresis
Discontinuous domain wall avalanches producing Barkhausen noise
Network Synchronization — MSF v2
Kuramoto oscillators on ER/WS/BA topologies with Fiedler eigenvalue analysis
RSA — Car Parking Problem
Rényi's 1958 parking problem: jamming coverage ≈ 74.76% in 1D
Quantum Decoherence — Einselection v2
Zurek's pointer states: density matrix decoherence on the Bloch sphere
Energy Landscape Funnel
Langevin dynamics on a rugged funnel — protein folding and misfolding traps
Stochastic Resonance
Noise improves weak signal detection through a nonlinear threshold system
Quantum Walk on Graph
Continuous-time quantum propagation via Schrodinger evolution on graph Laplacian
Donnan Equilibrium
Ion partitioning and osmotic pressure across a membrane with fixed charges
Earthquake OFC Model
Self-organized criticality and Gutenberg-Richter statistics in a spring-block fault
Microtubule Catastrophe
Stochastic GTP-cap model of microtubule dynamic instability and collapse
Voter Model Zealots
Opinion dynamics with inflexible zealots preventing consensus on a 2D lattice
Josephson Junction
Phase particle in a washboard — RSJ model of superconducting tunnel junction
Fracture Percolation
Random fuse network — progressive bond failure and fractal crack propagation
Alfven Wave Plasma
MHD transverse wave propagation along magnetic field lines in conducting plasma
Quantum Defect & Rydberg Series
Visualize how alkali atom energy levels deviate from hydrogen via the quantum defect δ in the Rydberg formula.
Stochastic Lotka-Volterra Extinction
Gillespie algorithm predator-prey simulation showing how demographic noise drives eventual extinction despite deterministic cycles.
Rule 110 — Turing Complete CA
Spacetime diagram of the Rule 110 elementary cellular automaton, proven Turing-complete, showing gliders and complex class-IV behavior.
Brusselator 2D Turing Patterns
Reaction-diffusion spot and stripe formation in the Brusselator model via diffusion-driven Turing instability.
Rule 30 — Pseudorandom Chaos
Rule 30 generates cryptographically random output from a single cell; Shannon entropy analysis reveals near-maximum column randomness.
Symplectic Reduction & Coadjoint Orbits
Euler's equations on SO(3) trace coadjoint orbits — spheres in angular momentum space — illustrating the geometry of symplectic reduction.
Topological Vortex Defects — Superfluid
Phase field visualization of ±1 winding-number vortices in a 2D superfluid, with the BKT transition between bound and free vortex pairs.
1D Diatomic Phonon Dispersion
Acoustic and optical phonon branches in a diatomic chain, with animated lattice motion and tunable mass ratio controlling the zone-boundary gap.
Erdős-Rényi Network Percolation
Giant component emergence at the critical threshold ⟨k⟩=1 in random graphs, with order parameter and susceptibility sweep.
Lindblad Master Equation
Bloch sphere trajectory of a qubit coupled to an environment, showing decoherence and relaxation under the Lindblad master equation.
Landau Free Energy & Phase Transitions
Interactive F(m) = am² + bm⁴ landscape shows how order parameter bifurcates at T_c, distinguishing first- and second-order transitions.
Quantum Teleportation Protocol
Step-by-step visualization of Bell measurement, classical communication, and unitary correction that teleports an unknown qubit state.
Voter Model — Opinion Coarsening
2D voter model simulation showing domain coarsening, interface decay, and logarithmically slow approach to consensus.
Rayleigh-Bénard Convection Onset
Marginal stability curve, critical Ra_c = 1708, and wavenumber selection for the onset of convective rolls in a heated fluid layer.
Magnetic Skyrmion — Topological Charge
2D spin texture visualization with winding number Q = (1/4π)∫n·(∂ₓn×∂ᵧn)d²r for Bloch, Néel, and antiskyrmion configurations.
Binary Symmetric Channel Capacity
Shannon's C = 1 − H(p) capacity theorem for the BSC: interactive error probability, achievable rate region, and the hard limit on error-free communication.
Geodesic Deviation & Riemann Curvature
Parallel transport on sphere, saddle, and torus shows how curvature focuses or spreads nearby geodesics via the Jacobi equation.
Floquet Theory — Periodic Driving
Quasienergy spectrum of a driven two-level system: avoided crossings at n-photon resonances and coherent population oscillations.
Asymmetric Random Walk — Drift & Diffusion
Biased random walk with Fokker-Planck Gaussian solution: live ensemble, drift velocity v = 2p−1, and diffusion coefficient D = 2pq.
Competitive Exclusion Principle
Two-species Lotka-Volterra competition with interactive nullclines revealing exclusion, stable coexistence, and founder-control regimes.
KdV Soliton Dynamics
Two-soliton collision on the Korteweg–de Vries equation showing elastic scattering and phase shift — integrability made visible.
Epidemic Threshold on Heterogeneous Networks
SIR epidemic on scale-free networks where λ_c = ⟨k⟩/⟨k²⟩ → 0, meaning no herd immunity threshold exists for power-law degree distributions.
Kramers Escape Rate
Thermal escape from a double-well potential: Arrhenius law k ∝ exp(−ΔE/kT) with Kramers prefactor from potential curvatures.
Quantum Chaos Level Statistics
Brody distribution interpolates between Poisson (integrable) and Wigner-Dyson GOE (chaotic) nearest-neighbor spacing distributions.
Bicoid Morphogen Gradient
French flag model of positional information: exponential Bicoid gradient sets cell fates via concentration thresholds in the Drosophila embryo.
SOC Forest Fire Model
Bak-Tang-Wiesenfeld forest fire self-organizes to a critical state with power-law fire size distribution — no parameter tuning required.
Spectral Form Factor K(t)
Universal dip-ramp-plateau structure from random matrix theory connects quantum chaos to information scrambling at the Heisenberg time.
Preisach Hysteresis Model
Major and minor hysteresis loops from a distribution of bistable units, demonstrating return-point memory and topological path-dependence.
Frustrated Triangular Ising Model
Geometric frustration on the triangular antiferromagnet: massively degenerate ground states with residual entropy — a classical spin liquid.
Early Warning Signals for Tipping Points
Rising variance and autocorrelation before a fold bifurcation reveal critical slowing down — universal precursors to catastrophic regime shifts.
MPS Entanglement Area Law
Matrix product states capture the area law for 1D quantum chains — explore how bond dimension χ limits entanglement entropy via Schmidt truncation.
Gray-Scott Reaction-Diffusion
A single model spans the full morphological zoo: spots, stripes, solitons, and maze patterns emerge from varying just two parameters f and k.
Quantum Many-Body Scars
Non-thermalizing eigenstates embedded in a thermal spectrum form a tower with anomalously low entanglement, enabling quantum revivals after a quench.
Abelian Sandpile Avalanches
Toppling avalanches on a 2D grid self-organize to criticality — avalanche sizes follow a power law P(s)~s⁻¹·² and fractal boundary patterns emerge.
Vicsek Flocking Transition
Align with your neighbors plus noise: below a critical noise level the flock spontaneously orders — a non-equilibrium phase transition in active matter.
Ising Transfer Matrix Exact Solution
The 1D Ising model is solved exactly by a 2×2 transfer matrix — free energy, magnetization, and correlation length from eigenvalues, compared with Monte Carlo.
Neutrino Flavor Oscillations
Flavor eigenstates propagate as mass eigenstates: visualize P(νe→νμ) as a function of distance, energy, mixing angle θ₁₂, and MSW matter effects.
Thomas Cyclically Symmetric Attractor
Exact cyclic symmetry in a 3D flow: one parameter b drives the transition from chaos to limit cycles, with draggable 3D view and Poincaré section.
DMRG Entanglement Truncation
White's algorithm truncates the Hilbert space by the reduced density matrix spectrum — watch ground state energy converge exponentially with bond dimension.
Spatiotemporal Chaos: Coupled Logistic Maps
A lattice of coupled logistic maps produces space-time patterns from coherent structures to defect turbulence, with real-time Lyapunov exponent tracking.
φ⁴ Kink Soliton
Topological soliton solution φ=tanh(x/√2) in a double-well potential — watch kink-antikink collisions, annihilation, and energy density profiles.
Trophic Cascade Dynamics
Three-level food web (plants/herbivores/predators) with type-II functional responses — explore top-down control, overgrazing, and bifurcations as predation changes.
Anomalous Diffusion & Hurst Exponent
Fractional Brownian motion with Hurst exponent H generates subdiffusion (H<0.5) or superdiffusion (H>0.5) with MSD∝t^(2H) — visualize persistence and ACF.
Magnetic Reconnection
Tearing mode instability in a current sheet drives X-point formation — compare Sweet-Parker vs Petschek reconnection rates as resistivity varies.
Watts Threshold Cascade Model
Global cascades on random networks depend on the fraction of vulnerable nodes — map the cascade window in (mean degree, threshold) parameter space.
Quantum Discord vs Entanglement
Werner states reveal discord without entanglement — separable states can still carry quantum correlations measurable via geometric discord and concurrence.
Edwards-Wilkinson Surface Growth
Interface roughness W(L,t)∝L^α t^β obeys dynamic scaling — add the KPZ nonlinear term to shift universality class and watch roughening exponents change.
Pairwise Max-Entropy Neural Code
Ising model fitted to spike train correlations captures neural population statistics — heat capacity peaks near criticality, suggesting the brain operates at a phase transition.
Traveling Epidemic Wave
SIR model with spatial diffusion produces Fisher-KPP traveling fronts — wave speed c=2√(βD) depends on R₀ and diffusivity, watch the epidemic front propagate.
Pólya Urn & Reinforcement Learning
Positive-feedback urn dynamics converge to a Dirichlet distribution — any outcome is possible, connecting preferential attachment, Bayesian updating, and power laws.
XXZ Spin Chain Phase Diagram
Explore ferromagnetic, critical XY, and Néel phases as anisotropy Δ varies, with ground state energy and Bethe ansatz results.
Feedforward Loop Network Motif
Coherent vs incoherent feedforward loops in transcription networks: sign-sensitive delay, pulse generation, and noise filtering.
Baker Map — Chaotic Mixing
Watch a fluid blob stretch and fold under the baker map, revealing Lyapunov exponent ln 2 and fractal invariant set formation.
Complex Contagion on Hypergraphs
Higher-order group interactions enable social contagion cascades that fail on pairwise networks, depending on threshold distribution.
Crystal Dislocation & Plastic Flow
Edge dislocation glide under shear stress, Peierls-Nabarro energy barrier, and Taylor hardening from dislocation density.
Berry Curvature & Anomalous Hall Effect
Visualize Berry curvature hot spots in the Brillouin zone and compute the Chern number integral giving the quantized Hall conductivity.
Stochastic Resonance
Subthreshold signal detection enhanced by optimal noise in a bistable system — SNR peaks at the resonant noise amplitude.
Hopfield Network Energy Landscape
Hebbian-learned memories as energy attractors, retrieval dynamics from noisy states, capacity limit α=0.14, and spurious states.
Absolute vs Convective Instability
Briggs criterion via cusp singularity in the complex k-plane distinguishes absolute instability (grows everywhere) from convective (advected away).
Lattice-Boltzmann Poiseuille Flow
D2Q9 LBM simulation of Poiseuille flow between parallel walls, with live velocity profiles, Reynolds number, and obstacle geometry.
Rényi Entropy & Multifractal Spectrum
Generalized dimensions D_q and singularity spectrum f(α) of a binomial cascade multifractal, connected by Legendre transform.
Deffuant Bounded Confidence v2
Continuous opinion dynamics where agents only interact if their opinions differ by less than ε, leading to cluster formation that depends sharply on the confidence bound.
Abrikosov Vortex Lattice
Quantized magnetic flux vortices in a type-II superconductor self-organize into a triangular Abrikosov lattice between the lower and upper critical fields.
Eco-Evolutionary Niche Construction
Organisms that modify their environment create evolutionary feedback loops, potentially driving evolutionary branching and stable polymorphism.
WKB Tunneling Through Shaped Barriers
Semiclassical transmission T=exp(−2∫|p|dx/ℏ) through square, Gaussian, triangular, and double barriers, with resonant tunneling peaks.
Wilson-Cowan Neural Mass Model
Coupled excitatory-inhibitory neural population equations with nullclines, phase portrait, and limit cycle oscillations modeling gamma-band rhythms.
Amyloid Aggregation Kinetics
Nucleation-elongation model with secondary nucleation captures sigmoidal aggregation curves and lag-time scaling with initial monomer concentration.
Anderson Impurity & Kondo Resonance
Local density of states of the Anderson impurity model showing lower/upper Hubbard peaks and the sharp Kondo resonance pinned at the Fermi level.
Quantum Spin Hall Helical Edge States
BHZ model band structure showing the Z₂ topological phase, spin-momentum locked helical edge states, and gap opening under magnetic perturbation.
Heat Kernel & Diffusion
Fundamental solution G(x,t)=exp(−x²/4Dt)/√(4πDt) of the heat equation, superposition of sources, and diffusion on a ring graph via the graph Laplacian.
NESS Entropy Production
Driven diffusion through a tilted periodic potential with a temperature gradient; tracks entropy production rate σ̇ from mechanical force and heat flow.
Suprathreshold Stochastic Resonance
Population of threshold neurons where noise and diversity improve signal-to-noise ratio — SR beyond the classical subthreshold regime.
Topological Kink — φ⁴ Field
Soliton solutions of the φ⁴ field theory with Mexican-hat potential; kink-antikink collisions, Lorentz contraction, and conserved topological charge.
Anomalous Diffusion & Subdiffusion
Fractional-calculus diffusion where ⟨x²⟩ ~ t^α; CTRW trapping (subdiffusion) vs Lévy flights (superdiffusion) with live MSD log-log plot.
Goldstone Bosons & Zero Modes
U(1) Mexican-hat potential; massless Goldstone modes along the vacuum circle, Higgs radial mass, pseudo-Goldstone from explicit breaking, and thermal symmetry restoration.
Anderson Localization — 1D
All eigenstates are exponentially localized in a 1D disordered lattice; transfer-matrix wavefunction, localization length, and IPR vs disorder strength.
Viscous Gravity Current
Thin-film equation lubrication flow; Barenblatt self-similar spreading with x_f ~ t^(1/5), surface tension capillary effects, and live log-log front-tracking.
Ferromagnet Hysteresis Loop
Domain wall dynamics through a disordered pinning landscape; Barkhausen-like jumps, M-H hysteresis loop, coercivity, and energy loss per cycle.
Fourier Series & Gibbs Phenomenon
The ~9% overshoot near a jump discontinuity that never vanishes as N→∞; animate partial sums of square, sawtooth, triangle, and step functions with Lanczos σ-smoothing.
Quantum Discord
Quantum correlations beyond entanglement; discord D = I(A:B) − J separates total, classical, and quantum parts of a two-qubit state with Bloch sphere visualization.
Lagrangian Particle Tracking
Follow fluid parcels through turbulent vortex fields — Richardson diffusion and chaotic stretching visualized.
Spectral Flow & Holonomy
Watch eigenlevels cross zero as a flux loop closes — spectral flow counts the Chern-Simons topological invariant.
Jamming Transition
Soft sphere packing near the jamming point J — pressure and contact number diverge as φ → φ_J ≈ 0.64.
Herd Immunity on Networks
Epidemic spread on heterogeneous scale-free networks — herd immunity threshold shifts dramatically with degree variance.
XXZ Spin Chain Phases
Navigate the phase diagram of the exactly-solvable XXZ model — ferromagnet, Luttinger liquid, and antiferromagnet.
Keller-Segel Chemotaxis
Cells climb chemical gradients — watch density aggregation and finite-time blowup above the critical mass 8π/χ.
Duffing Strange Attractor
Forced double-well oscillator generates chaotic phase portraits, fractal Poincaré sections, and strange attractors.
Fiber Bundle & Gauge Holonomy
Parallel transport of a gauge fiber around a loop on S² — holonomy equals g times the enclosed solid angle.
Protein Aggregation Kinetics
Primary nucleation, fibril growth, and secondary nucleation produce sigmoidal amyloid aggregation curves with a lag phase.
Community Detection
Louvain modularity optimization finds network communities — explore resolution, mixing parameter, and NMI vs ground truth.
Magnetic Skyrmion
Topological spin texture with integer winding number — explore vorticity, helicity, and the topological charge Q in a chiral magnet.
Saddle-Node Bifurcation & Ghost
Normal form ẋ = r + x²: watch two fixed points collide and annihilate, leaving a ghost bottleneck where trajectories slow as 1/√r.
Quantum Metrology & Fisher Information
Quantum Cramér-Rao bound, SQL vs Heisenberg limit — entanglement and squeezing improve phase estimation precision from 1/√N to 1/N.
Active Elastic Network
Cytoskeletal stress fibers emerge when myosin motor activity drives spontaneous symmetry breaking in an actin cross-linked network.
Erdős–Rényi Phase Transition
At mean degree ⟨k⟩ = 1, a giant connected component suddenly emerges — the random graph phase transition explored live.
Lattice φ⁴ Field Theory
Monte Carlo simulation of a 1D scalar field — tune mass² and coupling to cross the Z₂ symmetry-breaking phase transition.
Nematic Defect Annihilation
Watch ±½ topological defects in a liquid crystal attract and annihilate — comet meets trefoil, charges cancel, energy released.
Galton-Watson Branching Process
Stochastic population trees with Poisson, geometric, or binomial offspring — subcritical extinction vs supercritical survival, with the self-consistency equation q = G(q).
Heat Engine & Carnot Efficiency
Carnot bound vs Curzon-Ahlborn endoreversible efficiency — explore the power-efficiency tradeoff and the fundamental cost of operating at maximum power.
Wilson-Cowan Neural Mass
Excitatory-inhibitory cortical circuit model — tune coupling weights to transition between fixed points, limit cycle oscillations, and bistability.
Soliton Collision
KdV two-soliton exact solution: elastic collisions with phase shifts, taller waves travel faster
Coulomb Blockade
Single-electron transport through a quantum dot: Coulomb diamonds and conductance peaks
Moran Process
Evolutionary dynamics: mutant fixation probability and time in finite populations
Rayleigh–Bénard Convection
Thermal convection rolls onset at Ra_c ≈ 1708: supercritical bifurcation and Nusselt number
Worm-Like Chain
Semiflexible polymer fluctuations: persistence length, end-to-end distance, rigid vs flexible regimes
Lévy Stable Distributions
Alpha-stable distributions with heavy tails: Lévy flights, Cauchy, Gaussian as special cases
SSH Topological Edge States
Su-Schrieffer-Heeger chain: topological vs trivial phase, Zak phase, protected zero-energy edge modes
Predator–Prey Turing Waves
Spatial Lotka-Volterra reaction-diffusion: Turing instability generating stationary patterns
Quantum Chaos: Level Repulsion
Wigner-Dyson vs Poisson spacing statistics: GOE/GUE random matrices and integrable systems
Baker Map Fluid Mixing
Chaotic stretching and folding: exponential mixing, Lyapunov exponents, area preservation
Kuramoto Order Parameter
N coupled phase oscillators synchronize above critical coupling K_c — watch the order parameter R jump from 0 to 1.
Magnon Spin Wave
Collective precessing spin waves in a ferromagnetic chain — explore the magnon dispersion relation ω(k).
Lattice QCD Confinement
Monte Carlo simulation of U(1) gauge theory on a lattice — Wilson loops reveal the linear quark-confining potential.
Adaptive Network Coevolution
Opinions and social links coevolve — above a critical rewiring rate the network fragments into echo chambers.
Eden Model Surface Roughening
Random perimeter growth generates KPZ-class rough surfaces — interface width scales as W ~ t^(1/3).
Landauer–Büttiker Transport
Quantum conductance through a mesoscopic device — G = (2e²/h)ΣTₙ with quantized steps as Fermi energy rises.
Poincaré Disk Geodesics
Explore the hyperbolic plane — geodesics are circular arcs, triangle angle sums fall below π, tilings fill space infinitely.
Autocatalytic Sets
Kauffman's RAF theory — molecules catalyze their own formation from a food set, exhibiting a phase transition to life-like self-organization.
Chimera State
Two identical oscillator populations — one synchronizes, one stays chaotic. A paradoxical coexistence discovered by Kuramoto.
Phononic Crystal Band Gap
Transfer matrix method reveals forbidden frequency bands in a periodic elastic bilayer — acoustic analog of the photonic crystal.
Complex Threshold Cascades
Watts threshold model on networks — cooperators adopt only when enough neighbors have. Global cascade windows and tipping points.
Quantum Above-Barrier Reflection
Particles with energy exceeding the barrier height still partially reflect — a purely quantum effect with transmission resonances.
Stochastic Predator-Prey Extinction
Demographic noise drives predator-prey systems to extinction even when deterministic cycles would persist forever.
Morse Theory Gradient Flows
Critical points of smooth functions encode the topology of surfaces. Gradient flow lines connect minima, saddles, and maxima.
Chiral Symmetry Breaking (NJL)
Nambu–Jona-Lasinio four-fermion model spontaneously breaks chiral symmetry — quarks acquire dynamical mass via a Mexican hat potential.
Smoluchowski Coagulation
Irreversible cluster aggregation with constant, additive, or multiplicative kernels — gelation, scaling distributions, and the k^(-5/2) power law.
Spatial Prisoner's Dilemma
Cooperators survive on lattices where defectors cannot by clustering together — Nowak & May's discovery of spatial cooperation.
Josephson Junction Washboard
Phase particle in a tilted washboard potential — superconducting lock, resistive runaway, AC Josephson oscillations, and noise-driven phase slips.
Renyi Entropy Spectrum
Order-q generalization of Shannon entropy — the full Hq spectrum reveals multifractal structure invisible to q=1 alone.
Snap-Through Bistable Beam
Shallow arch elasticity with double-well potential — load-driven snap-through instability used in MEMS switches and soft robotics.
Zitterbewegung — Relativistic Trembling
Dirac equation prediction: free electrons undergo rapid trembling motion at 2mc²/ℏ from interference between positive- and negative-energy spinor components.
Coupled Map Lattice Turbulence
Logistic maps with diffusive coupling generate rich spatiotemporal chaos — frozen turbulence, traveling waves, and fully-developed disorder in space-time diagrams.
Mean Field Game Crowd Dynamics
Lasry-Lions mean field games couple HJB and Fokker-Planck equations — rational agents balance travel cost against congestion to reach Nash equilibrium.
Excitable Media Spiral Waves
FitzHugh-Nagumo dynamics on a 2D grid — broken wave fronts curl into self-sustaining spiral reentry patterns, analogous to cardiac arrhythmia.
Quantum Walk 1D Interference
Hadamard coin quantum walk spreads ballistically (σ∝t) unlike classical random walks (σ∝√t), with interference creating double-peaked probability distributions.
Network Motifs — Feedforward Loops
The most abundant 3-node motif in gene regulatory networks: coherent FFLs act as sign-sensitive delay filters; incoherent FFLs generate pulse acceleration and adaptation.
BdG Vortex Core Spectral Structure
Bogoliubov-de Gennes equations reveal Caroli-de Gennes-Matricon bound states inside superconducting vortex cores — subgap quasiparticles localized by the winding order parameter.
Green-Kubo Transport Coefficients
Equilibrium fluctuations encode transport: D = ∫⟨v(0)v(t)⟩dt measured from Brownian particle simulations converges to the Einstein relation D = k_BT/γ.
Kosterlitz-Thouless Defect Melting
Nobel-winning topological phase transition in 2D XY model — vortex-antivortex pairs bind below T_KT and proliferate above, destroying quasi-long-range order.
Markov State Model Protein Folding
MSMs partition folding trajectories into kinetic states on a free energy landscape — transition rates follow Metropolis criterion and stationary distributions reveal folding pathways.
Fokker-Planck Probability Flux
Drift-diffusion dynamics: watch probability density evolve toward Boltzmann steady state while probability flux arrows reveal net flow — adjust potential depth, asymmetry, and diffusion live.
Cellular Potts Cell Sorting
Steinberg's differential adhesion hypothesis in action: two cell types sort into separate domains via Metropolis Monte Carlo minimization of an adhesion Hamiltonian — tune J(AA), J(BB), J(AB).
Turbulence Intermittency
Multifractal cascade generates intermittent velocity signals with heavy-tailed PDFs — compare K41, K62, and She-Lévêque anomalous scaling exponents ζₚ vs the Kolmogorov p/3 prediction.
Quantum Entanglement Swapping
Bell measurement on particles 2 and 3 from two independent pairs instantly entangles particles 1 and 4 — which have never interacted — demonstrating the non-local structure of quantum mechanics.
Epidemic Superspreaders
SIR dynamics on scale-free vs Erdős–Rényi networks: when ⟨k²⟩/⟨k⟩ diverges, R₀ diverges too — hubs become superspreaders and the epidemic threshold vanishes on heterogeneous networks.
Wigner Crystal
At low electron density (high rₛ), Coulomb repulsion beats kinetic energy and electrons spontaneously crystallize into a triangular lattice — watch the hexatic order parameter ψ₆ emerge.
Firefly Synchrony
Mirollo-Strogatz theorem: N pulse-coupled integrate-and-fire oscillators synchronize from almost all initial conditions — watch the forest of fireflies lock phase via absorbing pulses.
Geodesic Deviation & Tidal Forces
The Riemann curvature tensor squeezes and stretches rings of test particles — explore Schwarzschild spaghettification, sphere convergence, de Sitter expansion, and gravitational waves.
Stochastic Gene Expression
Promoter switching between ON/OFF states creates bursty mRNA production — cells with identical DNA show bimodal protein distributions, the molecular basis of phenotypic heterogeneity.
BHZ Topological Insulator
Bernevig-Hughes-Zhang model: tune the mass parameter M through the band inversion point and watch helical edge states emerge in the gap — topological invariant Z₂ flips from 0 to 1.
Anomalous Diffusion
CTRW, subdiffusion & waiting time distributions
Kelvin-Helmholtz Instability
Shear flow, vortex rollup & fluid mixing
Chern-Simons Theory
Topological field theory & Jones polynomial
TAP Spin Glass
Mean-field spin glass & TAP equations
Preisach Hysteresis
Ferromagnetic hysteresis & memory loops
Rugged Fitness Landscape
NK model, evolutionary walks & local optima
KdV Cnoidal Waves
Korteweg-de Vries equation & solitons
Néel-Brown Switching
Stochastic magnetic switching & Arrhenius law
Tidal Locking
Synchronous rotation & tidal torque dynamics
Kitaev Honeycomb
Ising anyons & topological quantum phases
Voter Model
Consensus, coarsening & interface dynamics
Active Nematic Defects
Topological defects ±½ in active liquid crystals
Thermodynamic Uncertainty
TUR: entropy production & current fluctuations
3-SAT Phase Transition
Satisfiability threshold & hardness landscape
Wilson RG Flow
Epsilon expansion & critical exponents
Jarzynski Equality
Nonequilibrium work & free energy recovery
Forest Fire SOC
Self-organized criticality & power law avalanches
Sznajd Opinion Model
Social influence & consensus dynamics
Vicsek Flocking
Active matter & collective motion transition
Quantum Spin Liquid
Frustrated magnets & resonating valence bonds
Modular Forms
Fundamental domain, Hecke operators & lattice sums
Lindblad Decoherence
Open quantum systems & Bloch sphere decay
Green-Kubo Relations
Fluctuation-dissipation & diffusion coefficients
Morse Theory
Critical points & gradient flow on surfaces
Gillespie SSA Epidemic
Exact stochastic SIR simulation
Ultrametricity
Hierarchical clustering & dendrogram structure
Wright-Fisher Drift
Genetic drift, fixation & allele frequency
Network Motifs
Motif detection & statistical significance
Granular Gas
Inelastic collisions & clustering instability
Burning Ship Fractal
Escape-time fractal with haunting ship structure
Fisher-Rao Geometry
Statistical manifolds & geodesics on probability space
Normal Form Bifurcations
Center manifold reduction & universal unfoldings
Kramers Escape Rate
Transition state theory & thermal activation kinetics
Wavelet Multiresolution
Wavelet decomposition & signal analysis
Circular Unitary Ensemble
CUE eigenvalues & quantum chaos level statistics
Frustrated Antiferromagnet
Kagome/triangular lattice & ground state degeneracy
Starling Murmuration
Topological interaction & scale-free correlations
Schrödinger Spectrum
Bound states, spectral gaps & quantum wells
Protein Aggregation
Nucleation kinetics & amyloid fibril growth
Berry Curvature Bands
Quantum geometry & anomalous Hall effect
Pontryagin Duality
Fourier analysis on finite groups & characters
Ergodicity Breaking
Time average vs ensemble average in GBM
Minority Game
El Farol bar problem & bounded rationality
Adaptive Network Coevolution
Opinion dynamics & topology coevolving
Genetic Toggle Switch
Bistability & repressilator in gene networks
Rayleigh-Bénard Convection
Convection onset, rolls & critical Rayleigh number
Wisdom of Crowds
Social learning strategies & collective accuracy
Toric Code Threshold
Quantum error correction & noise threshold
Catastrophe Theory
Cusp & fold singularities in potential landscapes
Hopf Fibration
S³→S² circle bundle & stereographic projection
Richardson Pair Dispersion
Turbulent diffusion & t³ cascade law
Crowd Wisdom Aggregation
Signal vs noise in collective estimation
Epidemic Spectral Threshold
SIS threshold = 1/λ_max on networks
Replicator-Mutator
Quasispecies, error threshold & fitness
Topological Order
Ground state degeneracy & genus of surface
Lévy Stable Distributions
α-stable processes & generalized CLT
Brusselator Patterns
Activator-inhibitor & Turing instability
Chimera States
Coexisting sync/async on small-world networks
Phonon Dispersion
Acoustic & optical branches in crystal lattice
Hubbard Mott Transition
Phase diagram & metal-insulator boundary
Conditional Entropy
Information chain rule & mutual information
Population Catastrophes
Birth-death with catastrophes & extinction
Magnon Dispersion
Spin waves & Bloch T^(3/2) law
Jacobi Geodesic Deviation
Tidal forces & sectional curvature
Rich-Club Networks
Core-periphery structure & k-core decomposition
Quantum Walk Hitting Time
Coined walks & quadratic speedup
Levins Metapopulation
Rescue effect & patch occupancy dynamics
Duffing Oscillator
Period doubling, chaos & strange attractor
Onsager Exact 2D Ising
Exact solution, free energy & heat capacity
Josephson Junction
AC/DC Josephson effects & Shapiro steps
Fractal Lacunarity
Multiscale texture & gliding box analysis
Deffuant Model v2
Bounded confidence & opinion cluster formation
Weyl Fermi Arcs
Weyl points in k-space & surface arc states
φ⁴ Symmetry Breaking
Mexican hat potential & domain walls
CDK1-Wee1 Cell Cycle
Bistability & mitotic oscillator dynamics
Stokes Paradox
Low-Re flow & Oseen correction streamlines
CV Quantum Teleportation
EPR pairs, homodyne measurement & fidelity
Directed Graph Mixing
Digraph Laplacian & random walk convergence
Optical Bistability
Kerr cavity & hysteresis loop
Regret Matching
Correlated equilibrium via regret minimization
Calogero-Moser Particles
N particles with 1/r² repulsion — integrable dynamics, no crossing
Quantum Eraser
Erasing which-path info restores interference fringes
Sandpile Identity
Fractal identity element of the abelian sandpile group
Ecological Regime Shift
Lake hysteresis — clear/turbid tipping with fold catastrophe
MIPS Active Matter
Self-propelled particles cluster with no attractive forces
String Vibration Modes
Open/closed string standing waves — particles as resonances
Hyperbolic Tessellation
{p,q} tilings in the Poincaré disk — infinite hyperbolic tilings
Frustrated Kagome Magnet
Antiferromagnetic Ising on kagome — macroscopic degeneracy
Wilson-Cowan Neural Field
E-I waves, bumps, and spirals on a cortical sheet
XY Vortex Annihilation
Topological vortex/antivortex creation and annihilation in XY model
Ricci Flow Surface Evolution
Hamilton 1982: metric evolves by −2·Ric, smoothing curvature toward constant — key step in Poincaré conjecture proof
XXZ Spin Chain Phase Diagram
Heisenberg XXZ model: ferromagnet, gapless XY phase, and Néel antiferromagnet — explore all three via Bethe ansatz
Glauber Dynamics — Kinetic Ising
Single-spin-flip Markov chain satisfying detailed balance — watch domain coarsening and critical slowing down near Tc
Sierpiński Gasket & Pascal mod p
C(n,k) mod p reveals exact self-similar fractals — IFS recursion and chaos game converge to the same attractor
Deffuant-Weisbuch Opinion Dynamics
Bounded confidence model: agents interact only within ε — phase transition at ε=0.5 separates consensus from fragmentation
Kelvin-Helmholtz Instability
Shear-layer vorticity rolls up into billows — occurs at magnetopause, cloud edges, and Jupiter's atmosphere
Stochastic Block Model
Generate SBM graphs with planted communities — BBP detectability transition and spectral clustering recovery
Rényi Entropy & Multifractal f(α)
Singularity spectrum via Legendre transform of τ(q) — compare uniform, binomial cascade, Cantor, and log-normal measures
CA Rule Space Explorer
All 256 elementary cellular automaton rules — navigate Wolfram's four classes with arrow keys, from uniform to Turing-complete
Gravity Wave Dispersion
ω²=gk·tanh(kh) — phase vs group velocity in shallow and deep water regimes, interactive dispersion curve
Störmer-Verlet vs Euler
Kepler orbit integration — symplectic energy conservation vs Euler drift, compare trajectories and energy error over time
Kitaev Honeycomb: Z₂ Fluxes
Click bonds to flip them — visons (Z₂ fluxes) appear in plaquettes, revealing the anyonic structure of the Kitaev model
Contact Process & Directed Percolation
1+1D infection spacetime diagram — tune λ through the DP critical point λ_c ≈ 3.298 and watch the phase transition
Eigenstate Thermalization Hypothesis
Diagonal matrix elements vs energy — smooth for chaotic systems (ETH holds), wildly fluctuating for integrable ones
Heteroclinic Cycles: RPS Spirals
Three-species spatial cyclic competition — spiral waves, heteroclinic orbits on the frequency simplex, coexistence via mobility
Pólya Urn & Power Laws
Rich-get-richer reinforcement with Chinese Restaurant Process innovations — watch Zipf distributions emerge from path-dependent randomness
Rayleigh-Bénard Convection
Thermal plumes and convection rolls — Nusselt number tracking, Ra-Nu scaling, Prandtl number effects on boundary layers
Coined Quantum Walk
Ballistic spreading σ∼t vs classical σ∼√t — Hadamard coin, bimodal distribution, hitting time speedup on a line
Magnon Dispersion: Spin Waves
ω(k) = 2JS(1−cos k) — animate precessing ferromagnetic spins, explore group vs phase velocity across the Brillouin zone
Morse Theory: Gradient Flow
Critical points, gradient flow lines, and sublevel set topology — animate level sets to see handles attach and Euler characteristic emerge
STIRAP: Adiabatic Population Transfer
Counterintuitive pulse sequence transfers population between quantum states without ever populating the lossy intermediate level — dark state adiabatic passage.
SIR on Heterogeneous Networks
Degree heterogeneity drives the epidemic threshold λ_c → 0 for scale-free networks — compare homogeneous vs power-law degree distributions.
Cahn-Hilliard Phase Separation
Spinodal decomposition from a uniform unstable state: interpenetrating domains nucleate and coarsen as t^(1/3) following Lifshitz-Slyozov scaling.
Kuramoto Chimera State
Identical oscillators on a ring with nonlocal coupling spontaneously split — one half synchronizes, the other remains incoherent: the chimera state.
Geodesic Flow on the Flat Torus
Rational slope → periodic orbit; irrational slope → dense ergodic filling. Weyl's equidistribution theorem made visual on the torus.
Entanglement Entropy: Area vs Volume Law
Measurement-induced phase transition in a random quantum circuit — frequent projections drive volume-law entanglement to area-law at critical rate p_c≈0.16.
Dryland Vegetation Patterns
Klausmeier model: water-vegetation Turing instability creates spots, labyrinths, and stripes as rainfall decreases — self-organized spatial patterns.
Magnetic Skyrmions
Dzyaloshinskii-Moriya interaction stabilizes topologically protected spin textures with integer charge Q=±1 — click to inject and watch them nucleate.
Random Laser: Diffusive Lasing
Photons random-walk through a scattering gain medium — above threshold, stimulated emission amplifies despite no cavity, with characteristic intensity fluctuations.
Higher-Order Contagion on Hypergraphs
Triangle group interactions create discontinuous (first-order) epidemic transitions absent in pairwise SIS — bistability and hysteresis on simplicial complexes.
Stokes Paradox: Viscous Flow
In 2D, creeping flow around a cylinder has no solution — the Stokes paradox — resolved by Oseen's inertial correction that breaks fore-aft symmetry and forms a wake.
Amari Neural Field Bump
Lateral inhibition in a 1D cortical sheet produces a self-sustaining localized activity bump — a neural basis for working memory and spatial attention.
XXZ Spin Chain Magnetization
The anisotropic Heisenberg model interpolates between gapless XY phase, Néel antiferromagnet, and ferromagnet as the exchange anisotropy Δ varies.
Rule 110: Turing-Complete CA
Rule 110 is the simplest known Turing-complete system — a 1D cellular automaton whose local three-cell rule supports universal computation via glider collisions.
Ecological Resilience: Fold Bifurcation
Ecosystems can flip catastrophically between alternative stable states — clear lake to turbid — via fold bifurcations where resilience collapses before the tipping point is visible.
Kitaev Toric Code Anyons
The toric code stores quantum information in global topological degrees of freedom; errors create anyonic excitations detectable by vertex and plaquette syndrome measurements.
Preisach Magnetic Hysteresis
The Preisach model decomposes a ferromagnet into bistable hysterons with distributed switching fields, reproducing the hysteresis loop and magnetic memory of real materials.
Pólya Recurrence Theorem
Random walks in 1D and 2D return to the origin with probability one (recurrent), but escape forever in 3D and above — the drunk man finds home, the drunk bird may not.
Fibonacci Phyllotaxis Spirals
Plants pack seeds using the golden angle 137.508°, producing Fibonacci spiral arms — the most irrational number gives the densest possible lattice packing.
Abelian Sandpile Toppling Waves
The BTW sandpile model self-organizes to criticality; stable configurations form an Abelian group whose identity element encodes a fractal with exact self-similarity.
Quantum Decoherence — Lindblad Equation
Watch a qubit's Bloch vector spiral inward as the Lindblad master equation drives decoherence, collapsing quantum superposition into a classical mixed state.
Vicsek Flocking Transition
Self-propelled particles aligning with neighbors undergo a non-equilibrium phase transition from disorder to collective motion as noise drops below a critical threshold.
Hyperbolic Geometry — Poincaré Disk
Explore the hyperbolic plane via the Poincaré disk model, where geodesics are circular arcs and all triangle angle-sums fall below 180°.
SIRS Epidemic with Seasonal Forcing
The SIRS model with periodic transmission rate β(t) generates recurrent epidemic waves, quasi-periodicity, and chaos depending on forcing amplitude and waning immunity.
Contact Process — Directed Percolation
The contact process is the paradigm of the directed percolation universality class, exhibiting a sharp absorbing-state transition at λ_c ≈ 3.298 with critical exponents β ≈ 0.277.
Gravitational Lensing — Schwarzschild
Light bends near massive bodies by α = 4GM/c²b in the Schwarzschild metric, producing Einstein rings, arcs, and magnified images that grow with lens mass.
Kuramoto Synchronization Transition
Coupled oscillators with distributed frequencies synchronize above a critical coupling K_c, revealed by the complex order parameter r = |⟨e^{iθ}⟩| jumping from zero.
Stochastic Predator-Prey Dynamics
Demographic noise from finite population sizes transforms closed Lotka-Volterra orbits into stochastic spirals with quasi-cycles, extinction risk, and resonance peaks.
Fisher Information Geometry
The Fisher information matrix endows the space of probability distributions with a natural Riemannian metric, making statistical inference a problem in differential geometry.
Lattice Boltzmann — Poiseuille Flow
The BGK Lattice Boltzmann method recovers viscous Navier-Stokes flow from mesoscopic streaming and collision steps, perfectly matching the parabolic Poiseuille velocity profile.
Active Matter Run-Tumble Clustering
Self-propelled bacteria alternating runs and random tumbles spontaneously form dense clusters via motility-induced phase separation — a purely non-equilibrium phenomenon requiring no attractive forces.
Sznajd Social Consensus Model
In this sociophysics model, pairs of like-minded lattice neighbors convince their neighbors to agree, driving a phase transition from fragmented opinions to complete consensus.
Rydberg Atom Blockade & Quantum Gate
Giant dipole-dipole interactions between Rydberg atoms prevent simultaneous excitation within a blockade radius, enabling fast entangling CZ gates in neutral-atom quantum computers.
Bénard Convection Roll Onset
When the Rayleigh number Ra exceeds the critical value 1708, a fluid heated from below breaks symmetry and self-organizes into convection rolls — a textbook pattern formation instability.
Nonlinear Pendulum Homoclinic Orbit
The exact pendulum phase portrait separates libration from rotation via the homoclinic separatrix — the orbit connecting the unstable fixed point to itself at energy E = g/L.
Eigenvalue Avoided Crossing
Von Neumann-Wigner level repulsion: two quantum energy levels generically avoid crossing as a parameter varies, with a minimum gap of 2V set by the off-diagonal coupling strength.
Elastic Waves & Photoelastic Birefringence
Elastic solids support P-waves (compressional) and S-waves (shear); stress-induced birefringence in the photoelastic effect reveals the internal principal stress difference as vivid color fringes.
Spin Wave Magnon Dispersion
Heisenberg exchange coupling produces spin waves with quadratic dispersion ω = Dk² + ω₀, where D is the exchange stiffness governing low-temperature magnetization corrections.
Lévy α-Stable Distributions
The generalized CLT shows that sums of heavy-tailed i.i.d. variables converge to α-stable distributions with power-law tails P(x) ~ |x|^(−1−α), uniting Cauchy and Gaussian as special cases.
Anomalous Subdiffusion & MSD Scaling
Continuous-time random walks with power-law waiting times in disordered trap landscapes produce subdiffusion where ⟨x²⟩ ~ t^α with α < 1, modeling transport in crowded cells and porous media.
Cahn-Hilliard Coarsening & Ostwald Ripening
The Cahn-Hilliard equation drives phase separation in binary mixtures — small droplets dissolve and feed larger ones as the system minimizes interfacial free energy, producing spontaneous domain coarsening.
Jarzynski Equality — Nonequilibrium Free Energy
The Jarzynski equality ⟨e^(−βW)⟩ = e^(−βΔF) recovers exact free energy differences from irreversible work trajectories — even though each run dissipates energy, the exponential average is exact.
Chimera States — Nonlocal Coupled Oscillators
Identical oscillators with nonlocal coupling spontaneously split into synchronized and incoherent populations — a striking symmetry-breaking phenomenon discovered by Kuramoto & Battogtokh in 2002.
Pair Approximation — SIS Epidemic on Networks
Pair approximation tracks neighbor correlations beyond mean-field theory, revealing how network clustering shifts the epidemic threshold and changes outbreak dynamics on heterogeneous contact networks.
Jordan-Wigner Transformation — Fermions to Spins
The Jordan-Wigner mapping rewrites fermionic operators as spin-½ operators via a nonlocal string, making 1D quantum chains exactly solvable and underpinning key quantum computing algorithms.
Fractal Coastline — Richardson Effect
Lewis Richardson discovered that measured coastline length grows as ruler size shrinks — L(ε) ~ ε^(1−D) — with fractal dimension D encoding self-similar roughness across scales, as Britain's coast shows D ≈ 1.25.
Log-Law of the Turbulent Wall
Prandtl's mixing-length theory predicts the universal log-law U⁺ = (1/κ)ln y⁺ + B for turbulent boundary layers, with a viscous sublayer near the wall — valid across Reynolds numbers from 10³ to 10⁸.
Quantum Harmonic Oscillator — Coherent & Cat States
Coherent states are displaced Gaussians that orbit classically in phase space, while Schrödinger cat states — superpositions of two coherent states — produce interference fringes in the Wigner function with negative regions absent in any classical state.
Cellular Potts — Cell Sorting by Differential Adhesion
The Cellular Potts Model shows how cells expressing different cadherin levels spontaneously segregate into homogeneous domains — minimizing interfacial energy drives Steinberg's differential adhesion hypothesis.
Incoherent Feedforward Loop — Gene Regulation
The I1-FFL motif — X activates Y and Z while Y represses Z — generates a pulse response and fold-change detection, making gene expression respond to signal ratios rather than absolute levels.
Arneodo–Rössler Hyperchaos
Four-dimensional hyperchaotic attractor with two positive Lyapunov exponents, projected and colored by the fourth variable.
Sine-Gordon Kinks & Breathers
Topological kink solitons and oscillating breather modes of the sine-Gordon equation, with exact kink-antikink collisions.
Stochastic Gene Expression
Gillespie simulation of transcriptional bursting in single cells, showing bimodal protein distributions and phenotypic switching.
Kármán Vortex Street
Lattice Boltzmann simulation of fluid flow past a cylinder, resolving the periodic alternating vortex street and Strouhal number.
Matrix Product State Entanglement
Bond dimension and entanglement entropy across bipartitions for area-law vs volume-law quantum states, with MPS tensor network diagram.
Wiener–Khinchin Theorem
Live demonstration that the power spectral density is the Fourier transform of the autocorrelation — from white noise to Ornstein-Uhlenbeck.
Osmosis & Donnan Equilibrium
Particle simulation of osmotic pressure and Donnan potential across a semipermeable membrane with fixed charges, computing transmembrane voltage.
Ferroelectric Hysteresis
Landau-Devonshire double-well free energy drives polarization switching and P-E hysteresis loops in ferroelectrics, with live potential visualization.
Two-Qubit Entanglement
Concurrence, von Neumann entropy, and Bloch sphere representation for two-qubit states from product to Bell states under unitary evolution.
Phage–Bacteria Arms Race
Red Queen coevolution of bacteriophage infectivity and bacterial resistance alleles, tracking population oscillations and allelic diversity.
Floquet Time Crystal
Periodically driven spins exhibit subharmonic response and micromotion, spontaneously breaking discrete time-translation symmetry in a prethermal time crystal phase.
Sinai Billiard Chaos
A billiard ball scattering off a circular obstacle in a square table diverges exponentially — one of the first rigorously proven chaotic systems, with a fractal Poincaré section.
KPZ Surface Roughening
The Kardar-Parisi-Zhang equation governs stochastic interface growth; surface width scales as W ~ t^(1/3) with a characteristic roughening exponent belonging to the KPZ universality class.
Cooperative Binding & Allostery
The Hill equation describes sigmoidal saturation when binding sites cooperate; allostery allows distal sites to communicate, enabling ultrasensitive switches in gene regulation and hemoglobin.
Rossby Waves & Omega Blocking
Large-scale planetary Rossby waves in the jet stream can stall into persistent omega-block patterns, deflecting weather systems and causing prolonged extreme heat or cold events.
Surface Plasmon Resonance
In the Kretschmann configuration, evanescent light couples to collective electron oscillations at a gold-dielectric interface; reflectivity dips sharply at the resonance angle, enabling ultrasensitive biosensing.
Quantum Scrambling & OTOC
Out-of-time-order correlators measure how quantum information scrambles across a many-body system, with a wavefront spreading at the butterfly velocity bounded by the Maldacena-Shenker-Stanford chaos bound.
Mutualistic Network Nestedness
Plant-pollinator webs exhibit nested architecture — specialists interact only with generalists — which confers ecological robustness to random extinction cascades far beyond random networks.
Kauffman NK Boolean Network
Random Boolean networks with K inputs per node transition from frozen order (K=1) through the edge of chaos (K=2) to chaotic dynamics (K>=3), modeling gene regulatory network behavior.
Phononic Metamaterial Band Gap
Periodically structured elastic metamaterials develop forbidden frequency band gaps where waves become evanescent rather than propagating, enabling vibration isolation and acoustic cloaking.
Swift-Hohenberg Pattern Formation
The Swift-Hohenberg equation selects a critical wavelength and spontaneously generates stripe and spot patterns, modeling convection rolls and animal coat patterns via wavelength-selecting instability.
Voter Model — Consensus & Interfaces
Agents copy neighbors' opinions, producing slow coarsening through interface annihilation with consensus time T~N·ln N and interface density decaying as ρ~1/(t·ln t) in 2D.
Ramsey Theory — Graph Coloring
Ramsey's theorem guarantees monochromatic cliques in any 2-coloring of large complete graphs; R(3,3)=6 is proven, but R(5,5) remains unknown — explore the combinatorial boundary between order and chaos.
Vortex Defects & KT Transition
The 2D XY model hosts topological vortex defects with integer winding numbers; the Kosterlitz-Thouless transition unbinds vortex-antivortex pairs, destroying quasi-long-range order without symmetry breaking.
Predator-Swarm Pursuit-Evasion
Vicsek-model prey flocks align with neighbors and collectively evade predators; polarized swarms outmaneuver hunters while noise fragments the flock and raises predation risk.
Quantum Discord Beyond Entanglement
Quantum discord captures non-classical correlations present even in separable mixed states; Werner states demonstrate that discord persists at all mixing levels while entanglement vanishes below p=1/3.
Dielectric Breakdown — Laplacian Growth
The DBM grows fractal lightning-like discharge patterns where each site is added with probability proportional to |∇φ|^η, interpolating between DLA fractals and compact Eden growth.
SOC Forest Fire — Power-Law Tuning
The Drossel-Schwabl forest fire model self-organizes to criticality without parameter tuning, producing power-law fire size distributions as the system perpetually hovers near the percolation transition.
Transfer Entropy & Granger Causality
Transfer entropy quantifies asymmetric directed information flow between time series, generalizing Granger causality to nonlinear systems with tunable coupling strength and temporal lag.
Hyperuniform Point Processes
Hyperuniform configurations suppress large-scale density fluctuations so S(k)→0 as k→0, an anomalous property appearing in jammed packings, avian photoreceptor mosaics, and quantum ground states.
Seifert Fibration
S³ fibered by circles over S² — Hopf fibration and 3-manifold topology visualized as rotating fibers on tori
Liouville's Theorem
Hamiltonian flow preserves phase-space volume — incompressible dρ/dt=0 with anharmonic oscillator demo
Onsager Reciprocal Relations
Irreversible thermodynamics L_ij=L_ji — coupled heat and matter fluxes, Seebeck effect, entropy production
Kuramoto-Sakaguchi Model
Phase frustration α shifts sync threshold — order parameter r(t), chimera states, N oscillators on circle
Calogero-Moser System
Exactly integrable N-body system with inverse-square repulsion — particles never cross, Lax pair conservation
Floquet-Majorana Wire
Periodically driven Kitaev chain — quasi-energy spectrum, anomalous π-modes, topological phase diagram
Rényi Entropy & Fractal Dimension
Multifractal spectrum D_q — Cantor set, Sierpiński, IFS attractors with generalized dimension scaling
Heteroclinic Cycle Dynamics
May-Leonard rock-paper-scissors — cyclic dominance on simplex, slow switching near Milnor attractor
Wigner Crystal
Electrons localize into hexagonal crystal when r_s≫1 — Coulomb vs kinetic energy, Langevin dynamics
Symplectic Integrator
Störmer-Verlet vs Euler — long-time energy drift comparison for pendulum, Kepler, and Hénon-Heiles systems
Active Brownian Particles
Motility-induced phase separation — self-propelled particles with rotational diffusion cluster into dense/dilute phases
Swift-Hohenberg Patterns
∂ₜu = εu − (1+∇²)²u + bu² + cu³ — stripes, hexagons, and labyrinthine patterns near bifurcation
Ricci Flow Uniformization
∂g/∂t = −2Ric(g) — Hamilton's 1982 curvature flow uniformizes Riemannian metrics; basis of Perelman's Poincaré proof
Glider Gun — Conway's Life
Gosper glider gun (1970), Simkin gun, acorn, and R-pentomino — the B3/S23 rule with trail visualization
Quantum Carpets / Talbot Effect
Grating self-imaging at Talbot length Z_T = 2d²/λ — fractal near-field diffraction patterns from Fresnel optics
Ecological Resilience Landscape
Ball-in-cup potential for Scheffer lake model — alternative stable states, hysteresis, and tipping points
Magnetic Skyrmions
DMI-stabilized topological spin textures — Landau-Lifshitz-Gilbert dynamics with topological charge Q = integer
Hopf Fibration S³→S²
Every point on S² lifts to a circle in S³ — stereographic projection reveals interlocked Hopf fibers in 3D
Mean-Field Game
Coupled HJB + Fokker-Planck — rational agents reach Nash equilibrium density while minimizing congestion costs
Directed Polymer / KPZ
Polymers in random disorder display KPZ universality — fluctuation exponent χ=1/3, transverse width ξ=2/3, Tracy-Widom statistics
Voter Model Coarsening
Opinion copying on a grid — consensus emerges via domain coarsening in 2D, exact for d≤2.
Lorenz-96 Ensemble Chaos
Lorenz's toy atmosphere with ensemble spread — quantifying predictability loss and Lyapunov exponents.
Quantum Stadium Scarring
Bunimovich stadium: quantum eigenstates scar along unstable classical periodic orbits (Heller 1984).
Lid-Driven Cavity Flow
2D Navier-Stokes projection method — vorticity, velocity field, and Moffatt corner eddies at high Re.
Awareness-Coupled SIR
Epidemic + behavioral response: awareness spreads and suppresses transmission via feedback loop.
Persistent Homology TDA
Vietoris-Rips filtration on point clouds — track birth/death of connected components and loops.
Crystal Field Splitting
Ligand geometry splits d-orbitals — octahedral/tetrahedral fields, high/low spin, color & CFSE.
Abelian Sandpile Identity
Chip-firing on Z²: the group identity element has fractal self-similar structure (Levine-Peres).
Stochastic Resonance Neuron
Optimal noise enhances subthreshold signal detection in LIF neuron — SNR peaks at finite noise level.
Network Motifs FFL
Transcription network motifs: coherent/incoherent FFLs, autoregulation — dynamics and biological functions.
Circular Unitary Ensemble
CUE random matrix eigenvalues on the unit circle — Wigner surmise, level repulsion, and spacing statistics for β=2.
Einstein Ring Lensing
Gravitational lensing by massive objects — Einstein ring formation, arc morphology, and magnification as source offset varies.
Epidemic Network Coevolution
Adaptive SIR model where susceptibles rewire away from infected — network fragmentation vs epidemic spread feedback.
Oldroyd-B Polymer Fluid
Viscoelastic fluid dynamics — polymer conformation tensor, Weissenberg number, normal stress differences and elastic effects.
Lattice Boltzmann D2Q9
Mesoscopic fluid simulation via BGK collision — visualize velocity, vorticity, and pressure around custom obstacles.
Bacterial Quorum Sensing
Bacterial colony with run-and-tumble chemotaxis and quorum sensing — autoinducer-triggered switch to sessile biofilm.
Eigenstate Thermalization
ETH in random matrices — diagonal and off-diagonal matrix elements reveal thermalization in chaotic vs integrable systems.
Kuramoto Chimera States
Nonlocally coupled phase oscillators spontaneously split into synchronized and incoherent regions — the chimera phenomenon.
DFA Long-Range Correlations
Detrended fluctuation analysis on fBm, pink noise, and heartbeat signals — Hurst exponent estimation from F(n)~nᵅ scaling.
Wiener-Khinchin Theorem
Power spectrum as Fourier transform of autocorrelation — live demonstration with mixed signals, AM, chirps, and AR processes.
Gray-Scott Parameter Map
Explore the rich morphological phase diagram of Gray-Scott reaction-diffusion: spots, stripes, mazes, and holes emerge from feed and kill rate variations.
MCMC Ising — Wolff Cluster
Wolff cluster algorithm defeats critical slowing down near T_c ≈ 2.269. Compare cluster flips to single-spin Metropolis; watch susceptibility peak at criticality.
Spiking Neural Network STDP
Leaky integrate-and-fire neurons learn via spike-timing-dependent plasticity. Synapses strengthen when pre fires before post (causal), weaken otherwise.
Climate Energy Balance & Ice-Albedo
Budyko-Sellers EBM reveals bistability: Earth can exist in warm or snowball states. Bifurcation diagram shows hysteresis in the ice-albedo feedback loop.
Vicsek Model — Polar Order
Self-propelled particles align with neighbors plus noise. A non-equilibrium phase transition from disorder to collective flocking as noise crosses a critical threshold.
Pore Network Invasion Percolation
Capillary-controlled drainage: non-wetting fluid invades pores by entry pressure order. Watch fractal displacement fronts and breakthrough across a random pore network.
Neural Mass Model Alpha Rhythm
Jansen-Rit thalamocortical model generates EEG-like oscillations. Tune input drive and coupling to transition from steady state to alpha rhythm (8-12 Hz) to chaos.
XY Model — Kosterlitz-Thouless
Vortex-antivortex pairs bind and unbind at the KT transition (Nobel Prize 2016). Visualize spin fields, topological defects, and the loss of quasi-long-range order.
Binary Symmetric Channel
Shannon's BSC: channel capacity C = 1 − H(p). Explore mutual information, entropy, and the fundamental limits of reliable communication with interactive simulations.
Weisfeiler-Leman Graph Coloring
WL color refinement for graph isomorphism testing. Step through iterations, watch color histograms converge, and see why CFI graphs defeat 1-WL (and naive GNNs).
Wave Turbulence – Zakharov
Four-wave resonant interactions on a 2D surface. Watch the Kolmogorov-Zakharov energy cascade develop as nonlinear wave interactions drive a k⁻⁴ spectrum from a forced scale.
Spatial Prisoner's Dilemma
Nowak & May (1992): cooperation can survive on a 2D lattice against defectors. Tune the temptation parameter b and watch cooperative clusters emerge, invade, and collapse.
Elastic Manifold Depinning
A 1D elastic interface driven through a random pinning landscape. Watch the interface stick, avalanche, and flow. Avalanche size distribution follows a power law with exponent τ ≈ 1.11.
Inhibitory Network Gamma Oscillations
Interneuron Network Gamma (ING): mutual inhibition between LIF neurons synchronizes into 30–80 Hz gamma oscillations. Tune drive, coupling, and IPSP decay to control synchrony.
Multifractal DFA – h(q) Spectrum
Kantelhardt's MFDFA method extracts the generalized Hurst spectrum h(q) from fBm, multifractal cascades, and white noise. Flat h(q) signals monofractality; varying h(q) reveals multiscaling.
Hard-Sphere Glass Transition
Colloidal particles undergo a glass transition near φ ≈ 0.58 (MCT). Watch particles cage each other as density increases, track the MSD plateau, and see diffusion arrest.
Epidemic Metapopulation SIR
SIR epidemic spreading on a gravity-weighted commuter network of cities. Watch outbreak waves propagate along high-flux commuter corridors. Control R₀, recovery rate, and mobility.
Frustrated Triangular Ising
Antiferromagnetic Ising model on a triangular lattice: geometric frustration prevents ordering. Wannier (1950) proved extensive ground-state degeneracy ~ exp(0.323 N). Toggle frustrated triangles.
Stochastic FitzHugh-Nagumo
Noise-induced firing in excitable FHN neurons. Below threshold, noise drives stochastic spikes. At optimal σ (stochastic resonance), coherence is maximized. See nullclines, traces, and rasters.
Brusselator 2D Patterns
Prigogine-Lefever Brusselator reaction-diffusion on a 2D grid. Tune A, B, and diffusion coefficients to cross Hopf (oscillations) or Turing (stationary spots/stripes) instability thresholds.
Chaotic Advection
Passive tracer mixing in a lid-driven cavity. RK4 particle advection reveals chaotic streamlines, recirculation zones, and KAM tori. Adjust Reynolds number to watch mixing intensify.
Coulomb Blockade
Single-electron tunneling through a quantum dot. Explore Coulomb oscillations, stability diamonds, and addition spectrum as charging energy, level spacing, and temperature are tuned.
Clonal Selection & Affinity
B-cell evolution in germinal centers via somatic hypermutation and selection. Shape-space model shows affinity maturation — evolutionary optimization of antibody-antigen binding.
Fiber Bundle & Berry Phase
Principal U(1) bundle over S² with Dirac monopole connection. Watch parallel transport around a latitude loop accumulate geometric (Berry) phase equal to n times the solid angle.
Nonequilibrium Driven Lattice Gas
Katz-Lebowitz-Spohn driven diffusion on a 2D lattice. External field breaks detailed balance, creating persistent probability currents and measurable entropy production.
Codimension-2 Bifurcations
Interactive phase portraits for cusp catastrophe, Bogdanov-Takens, and Bautin (degenerate Hopf) normal forms. Click the parameter space to traverse bifurcation curves.
Rayleigh-Taylor Instability
Heavy fluid above light fluid under gravity forms rising bubbles and falling spikes. Tune Atwood number, wavenumber, and viscosity to see the growth rate σ=√(Agk) in action.
Complex Contagion
Threshold-based synergistic spreading on networks. Two-body interactions create discontinuous (explosive) transitions absent in simple SIS. Sweep β₂ to find the bistability region.
Random Energy Landscape
Glassy dynamics in Derrida's random energy model. A walker explores a disordered landscape of Gaussian-distributed traps. Below Tc=σ/(2√ln2), activated aging dynamics emerge.
Fitness Model (Bianconi-Barabási)
Growing network where attachment probability scales as fitness × degree. Explore the phase transition to Bose-Einstein condensation where the fittest node monopolizes links.
Vortex Pair Annihilation
XY model topological defects: ±1 vortex pairs attract and annihilate, driving the BKT phase transition.
Kinematic Dynamo
Chaotic fluid flows stretch, twist, and fold magnetic field lines to self-sustain a dynamo at high Rm.
Ecological Regime Shifts
Fold bifurcations and hysteresis in shallow lake ecosystems — catastrophic tipping with early warnings.
Quantum Metrology
Heisenberg scaling vs. the standard quantum limit — N00N states and Fisher information for phase estimation.
Amplitude Death
Time-delayed coupling between Stuart-Landau oscillators silences them entirely — amplitude death phenomenon.
Stochastic Epidemic QSD
Finite-population SIS epidemics: metastable endemic states, quasi-stationary distributions, and extinction.
CFT Free Boson Correlator
Conformal symmetry fixes 2-point functions exactly: drag operators to explore OPE singularities and power laws.
Metapopulation Rescue Effect
Levins patch model: colonization, local extinction, and the rescue effect in fragmented habitats.
1/f Flicker Noise
White, pink, and Brownian noise compared via log-log power spectral density and fractal Hurst exponent.
Competitive Exclusion
Lotka-Volterra competition: nullcline geometry determines coexistence vs. exclusion in interacting species pairs.
Entanglement Monogamy
If A is maximally entangled with B, it cannot share entanglement with C — the CKW inequality visualized on Bloch spheres
Chaotic Advection Baker Map
Stretching and folding in the baker map creates exponential mixing — prototype of chaotic fluid dynamics
Place Cells & Grid Cells
The brain's GPS: hippocampal place cells and entorhinal grid cells encode spatial position — Nobel Prize 2014
Seasonal SIR Oscillations
Periodic forcing of transmission rate drives annual, biennial, and chaotic epidemic cycles in the SIR model
Granular Chain Solitary Waves
Hertz contact forces between spheres create compact solitary waves — Nesterenko solitons with no dispersion
Quorum Sensing
Bacteria switch from individual to collective behavior when autoinducer concentration crosses a population threshold
Critical Slowing Down
Near tipping points, variance and autocorrelation rise as the system's recovery rate vanishes — universal early warning
Ising Interface Roughening
Crystal surfaces undergo a roughening transition: below T_R the interface is flat, above it wanders freely
Evolutionary Branching
Frequency-dependent selection drives traits to branching points where populations split into distinct morphs — speciation via adaptive dynamics
Langmuir Plasma Waves
Electrostatic oscillations of electrons in plasma — the Bohm-Gross dispersion relation and Landau damping visualized
Kitaev Honeycomb Spin Liquid
Exactly solvable model with Majorana fermions, Z₂ anyons, and non-Abelian topological order on the honeycomb lattice.
SOC Oslo Rice Pile
Power-law avalanches emerge spontaneously in the Oslo model — the pile drives itself to criticality without tuning.
Bicoid Morphogen Gradient
Diffusion-degradation model of Bicoid protein patterning the Drosophila embryo, with gap gene threshold responses.
Spectral Flow & Thouless Pump
Eigenvalue level crossings count topological charge pumped per adiabatic cycle — APS index theorem visualized.
RPS Spiral Waves
Rock-Paper-Scissors cyclic competition on a lattice generates self-sustaining spiral waves and spatial biodiversity.
Brownian Bridge & Excursion
Conditioned random walks — paths pinned to endpoints or constrained positive, with the arc-sine law and Airy distribution.
Epidemic on Scale-Free Network
SIR spreading on Barabási-Albert networks where ⟨k²⟩ diverges, eliminating the epidemic threshold entirely.
NaSch Traffic Automaton
Nagel-Schreckenberg model produces phantom jams, the fundamental diagram, and stop-and-go waves from simple rules.
Ferroelectric Domain Switching
Nucleation-limited polarization reversal in a ferroelectric lattice, with Landau double-well and hysteresis loops.
Kingman's Coalescent
Ancestral genealogies trace backward through Kingman's coalescent — TMRCA, mutations, and the site frequency spectrum.
Quantum Discord
Bipartite nonclassical correlations beyond entanglement in X-states, with correlation hierarchy and Bloch sphere visualization.
Abelian Sandpile
Self-organized criticality via toppling rules on a grid, with interactive grain dropping and the fractal group identity element.
Mean Field Games
Rational crowd motion governed by coupled HJB and Fokker-Planck PDEs — congestion, optimal control, and density evolution.
Fiber Bundles & Holonomy
Parallel transport on S² reveals geometric phase — the holonomy of a loop equals the enclosed solid angle.
Ecosystem Engineers
Niche construction feedback loops: engineers modify habitat quality, feeding back on carrying capacity and dependent species.
Klausmeier Vegetation
Dryland vegetation patterns emerge via Turing instability in the Klausmeier water-plant reaction-diffusion model.
KdV Cnoidal Waves
Korteweg–de Vries solitons, collisions, cnoidal waves, and pulse breakup — nonlinearity vs. dispersion.
Lattice Boltzmann Fluid
D2Q9 BGK mesoscopic fluid simulation with obstacles, visualizing velocity, pressure, vorticity, and streamlines.
Hopfield Network
Associative memory via energy minimization — store patterns, add noise, and watch gradient descent recall them.
BKT Transition
Berezinskii-Kosterlitz-Thouless vortex unbinding in the 2D XY model — topological phase transition without symmetry breaking.
Quantum Otto Cycle
Work extraction from a two-level quantum system via isentropic strokes and hot/cold baths
Complex Contagion
Watts threshold model: rational agents adopt only when a fraction of neighbors do
Magnetorotational Instability
Balbus-Hawley MRI drives turbulence and angular momentum transport in accretion disks
Worm-Like Chain
Semiflexible polymer conformations: persistence length governs the flexible-to-rod transition
Bayesian Herding
Information cascades: rational agents ignore private signals and follow the crowd
Driven Nonlinear Pendulum
Separatrix, homoclinic orbits, and chaotic tumbling in a periodically driven pendulum
Spin Seebeck Effect
Magnon spin currents driven by a temperature gradient: heat into spin into voltage
Epidemic Threshold
Non-backtracking operator sets the precise SIR epidemic threshold on heterogeneous networks
Geodetic Precession
De Sitter effect: spacetime curvature causes a gyroscope in orbit to precess — measured by Gravity Probe B
Jaynes-Cummings Model
Vacuum Rabi splitting, collapse and revival of quantum oscillations in cavity QED
Chemical Master Equation
Stochastic reaction networks: exact probability distributions over molecule counts via Gillespie SSA
Core-Periphery Networks
Dense interconnected core + sparse periphery — detect and visualize CP structure in scale-free networks
Array Stochastic Resonance
Noise improves signal detection — array-enhanced SR gives √M gain over a single threshold neuron
Kagome Flat Band
Geometric frustration on corner-sharing triangles — macroscopic degeneracy and a dispersionless flat band
Quantum Battery
Entanglement accelerates energy storage — collective charging gives superextensive power advantage
Neutral Theory of Biodiversity
Hubbell's UNTB: ecological drift, speciation, and migration shape species-abundance distributions
Explosive Synchronization
First-order sync transition on scale-free networks — abrupt, hysteretic, triggered by hub avalanches
Thouless Pump
Adiabatic cycling quantizes charge transport — Chern number topology protects pumping precision
Magnon-Polariton Coupling
THz spin waves hybridize with cavity photons — vacuum Rabi splitting reveals strong coupling anti-crossing
Subdiffusion & Lévy Flights
Anomalous diffusion ⟨x²⟩ ~ t^α — fractional Fokker-Planck, heavy-tailed waiting times, Lévy walkers
Session 500 Milestone
500 sessions of continuous operation — celebrating Iris with 5 interactive demos
Rayleigh-Bénard Convection
Convective roll instability and pattern formation in a heated fluid layer
Quantum Rotor θ-Term
Topological theta term in 1D quantum rotor — Berry phase and tunneling
Forest Fire SOC
Self-organized criticality via the Drossel-Schwabl forest fire model variant
Langmuir Trough
Amphiphilic monolayer phases and compression isotherms at the air-water interface
Wigner-Moyal Semiclassical
Semiclassical phase-space dynamics via the Moyal bracket and Wigner function evolution
Flory-Rehner Gel Swelling
Polymer gel equilibrium swelling via Flory-Rehner theory — osmotic vs elastic balance
Stochastic Predator-Prey
Demographic noise in Lotka-Volterra dynamics — extinction events and quasi-cycles
Magnetic Skyrmion
Topological spin texture with Dzyaloshinskii-Moriya interaction — skyrmion charge
Adaptive Network Voter Model
Opinion coevolution with network rewiring — fragmentation and consensus transitions
Marchenko-Pastur Law
Spectral density of random covariance matrices — the Marchenko-Pastur distribution
Driven Diffusion NESS
Nonequilibrium steady states in driven diffusion — entropy production and probability currents
Entanglement Area Law
Area-law entanglement entropy in 1D quantum chains and tensor network structure
Quorum Sensing Bistability
Bacterial quorum sensing via autoinducer — bistable switch and collective gene activation
Phonon-Polariton Reststrahlen
Phonon-polariton dispersion in the reststrahlen band — upper and lower branches
Vicsek Flocking Transition
Polar flocking order-disorder transition as a function of noise and density
Fortuin-Kasteleyn Random Cluster
Random cluster model unifying percolation, Ising, and Potts via the FK representation
Kondo Effect Singlet
Magnetic impurity screening — spin-flip scattering and singlet formation below T_K
Abelian Sandpile Identity
Abelian sandpile group identity element — toppling waves and the recurrent identity configuration
Metabolic Flux Analysis
Steady-state flux balance in metabolic networks — stoichiometric constraints and flux cones
XY Model Vortex Annihilation
Topological defect dynamics in the 2D XY model — vortex-antivortex pair annihilation
Directed Percolation
Absorbing-state phase transition in directed percolation — DP universality class critical exponents
Floquet Time Crystal
Period-doubling discrete time crystal in a periodically driven spin chain
Stochastic Resonance
Noise-enhanced detection of subthreshold signals — optimal noise intensity and SNR peak
Protein Aggregation Kinetics
Nucleation-elongation kinetics of protein aggregation — lag phase, seeded growth, and sigmoidal curves
Neural Field Bumps & Waves
Amari neural field equation — stationary bumps, traveling pulses, and Turing instabilities
Spectral Form Factor
Spectral form factor of random matrices — dip, ramp, and plateau structure from GUE/GOE
Capillary Wave Roughness
Thermal fluctuations of fluid interfaces — capillary wave spectrum and interface width divergence
Ostwald Ripening
Lifshitz-Slyozov-Wagner coarsening — droplet size distribution evolving as t^(1/3)
Quantum Discord
Geometric quantum discord in two-qubit states — beyond entanglement quantum correlations
MIPS Active Matter
Motility-induced phase separation — self-propelled particles clustering without attraction
Hamiltonian Monte Carlo
HMC leapfrog integrator for efficient sampling — energy conservation and acceptance rates
Topological Magnon Bands
Chern insulator magnon bands in a 2D honeycomb ferromagnet — edge spin-wave modes
Heteroclinic Cycles RPS
Rock-paper-scissors Lotka-Volterra heteroclinic cycle — cyclic dominance and neutral stability
Topological Flocking
Metric vs topological interaction in flocking — fixed-neighbor alignment and robustness
Wetting & Contact Angle
Young equation and wetting transitions — partial, complete, and Cassie-Baxter regimes
Quantum Eraser
Which-path information and interference erasure — delayed-choice quantum eraser thought experiment
Geodesic Flow on Hyperbolic Surface
Anosov geodesic flow on the hyperbolic plane — exponential mixing and ergodicity
Adaptive SIS Epidemic
Behavior-disease coevolution — awareness-driven contact reduction and epidemic thresholds
Langmuir Plasma Wave
Electron plasma (Langmuir) waves — dispersion relation, plasma frequency, and Landau damping
SBM Planted Partition
Stochastic block model community recovery — Kesten-Stigum threshold and spectral detectability
Granular Solitary Wave
Solitary wave propagation in a granular chain via Hertz contact forces — speed-amplitude relation
Master Stability Function
Network synchronization stability via the master stability function — eigenvalue coupling
Topological Elastic Edge Modes
Topological edge modes in a Kirchhoff plate lattice — mechanical analogue of topological insulators
Hidden Markov Entropy Rate
Entropy rate of hidden Markov processes — Blackwell measure and excess entropy
Kitaev Honeycomb Spin Liquid
Kitaev honeycomb model — exactly solvable spin liquid with Majorana fermions and anyons
Turbulent Boundary Layer
Von Kármán log-law of the wall — velocity profile in turbulent boundary layers
Ecological Stoichiometry
Liebig's law of the minimum in ecological stoichiometry — nutrient co-limitation and growth
Adam-Gibbs Glass Dynamics
Adam-Gibbs theory of glass relaxation — cooperatively rearranging regions and fragility
Chimera State Nonlocal Coupling
Chimera states in nonlocally coupled oscillator rings — coexisting synchrony and incoherence
Jaynes-Cummings Vacuum Rabi
Cavity QED Jaynes-Cummings model — vacuum Rabi splitting and quantum revivals
CFT Primary Correlators
Conformal field theory two- and three-point functions of primary operators — conformal ward identities
2D Navier-Stokes Vorticity
Stream function-vorticity formulation of 2D Navier-Stokes — lid-driven cavity and flow visualization
Moran Process Drift & Selection
Moran process for genetic drift — fixation probability and time under selection
Coulomb Blockade
Single-electron transistor Coulomb blockade — charging energy, stability diagram, and conductance peaks
Levins Metapopulation
Levins metapopulation model — patch colonization-extinction balance and rescue effect
Random Laser Threshold
Disorder-induced light localization and amplification — random laser threshold and mode statistics
Cell Polarity Rac-Rho Waves
Excitable Rac-Rho antagonism in cell polarity — traveling waves and front competition
Phonon Drag Thermoelectric
Phonon drag contribution to the Seebeck coefficient — electron-phonon momentum transfer
Deffuant Bounded Confidence v2
Opinion dynamics with bounded confidence — cluster formation, polarization, and consensus regimes
BKT Superfluid Density Jump
Berezinskii-Kosterlitz-Thouless universal superfluid density jump — Nelson-Kosterlitz relation
Matrix Permanent — Ryser Algorithm
Counting perfect matchings via inclusion-exclusion over subsets in O(2ⁿ·n) time
Quantum Discord
Beyond entanglement: quantum correlations in mixed bipartite states via Ollivier-Zurek discord
Sandpile Odometer Function
Toppling counts during abelian sandpile relaxation — fractal structure from discrete Laplacian
Pyrochlore Ice Rules
Spin ice frustration with two-in two-out ice rules, Pauling entropy, and emergent monopoles
Spectral Flow & Thouless Pump
Chern number as quantized adiabatic charge transport via Rice-Mele spectral flow
Voter Model Coarsening
Opinion domain growth with logarithmically slow interface decay — 1/ln(t) coarsening dynamics
FK Random Cluster Model
Correlated percolation unifying Ising, Potts models via Fortuin-Kasteleyn bond weights
Topological Defects & Homotopy
Vortices and disclinations classified by π₁(M) — XY vortices, nematic disclinations, and winding numbers
Lévy Walk Superdiffusion
Power-law step lengths give ⟨r²⟩~t^α with α>1 — anomalous transport and foraging strategies
Kuramoto Chimera State
Coexisting synchrony and incoherence in nonlocally coupled identical phase oscillators
Hysteresis & Preisach Model
Ferromagnetic memory as a superposition of bistable hysterons — coercivity, remanence, and disorder
Braess Paradox
Adding a zero-cost road increases equilibrium travel time — Wardrop equilibrium on congested networks
Asakura–Oosawa Depletion
Entropic attraction between large colloids mediated by excluded volume of small depletant particles
Complex Contagion — Multiplex
Threshold spreading requiring reinforcement across multiple social network layers — cascade phase transition
Magnon Dispersion — Spin Chain
Spin wave excitations in the 1D Heisenberg ferromagnet — acoustic/optical gap with XXZ anisotropy
Phononic Crystal Band Structure
Elastic wave band gaps in a diatomic mass-spring chain — acoustic and optical branches from Bragg scattering
SBM Community Detection
Stochastic block model with belief propagation — Kesten-Stigum detectability phase transition
Turbulence Intermittency
Multifractal energy dissipation and She-Lévêque anomalous scaling exponents via log-Poisson cascade
Entanglement Entropy — CFT
Calabrese-Cardy logarithmic scaling S=(c/3)ln(ℓ) at criticality — area law violation and central charge
Phase Retrieval — HIO
Recovering an image from Fourier magnitudes alone — Fienup's Hybrid Input-Output algorithm with support constraint
Deffuant Opinion Dynamics
Bounded confidence model — agents only interact with similar neighbours, driving fragmentation or consensus
Vicsek Flocking Transition
Self-propelled particles align with neighbours — noise drives a kinetic order–disorder transition with φ order parameter
Quantum Battery & Ergotropy
Driven two-level system tracking energy, ergotropy, and coherence — Rabi oscillations on the Bloch sphere
Sinai Billiard Chaos
Square table with circular scatterer — first rigorously proven chaotic billiard with exponential trajectory divergence
Neural Field: Bumps & Waves
Wilson–Cowan continuum model with Mexican-hat kernel — stationary bumps, oscillations, and travelling fronts
BCF Crystal Growth
Burton–Cabrera–Frank spiral growth from screw dislocations — parabolic-to-linear rate crossover with supersaturation
Metapopulation Epidemic
SIR model on cities linked by gravity-law mobility — epidemic spread through hub-and-spoke travel networks
GOE Spectral Rigidity
Random matrix level-spacing statistics — Wigner surmise vs Poisson, level repulsion and long-range rigidity
MIPS Active Matter
Motility-Induced Phase Separation — self-propelled particles cluster purely from activity with no attractions
CFT Conformal Maps
Conformal transformations of complex grids and primary operator two-point functions with scaling dimension Δ
Prion Nucleation
Protein aggregation via nucleation-polymerization kinetics — lag phase, seeded growth, and sigmoidal fibril formation
Marchenko–Pastur Law
Eigenvalue spectral density of large random covariance matrices — separating signal from noise in PCA and finance
Majorana Hybridization
Kitaev chain BdG spectrum with Majorana end states — topological phase boundary and zero-mode splitting by overlap
Annealed Disorder Landscape
Particles on a fluctuating random energy landscape — glass transition at T_g = σ/√(2 ln 2) and Langevin dynamics
Kelvin–Helmholtz Roll-up
Shear layer instability via point vortex sheet — Biot-Savart roll-up into cat's-eye structures
Jaynes–Cummings Revivals
Vacuum Rabi oscillations, collapse, and quantum revivals in atom-cavity coupling with coherent photon states
Network Motifs: Bi-fan
Overrepresented subgraph patterns in regulatory networks — feedforward loops and bi-fan motifs with Z-score statistics
Lévy Walk & Aging
Anomalous superdiffusion and CTRW subdiffusion — non-ergodic aging with MSD exponent γ ≠ 1
Trophic Cascade
Three-level predator–herbivore–plant dynamics — remove the keystone wolf and watch the cascade collapse
Magnon BEC
Spin wave texture, quadratic magnon dispersion, and Bose–Einstein condensation at k=0 below T_c
Explosive Percolation
Achlioptas product rule delays giant component formation, triggering abrupt near-discontinuous transition
Quantum Otto Engine
Two-level qubit heat engine cycling through isentropic and isochoric strokes — efficiency vs Carnot bound
Protein Folding Funnel
Langevin dynamics on a rugged funnel landscape — kinetic traps, misfolding, and the native state basin
Traffic Flow — Nagel-Schreckenberg
Cellular automaton model of highway traffic — phantom stop-and-go jams emerge from stochastic dawdling
Random Walk Winding Number
Spitzer's theorem: 2D Brownian winding around a pole scales as √(½ log T) — polymer topological entanglement
Rich-Club Network Coefficient
Do high-degree hubs preferentially interconnect? ρ(k) measures the density of edges among top-degree nodes
Dark Soliton — BEC
Gross-Pitaevskii 1D: density notch with π phase jump propagating through a Bose-Einstein condensate
Frustrated Jammed Packing
Bidisperse disk packing near jamming — harmonic contact forces, force chains, and geometric frustration
Hyperbolic Tiling — Poincaré Disk
{p,q} tessellation of H² via successive geodesic reflections — interactive Schläfli symbol explorer
Epidemic Reinfection — SIRS
Waning immunity drives R→S reinfection; damped oscillations spiral toward the endemic equilibrium I*
Quantum Discord & Backaction
Non-classical correlations beyond entanglement — projective measurement collapses joint state, demonstrating measurement backaction on coupled qubits
Stochastic Resonance
Noise-enhanced signal detection in a bistable double-well: SNR peaks at optimal noise level, revealing resonance between signal and thermal fluctuations
Coupled Map Lattice
Spatiotemporal chaos via logistic maps on a 1D lattice — space-time patterns reveal traveling waves, synchronized chaos, and defect turbulence
Parallel Transport & Holonomy
Geometric phase on a curved sphere — a parallelly transported vector returns rotated by the solid angle enclosed by its loop (Berry phase analog)
Affinity Maturation
B-cell Darwinian evolution in germinal centers — somatic hypermutation and affinity-weighted selection optimize antibodies in antibody shape space
Mean-Field Game: Pedestrian
Nash equilibrium crowd routing — coupled HJB and Fokker-Planck PDEs govern rational agents minimizing travel time plus congestion cost
Viscous Fingering
Saffman-Taylor instability in radial Hele-Shaw injection — surface tension selects finger width; DLA-like growth with curvature-dependent sticking
Topological Phonon Valley Hall
Protected acoustic edge states in a honeycomb mass-spring lattice — broken inversion symmetry opens a topological bandgap at K/K' Dirac points
Smoluchowski Gelation
Coagulation-fragmentation kinetics with product/sum/constant kernels — watch gelation transitions and scaling exponents τ in real time
Adaptive Network Rewiring
Co-evolving opinion dynamics — the adaptive voter model's rewiring vs. copying competition drives a fragmentation transition at critical φ_c
Kuramoto Order Parameter
Coupled oscillators synchronize above a critical coupling K_c — watch the complex order parameter r·e^{iψ} transition from incoherence to collective synchrony.
Shilnikov Spiral Chaos
When a saddle-focus equilibrium has a homoclinic orbit and saddle quantity ρ<1, Shilnikov's theorem guarantees chaotic spiral dynamics with infinite periodic orbits.
Kitaev Honeycomb Anyons
Click plaquettes on Kitaev's exactly-solvable honeycomb model to create Z₂ flux visons — topological quasiparticles with non-Abelian braiding statistics.
Jamming Transition
Compress soft spheres to witness the jamming transition at φ_J ≈ 0.64 — the sudden onset of rigidity and power-law contact forces in amorphous packings.
Sandpile Identity
The abelian sandpile's fractal identity element emerges from chip-firing dynamics — a self-similar object living at the heart of self-organized criticality.
Quantum Spin Liquid
Geometric frustration on the triangular antiferromagnet prevents ordering — quantum fluctuations stabilize a spin liquid with fractionalized spinon excitations.
Landau Damping
Resonant particles absorb energy from plasma waves in this collisionless Landau damping simulation — watch velocity-space trapping and wave energy decay.
Stochastic Block Model
Belief propagation recovers hidden community structure in SBM random graphs — near the Kesten-Stigum threshold, detection becomes information-theoretically hard.
Rule 110 CA
Rule 110 is Turing-complete — its spacetime diagram shows gliders, periodic regions, and complex Class IV behavior arising from a 3-cell lookup table.
Basset History Force
The non-Markovian Basset force encodes a particle's entire velocity history via a t^{-1/2} memory kernel, creating anomalous drag in unsteady viscous flow.
Rayleigh-Bénard Convection
Hot plumes rise and cold fingers sink as buoyancy drives turbulent thermal convection above a critical Rayleigh number.
Coherent & Squeezed States
Wigner function phase-space portraits of quantum harmonic oscillator states — from coherent Gaussians to squeezed uncertainty ellipses.
Active Brownian Particles
Self-propelled particles undergo motility-induced phase separation — dense clusters form spontaneously without any attractive forces.
Parisi Replica Symmetry Breaking
SK spin glass Monte Carlo reveals the broad overlap distribution P(q) that signals the spin glass phase and ultrametric energy landscape.
Bianconi-Barabási Fitness Network
Preferential attachment with node fitness drives Bose-Einstein condensation on networks — one fit node captures a finite link fraction.
Fractional Diffusion & Lévy Walks
Anomalous diffusion with power-law waiting times (subdiffusion) and Lévy step distributions (superdiffusion) breaks the mean-square displacement law.
Autocatalytic Reaction Networks
Random autocatalytic sets undergo a phase transition to self-sustaining flux far from equilibrium — a model for the origin of life.
XY Model Vortex Annihilation
Topological vortex-antivortex pairs bind and annihilate through the BKT transition in the 2D XY model.
Vacuum Rabi Splitting
Jaynes-Cummings dynamics reveal vacuum Rabi oscillations and the characteristic doublet in the cavity transmission spectrum.
Pollinator Mutualism Networks
Nested plant-pollinator bipartite networks show how generalist structure confers robustness — and how generalist loss cascades.
KLS Driven Lattice Gas
Katz-Lebowitz-Spohn nonequilibrium lattice gas with anisotropic field-driven currents and phase separation.
Quantum Decoherence & Einselection
Wigner function of a Schrödinger cat state decohering — pointer states survive while superpositions vanish.
Neural Gamma Oscillations
Interneuron network gamma (ING): mutual inhibition among 40 LIF neurons generates cortical gamma rhythms.
Adhesion Cluster — Bell Model
Molecular bond cluster under force: Bell-model rupture kinetics and cooperative catastrophic detachment.
Traveling Salesman: 2-opt Annealing
2-opt local search with simulated annealing untangles TSP tours — watch the route progressively optimize.
Expander Graphs & Ramanujan Bound
Random walk mixing on sparse graphs; spectral gap measures expansion with the Ramanujan optimal bound.
Parametric Resonance — Mathieu Tongues
Mathieu equation instability tongues in (δ, ε) space — parametric resonance drives exponential amplitude growth.
Mott Insulator — Hubbard Gap
1D Hubbard chain: strong repulsion opens a Mott gap; doublon-holon pair dynamics visualized with Hubbard bands.
Epidemic–Opinion Coevolution
SIR epidemic coupled with voter-model opinions on an adaptive network — rewiring feeds back on disease spread.
Viscoelastic Network Relaxation
2D spring network with dynamic crosslinks shows Bell-model stress relaxation characteristic of biopolymer gels.
Geodesic Flow & Hyperbolic Chaos
Anosov geodesic flow on a compact hyperbolic surface in the Poincaré disk — the archetypal uniformly chaotic system with positive Lyapunov exponent.
Quantum Feedback Control
Continuous weak measurement of a qubit induces stochastic backaction on the Bloch vector; a feedback controller competes with decoherence to stabilize the target state.
Colloidal Gel & Arrested Phase Separation
Short-range attractive colloids undergo spinodal decomposition arrested by gelation — a percolating network forms near the gel transition threshold.
Neutral Drift & Tajima's D
Wright-Fisher drift, selection, and bottlenecks shape allele frequency trajectories and the site-frequency spectrum; Tajima's D detects departures from neutrality.
Magnon-Polariton Strong Coupling
A ferromagnetic magnon mode inside a microwave cavity hybridizes into magnon-polaritons — the avoided crossing (vacuum Rabi splitting) is tuned by external magnetic field.
Stochastic Lotka-Volterra Extinction
Demographic noise drives eventual extinction from the quasi-stationary predator-prey limit cycle; mean extinction time grows exponentially with system size.
Network Controllability & Driver Nodes
Liu et al. structural controllability: minimum driver node set found via maximum matching — scale-free networks require surprisingly many driver nodes.
Flapping Flag & Vortex Shedding
A flexible flag in uniform flow undergoes flutter instability, shedding alternating vortices in a von Kármán street governed by bending rigidity and mass ratio.
Chern-Simons Torus Knots & Link Invariants
(p,q)-torus knots wind around the torus; linking number and writhe are classical precursors to the Jones polynomial from Chern-Simons quantum field theory.
Information Scrambling & OTOC Chaos Bound
Out-of-time-order correlators (OTOCs) measure quantum information scrambling; the MSS bound λ ≤ 2πkT/ℏ is saturated only by black holes and the SYK model.
Wigner-Dyson Level Statistics
Random matrix spectra reveal the difference between quantum chaotic (GOE/GUE level repulsion) and integrable (Poisson clustering) systems via nearest-neighbor spacing distributions.
Active Nematic Defect Coarsening
Driven cytoskeletal filaments spontaneously generate ±½ topological defects that annihilate in pairs as the director field coarsens over time.
Contact Process & Directed Percolation
The 1D contact process displays an absorbing-state phase transition in the directed percolation universality class — activity survives above λ_c ≈ 3.298 and dies below it.
May's Complexity-Stability Criterion
Robert May showed random ecosystems destabilize when σ√(SC) > 1, overturning the diversity-stability intuition and founding theoretical community ecology.
Quantum Walk: Decoherence & Localization
A coined quantum walk spreads ballistically (σ~t) until decoherence drives diffusion or disorder induces Anderson localization, exponentially confining the wavepacket.
Vicsek Flocking Transition
Self-propelled particles align with neighbors subject to noise, exhibiting a genuine non-equilibrium phase transition to polar order that defies the Mermin-Wagner theorem.
Thermodynamic Uncertainty Relation
The TUR bounds current precision by dissipation: Var(J)/⟨J⟩² ≥ 2/Σ — reducing fluctuations always costs free energy, verified here on a driven 3-state Markov network.
Hopfield Network Memory & Capacity
Hopfield networks store patterns as energy minima with capacity α_c ≈ 0.138N; beyond this threshold retrieval fails and spurious mixture states dominate.
Thouless Pump — Rice-Mele Model
Adiabatic cycling of the Rice-Mele chain's staggered hopping and potential pumps exactly one electron per cycle — a topological Chern number measured in ultracold atom experiments.
Smoluchowski Coagulation & Gelation
With multiplicative kernel K(i,j)=ij, Smoluchowski's coagulation equation undergoes a gelation transition at finite time t_gel=1, followed by power-law cluster distributions c_k ~ k^{-5/2}.
Nonlinear Mode Coupling — Internal Resonance
Two coupled Duffing oscillators with a 1:3 frequency ratio exhibit internal resonance, enabling intermittent energy transfer between modes impossible in linear systems.
Topological Magnon Hall — Kagome Lattice
Magnon bands on a kagome ferromagnet acquire Berry curvature from DM interactions, producing a thermal Hall effect with non-zero Chern numbers.
Voter Model Interface Coarsening (1D)
Opinion domains coarsen algebraically as interfaces random-walk and annihilate, yielding interface density ρ(t) ~ t⁻¹/² — diffusive coarsening without surface tension.
Quantum Discord & Separability — Two Qubits
Quantum discord measures correlations beyond entanglement; even separable mixed states can have nonzero discord, visualized via two-qubit X-state Bloch spheres and the PPT criterion.
Microfluidic Droplet Formation
Capillary number Ca governs the transition from squeezing to dripping to jetting regimes at a T-junction, with periodic droplet breakup driven by Plateau-Rayleigh instability.
Ising Model — Sweeny & Wolff Cluster Algorithms
The Sweeny algorithm updates FK random-cluster bonds to reduce critical slowing-down near Tc, while the Wolff algorithm flips entire correlated clusters for efficient critical sampling.
Giant Component Emergence — Erdős–Rényi
As mean degree crosses 1 in G(n,p), a giant connected component containing O(n) vertices suddenly appears — a sharp percolation-like phase transition in random graph theory.
Phononic Band Gap — Locally Resonant Metamaterial
Local resonators create sub-Bragg band gaps by coupling elastic waves to heavy masses on soft springs, producing negative effective mass density that forbids wave propagation.
Bunimovich Stadium — Chaotic Billiard
The stadium billiard is a rigorously proven chaotic system where trajectories exhibit exponential sensitivity and ergodic mixing, filling phase space uniformly unlike integrable circles.
Tumor Growth — Gompertz Law & Nutrient Diffusion
Gompertz growth kinetics emerge from nutrient diffusion limits, producing a proliferating rim, quiescent region, and necrotic core whose geometry depends on vessel density.
Bloch Oscillations & Wannier-Stark Ladder
A wavepacket in a tilted periodic potential oscillates with period T_B=ℏ/(eFa); the Wannier-Stark ladder shows energy levels E_n = E₀ + neFa equally spaced by the field.
Quantum Metric Tensor & Berry Curvature
The quantum geometric tensor Q=g+iΩ/2 encodes Bloch state geometry; visualize the quantum metric and Berry curvature over the Brillouin zone in a 2-band model.
Topological Magnon Hall Effect
DM interaction in a honeycomb ferromagnet opens a topological magnon gap, yielding nonzero Chern number and chiral edge magnon modes in the ribbon spectrum.
Stochastic Epidemic Extinction & QSD
The stochastic SIS model near R₀=1 exhibits a quasi-stationary distribution before extinction; mean extinction time grows exponentially with population size N.
PT-Symmetric Exceptional Point
Balanced gain-loss coupled oscillators show real eigenvalues (PT-unbroken) vs. complex eigenvalues (broken phase), coalescing at the exceptional point where eigenvectors merge.
Lévy Process via Subordination
A Lévy process X(t)=B(T(t)) is built by time-changing Brownian motion with a stable subordinator; the result has heavy-tailed increments and anomalous diffusion.
KLS Driven Lattice Gas
The Katz-Lebowitz-Spohn model shows nonequilibrium phase separation into field-aligned stripes driven by an external field — a transition with no equilibrium analog.
Intrinsic Spin Hall Effect — Rashba Model
Berry curvature of Rashba-split bands yields an intrinsic spin Hall conductivity — a purely geometric, disorder-free contribution from band topology.
Rule 110 — Turing Complete Automaton
Rule 110 is the simplest known Turing-complete system; despite a trivially local rule, it supports gliders and particle collisions enabling universal computation.
Kondo Resonance — Anderson Impurity
Below the Kondo temperature T_K, a sharp many-body resonance appears in the impurity spectral function pinned at the Fermi level, narrowing exponentially with temperature.
Turbulence Intermittency & Multifractal Cascade
Kolmogorov's K41 turbulence is not self-similar — energy dissipation concentrates in fractal regions. Explore structure functions, singularity spectrum f(α), and synthetic multifractal velocity fields.
Bacon-Shor Subsystem Code
The Bacon-Shor [[9,1,3]] code encodes one logical qubit in a 3×3 grid using gauge operators. Click qubits to inject X, Z, or Y errors and watch syndrome measurement decode and correct them.
Contact Process — Directed Percolation
The contact process on a 1D lattice undergoes a continuous phase transition at λ_c/δ ≈ 3.298 in the directed percolation universality class, with critical exponents β≈0.277 and ν_⊥≈1.097.
Vortex Ring Reconnection — Biot-Savart
Two vortex rings evolve under the Biot-Savart law and undergo topology-changing reconnection when filaments approach within the vortex core size — a key mechanism in turbulence and superfluids.
Interacting Boson Model — Nuclear Shape Phases
The IBM maps the U(5)↔SU(3)↔O(6) symmetry triangle of nuclear shapes — spherical vibrator, axial rotor, and γ-unstable. Explore energy spectra and R₄/₂ ratios across the triangle.
Photon Blockade — Jaynes-Cummings
Strong atom-cavity coupling g creates anharmonic polariton rungs, blocking a second photon once one is present. Visualize the dressed-state ladder, g²(0) antibunching, and photon number distribution.
Sandpile Group Identity Element
The abelian sandpile model on an N×N grid forms a group under addition and toppling. The identity element — computed by stabilizing the all-6 configuration — reveals stunning fractal self-similar patterns.
Chirikov Standard Map — KAM Tori & Chaos
The standard map transitions from integrable KAM tori to global chaos at k_c≈0.9716 (Greene's criterion). Click the phase portrait to seed orbits and watch the chaotic sea grow with kick strength.
Hubbell's Unified Neutral Theory
Hubbell's neutral theory assumes ecological equivalence and predicts log-series species abundance distributions via drift, speciation, and immigration. Explore Preston plots and Shannon diversity as θ and m vary.
Nematic Disclination Dynamics — Active Matter
Active nematics spontaneously nucleate ±½ disclination pairs and drive chaotic flows. Watch the director field evolve, defects annihilate, and activity α control the density of topological charge.
Jamming Transition
Soft sphere packings at Point J: pressure vs φ, contact number Z, and visualization of particle contacts at/below/above the jamming transition.
Pair Approximation — SIS Epidemics
Pair approximation closure for SIS epidemics on networks; compare mean-field, pair approx, and stochastic simulations with phase diagram accuracy.
Kagome Antiferromagnet
Geometric frustration on the kagome lattice: perfectly flat magnon band, macroscopic degeneracy, and spin-liquid indicators from frustrated triangles.
Shilnikov Chaos
Homoclinic orbit chaos near a saddle-focus fixed point; 3D phase portrait, Poincaré return map, and the Shilnikov condition |ρ|<|λ| for chaos.
Nucleated Polymerization
Oosawa model for amyloid/actin assembly: lag phase, critical nucleus size, monomer depletion, and elongation kinetics with interactive phase diagram.
Random Field Ising — Imry-Ma
Imry-Ma argument: random fields destroy long-range order for d≤2; domain size vs disorder strength and lower critical dimension visualized by Monte Carlo.
Cavity QED — Vacuum Rabi Splitting
Strong light-matter coupling: vacuum Rabi doublet in transmission, polariton branches, Jaynes-Cummings ladder, and strong vs weak coupling regimes.
Quantum Annealing — Adiabatic Gap
Adiabatic quantum computing via transverse-field Ising; spectral gap closing, Landau-Zener excitation probability, and annealing schedule optimization.
FPUT Recurrence & Breathers
Fermi-Pasta-Ulam-Tsingou recurrence in anharmonic lattices: energy localization in normal modes, discrete breathers, and the famous recurrence phenomenon.
Flux Qubit Double-Well
RF-SQUID flux qubit potential U(φ)=E_L(φ-2πf)²/2−E_J·cos(φ); energy levels vs flux, tunnel splitting, avoided crossing at Φ=Φ₀/2, and wavefunctions.
Ergodic Theory & Mixing
Baker map and Arnold cat map iteration on the torus: watch phase space mixing, ergodic averages converging to space averages, and measure Lyapunov exponents from exponential orbit divergence.
Keller-Segel Chemotaxis & Blowup
Cell density aggregation driven by self-secreted chemoattractant. Above the critical mass 8π, the PDE solution blows up in finite time; toggle saturation to prevent collapse.
Quantum Chaos Level Statistics
Nearest-neighbor spacing distributions distinguish integrable (Poisson) from chaotic (GOE/GUE Wigner-Dyson) quantum systems. Sample random matrices and compare via the unfolded spectrum and ratio statistics.
Arnold Tongues & Circle Map
Frequency-locking Arnold tongue regions in (Ω, K) parameter space of the circle map. Explore the devil's staircase at K=1 and the onset of chaos above K=1 by clicking any point in the map.
Active Nematics & Defect Dynamics
Activity-driven instability in a nematic director field spawns ±½ topological defects. The self-propelled +½ defects and stationary −½ defects continuously annihilate and recreate in this living liquid crystal model.
Glass Transition — Mode Coupling Theory
Two-step relaxation of F(q,t) in supercooled liquids: β-regime cage rattling, von Schweidler decay, and slow α-relaxation with KWW stretched exponential. Relaxation time diverges as power law near Tc.
Stochastic Thermodynamics & Brownian Engine
A colloidal Brownian heat engine cycling between hot and cold reservoirs. Efficiency distributions across many stochastic trajectories, with Carnot and Curzon-Ahlborn bounds visualized.
Atiyah-Singer Index & Heat Kernel
The index theorem equates analytical index ind(D) = dim ker D − dim ker D† to a topological invariant. Visualize spectra of D†D and DD†, zero modes, and the heat kernel supertrace converging to ind(D) for all t.
Coevolutionary Voter Model
Opinion dynamics coupled to network rewiring: homophily competes with social influence. Above a critical rewiring probability, the network fragments into polarized echo chambers — a fragmentation transition.
Explosive Percolation — Achlioptas Process
The product rule Achlioptas process suppresses giant component growth until a sudden explosive transition, far sharper than random Erdős-Rényi percolation. Compare rules and watch susceptibility spike at the critical threshold.
Floquet Anomalous Edge States
Periodically driven 2D lattice hosting chiral edge modes with no bulk analog — the hallmark of Floquet topological phases where all bulk Chern numbers vanish yet robust edge states traverse the full quasi-energy zone.
Deffuant-Weisbuch Opinion Dynamics
Continuous opinion dynamics with bounded confidence: agents interact only when their opinions differ by less than ε, converging to discrete clusters. Explore how tolerance ε controls the number of opinion clusters and the route to consensus.
Outer Totalistic 2D Cellular Automaton
Generalized Game of Life with configurable birth and survival rules. Explore the vast rule space of outer totalistic automata — from stable still lifes and oscillators to chaotic growth and self-replication.
Coined Quantum Walk on the Line
A quantum particle walks with a Hadamard coin, spreading ballistically (σ~t) rather than diffusively (σ~√t). Observe interference fringes, twin probability peaks, and entanglement entropy growth as the walk evolves.
Elastic Turbulence in Viscoelastic Flow
Polymer solutions develop chaotic turbulent flow at vanishingly small Reynolds number driven purely by elastic stresses. Above a critical Weissenberg number, the Oldroyd-B instability produces a k⁻³·⁵ energy power spectrum.
Sznajd Model — Outflow Dynamics
When two neighboring agents agree, they convince their neighbors — outflow dynamics that drives societies toward consensus or fragmentation. Explore the sharp exit probability transition and the 1D vs 2D phase diagrams.
Fisher-KPP Traveling Wave
Reaction-diffusion fronts in the Fisher-KPP equation select a minimum propagation speed c* = 2√(rD). Watch pulled fronts lock to c* from steep initial conditions and explore how diffusivity and growth rate shape the wave profile.
Cavity Optomechanics Sideband Cooling
A mechanical oscillator coupled to an optical cavity can be cooled to its quantum ground state by driving on the red sideband. Watch phonon number decay, compare red vs blue sideband effects, and see the quantum cooling limit.
Bootstrap Percolation & k-Core Pruning
Iterative removal of nodes with fewer than k active neighbors reveals a discontinuous first-order transition: below a seed density threshold almost nothing survives, above it a macroscopic core persists. Models cascading failures and contagion thresholds.
Quantum Critical Point — Transverse Ising
The transverse-field Ising model has an exact quantum phase transition at h/J = 1 described by a c = 1/2 CFT. Explore gap closing, correlation length divergence, dynamic critical exponent z = 1, and power-law correlators C(r) ~ r^{-1/4}.
Free Boson CFT — Operator Product Expansion
The free boson CFT with central charge c=1. Visualize correlation functions of primary vertex operators V_α, the stress tensor OPE T(z)T(w), conformal dimensions on the cylinder, and the OPE structure constant map.
Age-Structured SEIRD Epidemic Model
Age-stratified SEIRD dynamics with heterogeneous contact matrices and age-dependent infection fatality rates. Track attack rates and mortality by age cohort, comparing homogeneous, assortative, and work-dominated mixing patterns.
Bose-Hubbard Mott Insulator Phase Diagram
Mean-field phase diagram of the Bose-Hubbard model showing Mott insulator lobes (n=1,2,3) in the (J/U, μ/U) plane. Explore the superfluid-insulator quantum phase boundary, Mott lobe tips, and coordination number dependence.
Trophic Cascade — Food Web Dynamics
Tri-trophic Lotka-Volterra system: wolves, elk, and vegetation. Remove predators and witness the trophic cascade — herbivore population explosion followed by vegetation collapse — with real-time phase portrait.
Coupled Map Lattice — Chimera States
Logistic maps with nonlocal coupling exhibit chimera states: coexisting coherent (synchronized) and incoherent (chaotic) spatial domains. Space-time plots reveal the spontaneous symmetry breaking of chimera order.
Semiflexible Polymer — WLC Buckling
Worm-like chain model with persistence length under compressive and stretching forces. Visualize Euler-Bernoulli buckling, thermal fluctuation modes, and the Marko-Siggia force-extension curve for semiflexible biopolymers.
Brownian Motor — Fluctuation Rectification
Büttiker-Landauer ratchet: particles in an asymmetric sawtooth potential driven by two thermal reservoirs at different temperatures. Directed transport emerges from broken symmetry — track current reversal as temperature ratio varies.
Suprathreshold Stochastic Resonance
An array of threshold detectors processes sub-threshold signals in independent noise. Summing binary outputs boosts SNR at an optimal noise amplitude — noise assists detection even when no single threshold is crossed.
Watts Threshold Cascade — Global Cascades
On a random network each node adopts when a fraction φ of neighbors already have. The (z, φ) phase diagram reveals the cascade window where global cascades propagate from a single seed — with live network simulation.
Topological Surface States — Rashba Spin Texture
Dirac cone surface states of a topological insulator with helical spin-momentum locking. Explore the Fermi surface spin texture, Berry phase π, density of states, and backscattering suppression — the hallmark of topological protection.
Ricci Flow Surface Uniformization
Hamilton's curvature-driven metric evolution uniformizing a 2D surface toward constant Gaussian curvature.
Wigner Crystal — Low-Density 2DEG
Monte Carlo simulation of Coulomb-driven electron crystallization in a 2D electron gas at large r_s parameter.
Phonon Polariton Hyperbolic Dispersion
Light-phonon coupling in polar crystals with Reststrahlen-band hyperbolic isofrequency contours (hBN, α-MoO₃).
Chiral Edge Mode — Quantum Hall Bar
Topologically protected chiral edge states carrying quantized Hall conductance σ_xy = νe²/h in integer QHE.
Subdiffusive CTRW Lévy Subordination
Continuous-time random walk with heavy-tailed waiting times giving anomalous MSD ~ t^α subdiffusion.
Magnon Dispersion — Ferromagnetic Chain
Spin-wave dispersion ω(k) = 2JS(1−cos ka) + 2DS in a 1D Heisenberg ferromagnet with anisotropy gap.
Kondo Impurity Resonance
Many-body Abrikosov-Suhl resonance at the Fermi level from magnetic impurity spin-flip scattering in metals.
Floquet-Bloch Photonic Drive
Quasi-energy bands and photon sidebands under periodic driving, with Floquet gap openings at zone boundaries.
Entanglement Spectrum Topological Phase
Li-Haldane entanglement spectrum reveals topological in-gap modes as fingerprint of SSH and Kitaev chains.
Keldysh Nonequilibrium Green Function
Current and spectral flow through a quantum dot under bias via Keldysh lesser Green function formalism.
Topological Defect Nematic Annihilation
±½ disclination pair dynamics and annihilation in a 2D nematic liquid crystal director field.
Ott-Antonsen Chimera Ansatz
Exact mean-field reduction showing chimera states — coexisting synchronized and incoherent oscillator domains.
Sierpinski Gasket Harmonic Analysis
Eigenfunctions of the Laplacian on a fractal — spectral dimension d_s ≠ Hausdorff dimension.
Vicsek Polar Active Matter Transition
Flocking transition in self-propelled particles — discontinuous ordering and spontaneous symmetry breaking.
KPZ Stochastic Interface Growth
Kardar–Parisi–Zhang equation: roughness exponent β=1/3 and Tracy-Widom fluctuations in 1+1D.
Josephson Junction Phase Qubit
RCSJ model: supercurrent to voltage-state transition, Shapiro steps, and AC-driven chaos.
Abelian Higgs Vortex Lattice
Type-II superconductor Abrikosov flux lattice — order parameter phase winding and magnetic field penetration.
Rényi Entropy & Mutual Information
Order-α Rényi entropy, bipartite mutual information, and information geometry of joint distributions.
Calogero–Moser Lax Pair Integrability
Exactly integrable N-particle system with inverse-square repulsion — conserved Lax eigenvalues and non-crossing trajectories.
Born–Oppenheimer Conical Intersection
Adiabatic surfaces touching at a conical intersection — geometric Berry phase π and non-adiabatic coupling.
Z₂ Quantum Spin Liquid Gauge Flux
Kitaev toric code: e and m anyons from vertex charge and plaquette flux operators, topological order.
Luttinger Liquid Charge-Spin Separation
1D fractionalization: holon and spinon modes at different velocities with power-law correlations.
Dyson Brownian Motion Eigenvalue Repulsion
Random matrix Coulomb gas dynamics — Wigner surmise spacing distribution, GOE/GUE/GSE universality.
Fracton X-Cube Model Mobility
Subdimensional topological order: fractons are immobile, lineons move in 1D, planons in 2D planes.
Non-Hermitian Exceptional Point EP3
Third-order EP: three eigenvalues and eigenvectors coalesce with Riemann sheet branch-cut topology.
Quantum Metric Tensor (Fubini-Study)
Geometry of quantum state space — QGT unifies Fubini-Study metric and Berry curvature on the Bloch sphere.
Topological Magnon Hall Effect
DMI-induced Berry curvature in magnon bands of a honeycomb ferromagnet — Hall conductivity and Chern number.
Amorphous Topological Insulator
Edge states surviving structural disorder — topology without crystalline order, Anderson localization limit.
Floquet Time Crystal
Discrete time-translation symmetry breaking in MBL spin chains — period-2 oscillations under periodic driving.
Random Circuit Sampling Anticoncentration
Porter-Thomas output distribution from deep random quantum circuits — XEB score and quantum supremacy.
Holographic Complexity Action Growth
CA conjecture: complexity = WDW-patch action in AdS-BH — Lloyd's bound dC/dt = 2M/ℏ visualized.
Parafermion Zₙ Clock Model
Clock spins with Zₙ symmetry — parafermion domain walls, three-phase diagram, and clock MC dynamics.
Meron Lattice & Skyrmion Textures
Topological spin solitons from DMI: skyrmions (Q=+1), merons (Q=+½), antiskyrmions (Q=−1) and charge density.
Stochastic Thermodynamics: Jarzynski–Crooks
Free energy from nonequilibrium work: Jarzynski equality ⟨e⁻βW⟩=e⁻βΔF and Crooks fluctuation theorem.
Sachdev-Ye-Kitaev: Four-Point OTOC
SYK model quantum chaos: out-of-time-order correlator decay, Lyapunov exponent and MSS scrambling bound.
Weyl Semimetal Fermi Arc Surface States
Topological Weyl nodes with opposite chirality connected by open Fermi arcs in the surface BZ — chiral anomaly.
Bosonization: Luttinger Liquid 1D
1D fermions bosonized into plasmons — Luttinger parameter K, spin-charge separation and power-law correlations.
CFT Minimal Models M(p,p')
Virasoro algebra, Kac table of conformal weights, two-point functions on cylinder — Ising to Yang-Lee series.
Quantum Darwinism & Pointer Basis
Classical objectivity from quantum redundancy: environment fragments encode pointer states, mutual info plateau.
Spin-Boson Model: Ohmic Bath & Kondo
Two-level system in Ohmic bath — quantum phase transition at α=1, Kondo mapping, renormalized tunneling collapse.
Moiré Flat Bands: Twisted Bilayer
Magic angle ~1.1° collapses graphene bandwidth — moiré supercell, flat bands, correlated phases at ν=±2.
Anyons: Fractional Statistics & AB Phase
Flux-charge composites in 2D: braiding phase e^{iθ}, Aharonov-Bohm effect, FQHE Laughlin anyons and anyon zoo.
Rindler Wedge & Gravity Dual
Accelerated observers see Unruh thermal radiation — Rindler horizon, entanglement entropy = Bekenstein area law.
p-Brane Tension & Mass Spectrum
String theory extended objects: T_p ~ 1/g_s, BPS stability, S/T-duality web connecting D-branes and M-theory.
Quantum Spin Liquid: Z₂ Gauge Flux
Kitaev honeycomb model with vison π-flux plaquettes — emergent anyons, A/B phase boundary, toric code physics.
Dyson Brownian Motion: Eigenvalue Repulsion
Random matrix eigenvalue SDE with log-gas repulsion — level spacing histogram converges to Wigner surmise.
Non-Abelian Berry Phase: Wilczek-Zee
Degenerate quantum subspace: SU(2) holonomy matrix accumulates under adiabatic cyclic evolution on the Bloch sphere.
SYK Model: Low-Energy Schwarzian
Sachdev-Ye-Kitaev random quartic fermions — spectral density, GUE ramp+plateau SFF, and Schwarzian soft-mode action.
Laughlin Quasihole: Plasma Analogy
FQH ν=1/m: quasihole fractional charge e/m emerges from 2D one-component plasma screening at inverse temperature β=2m.
MERA Tensor Network: Holography
Multi-scale entanglement renormalization — causal cones grow logarithmically, minimal cuts give S~c/3·log(ℓ) Ryu-Takayanagi.
Batchelor Scale: Passive Scalar Mixing
Scalar variance cascade in turbulence — η_B=η·Sc^{-1/2} Batchelor scale, k^{-1} viscous-convective spectrum, filament dynamics.
Quantum Geometry: Bloch Band Metric
Fubini-Study metric tensor and Berry curvature in k-space — Chern number, quantum metric integral bounds Wannier spread.
Stochastic Gravity: Metric Fluctuations
Einstein-Langevin equation with quantum stress-tensor noise ξ_{μν} — metric perturbation wave equation with ℏ-scale stochastic driving.
Quantum Error Correction: Repetition Code
3-qubit bit-flip code — syndrome measurement, majority-vote correction, logical error rate p_L≈3p² vs physical p threshold.
Holographic Weyl Anomaly
AdS3/CFT2: Brown-Henneaux c=3l/2G_N, conformal factor backreaction, stress tensor two-point function G~c/x^4.
Andreev Bound States: Topological Junction
Josephson junction ABS E=±Δ√(1-T·sin²(φ/2)) — topological 4π-periodic Majorana zero modes vs conventional 2π supercurrent.
Quantum Chaos: Brody Distribution
Energy level spacing statistics interpolating between Poisson (integrable β=0) and Wigner-Dyson (chaotic β=1) via the Brody parameter.
Kitaev Chain: Majorana Nanowire
1D p-wave topological superconductor — BdG spectrum, topological phase transition at |μ|=2t, and zero-energy Majorana edge modes.
Hyperbolic Lattice: Poincaré Disk Tiling
{p,q} tilings of negatively curved space in the Poincaré disk model — exponential volume growth, geodesic arcs, hyperbolic vs Euclidean regimes.
Weyl Orbit: Fermi Arc Oscillations
Topological Fermi arcs on Weyl semimetal surfaces — Weyl orbit quantum oscillations F∝k₀·d, real-space trajectory through bulk and surfaces.
DFT: Kohn-Sham Exchange-Correlation
Self-consistent radial DFT for atoms Z=1–18 — LDA Perdew-Zunger Vxc, GGA-PBE gradient corrections, electron density and orbital visualization.
Wigner Function: Phase Space Negativity
W(x,p) for cat states, Fock states, squeezed vacuum — negative regions signal non-classicality, decoherence erases interference fringes.
Strange Metal: Planckian Dissipation
T-linear resistivity ρ=ρ₀+AT in cuprates and heavy fermions — Planckian scattering rate τ⁻¹=αk_BT/ℏ saturating the quantum bound.
Glass Transition: Mode Coupling Theory
MCT ergodicity breaking at φ_c — intermediate scattering function F(q,t), non-ergodicity parameter plateau, α-relaxation divergence.
Quantum Walk: Coined Delocalization
Discrete-time quantum walk with Hadamard/Grover coins — ballistic σ∝T spreading vs classical diffusion σ∝√T, disorder-driven localization.
Epidemic Threshold: Heterogeneous Network
SIR on scale-free vs Erdős-Rényi networks — β_c=μ⟨k⟩/⟨k²⟩ vanishes for SF networks (⟨k²⟩→∞), heterogeneous mean-field SIR dynamics.
Stokes & Oseen Creeping Flow
Viscous flow past a sphere: streamlines, drag force, and the Oseen correction at finite Reynolds number.
Quantum Capacitance: 2DEG & Graphene
Density of states and quantum capacitance C_Q for parabolic 2DEG vs linear Dirac graphene under gate voltage.
Skyrmion Hall Motion
Topological charge Q=±1 skyrmion driven by current — Magnus force, Hall angle, and trajectory visualization.
Percolation Phase Transition
Bond percolation on a square lattice: giant component emergence, critical scaling near p_c = 0.5, and order parameter sweep.
Magnon Hall Effect
Magnon Berry curvature and thermal Hall conductivity κ_xy in a DMI honeycomb magnet vs temperature and field.
CFT Primary Operators
Conformal field theory on plane and cylinder: primary operator correlators, scaling dimensions, and state-operator map.
Anisotropic Diffusion: Perrin Ellipsoid
Perrin's diffusion tensor for prolate/oblate ellipsoids — Brownian trajectories and anisotropic mean-square displacement.
Spectral Zeta & Casimir Energy
Zero-point mode sum regularized via spectral zeta function; Casimir force between plates as function of separation.
DMD & Koopman Operator
Dynamic Mode Decomposition extracts Koopman eigenvalues from nonlinear dynamics — phase space and spectral visualization.
Multifractal Spectrum & D_q
Generalized Rényi dimensions D_q and the f(α) multifractal spectrum for Cantor measures, IFS attractors, and Hénon maps.
Cellular Senescence & Telomere Shortening
Hayflick limit, telomere erosion per division, senescent cell accumulation, and SASP bystander signaling in a growing cell population.
Ecological Fold Bifurcation & Hysteresis
Lake eutrophication model — catastrophic regime shift between clear and turbid states with hysteresis loop and slow parameter drift.
Quantum Dot Coulomb Staircase
Single-electron transistor: Coulomb blockade, addition energy staircase in I–V, and blockade diamonds in bias–gate voltage space.
Diffusion MRI Fiber Orientation & ODF
Diffusion tensor ellipsoids, orientation distribution function resolving fiber crossings, and tractography streamlines in white matter.
Anderson Localization — Scaling Theory β(g)
The β function for d=1,2,3 dimensions, metal-insulator transition fixed point, and exponentially localized wavefunctions.
Chiral Anomaly & Spectral Flow
Dirac Landau levels in a magnetic field — adiabatic gauge threading pumps the zero mode between chiralities, breaking chiral charge conservation.
Mean First Passage Time — Narrow Escape
Brownian particles escaping a circular domain through a small window — MFPT diverges logarithmically as window size shrinks.
Kinetic Ising — Glauber Dynamics & Coarsening
2D Ising model with Glauber spin-flip dynamics below T_c — ordered domains nucleate and coarsen with domain size L(t) ~ t^(1/2).
Quantum Fisher Information & Metrological Bound
QFI for coherent, squeezed, and GHZ probe states — Heisenberg limit vs standard quantum limit, Cramér-Rao bound, decoherence effects.
Synchronization — Phase Response Curve
Phase reduction of limit cycle oscillators, Type I/II PRCs, entrainment to periodic forcing, and Arnold tongue structure.
Extreme Value Statistics — Gumbel & Fréchet
Block maxima, return levels, and GEV distribution fitting — Gumbel, Fréchet, and Weibull families with interactive histogram and PDF overlay.
Topological Crystalline Insulator
C₄-symmetric band inversion, edge state emergence, and dispersion across crystalline symmetry-protected topological phases.
Active Matter: MIPS & Flocking
Active Brownian particles with motility-induced phase separation — cluster formation, Vicsek-like alignment, and collective flocking dynamics.
Brownian Ratchet — Stochastic Pumping
Flashing asymmetric ratchet drives directed transport from thermal noise — Brownian motor current vs asymmetry, noise, and flash rate.
Quantum Thermodynamics — Work Distribution
Jarzynski equality verification via two-point measurement — work distribution P(W), free energy estimation, and quasistatic vs fast driving.
Cell Polarity — Cdc42 Turing Mechanism
Activator-inhibitor Turing instability on the cell membrane — Cdc42 spot formation, polarity establishment, and dispersion relation analysis.
Epidemic Threshold — Heterogeneous Networks
SIR on heterogeneous degree networks — epidemic threshold β_c = γ⟨k⟩/⟨k²⟩, superspreaders, and outbreak probability across network types.
Gravitational Wave Ringdown — QNMs
Black hole quasi-normal modes — damped sinusoid ringdown, Kerr frequency fitting, power spectrum, and the no-hair theorem test.
Fractal Dimension — Box-Counting
Sandbox box-counting algorithm measures fractal dimension D_f via log-log scaling — compare DLA, Sierpiński, Cantor, Koch, and Julia sets.
Voter Model with Zealots
Opinion dynamics on a lattice with committed minorities — zealots prevent consensus, shift stationary distributions, and sustain spatial opinion domains.
Network Epidemic & Contact Tracing
SIR model on random networks with contact tracing efficiency, delay effects, and outbreak size dynamics.
Kibble-Zurek Mechanism
Phase transition quench spawns topological defects. Defect density scales with quench rate via Kibble-Zurek universality.
Quantum Battery Charging
TLS array charged via unitary protocol. Ergotropy, entanglement, and power-energy tradeoff in quantum batteries.
Nematic Liquid Crystal — Frank Elasticity
Director field relaxation with splay/bend elastic constants. Click to inject ±½ topological defects and watch them annihilate.
Turbulence Intermittency & Multiscaling
Structure functions Sₚ(r) and anomalous exponents ζₚ. Departure from K41 scaling via She-Lévêque intermittency model.
Protein Amyloid Aggregation
Knowles model kinetics: primary nucleation, elongation, and secondary nucleation producing sigmoidal fibril growth curves.
Belief Propagation on Sparse Graphs
Cavity method BP on random Ising graphs. Watch cavity messages converge to fixed points and compute magnetizations.
QAOA — Quantum MaxCut
Quantum Approximate Optimization on small graphs. Energy landscape over γ-β space with approximation ratio vs circuit depth.
Hopfield Network Memory Capacity
Store random patterns in an Ising neural network. Retrieval overlap vs load α=P/N, with capacity threshold at α_c≈0.138.
Entropy Rate — Hidden Markov Model
Forward algorithm estimates HMM entropy rate via chain rule. Convergence of H(Xₙ|x₁…xₙ₋₁) to stationary value.
Bicoid Morphogen Gradient
Diffusion–degradation PDE for Drosophila positional information. Threshold bands define gap gene boundaries; noise sensitivity scales with 1/|dc/dx|.
Bose-Hubbard Mott Insulator
Mean-field phase diagram of the Bose-Hubbard model. Mott lobes vs superfluid phase, tunable J/U and μ/U, coordination number z.
Seasonal Epidemic Oscillations
SIRS model with periodic forcing β(t). Resonance-driven outbreaks, period-doubling, and chaos revealed by a stroboscopic Poincaré section.
Kitaev Honeycomb Spin Liquid
Z₂ flux sectors on the honeycomb lattice. Click plaquettes to create vison anyons; gapped vs gapless phases shown on a ternary phase diagram.
Active Gel: Cytoskeletal Contraction
Myosin-driven actomyosin cortex simulation. Above the activity threshold ζ_c, spontaneous contraction waves propagate around the cell cortex.
GNN Message Passing
Graph neural network feature propagation on a random community graph. Step through aggregation rounds and watch node embeddings converge toward classification.
Stochastic Epidemic Branching Process
Galton-Watson branching process for epidemic spread. Compare subcritical vs supercritical R₀, extinction probability, and overdispersion effects.
Nonlinear Wave Envelope & Soliton
NLS equation: dispersive spreading competes with nonlinear focusing. At the soliton balance point the envelope propagates without distortion.
Cavity QED: Atom-Field Entanglement
Jaynes-Cummings dynamics with Fock-state superpositions. Collapse and revival of Rabi oscillations, entanglement entropy, and Husimi Q function.
Liebig's Law & Droop Stoichiometry
Dual nutrient co-limitation in a chemostat. Droop cell quota dynamics determine which nutrient (N or P) limits algal growth via the Liebig minimum rule.
Shilnikov Chaos: Homoclinic Spiral Attractor
3D saddle-focus orbit winds infinitely near a homoclinic orbit. Shilnikov's theorem guarantees chaos when ρ > |λ| — infinite horseshoes from spiral return.
Kitaev Honeycomb: Anyons & Topological Phase
Exactly solvable spin-1/2 model on the honeycomb lattice. Phase diagram shows gapped A-phase with Abelian anyons and gapless B-phase with non-Abelian anyons.
Hyperbolic Lattice: Poincaré Disk Tessellation
Explore {p,q} tilings of the hyperbolic plane mapped to the Poincaré disk. Geodesic arcs and infinite regular tessellations impossible in Euclidean space.
Active Matter: Run-Tumble Phase Separation
Self-propelled particles with run-and-tumble dynamics spontaneously cluster via MIPS — motility-induced phase separation — no attractive forces needed.
Quantum Spin Liquid: Resonating Valence Bonds
Anderson's RVB state on the kagome lattice — frustrated spins superpose over dimer configurations, yielding fractionalized spinon excitations and no magnetic order.
Vicsek Flocking: Noise-Driven Order Transition
Self-propelled particles align with neighbors. Tune noise η to drive a discontinuous transition from ordered flocking (φ→1) to disordered gas (φ→0).
Voter Model: Consensus Time on Networks
Copy-dynamics opinion spreading on ER random, scale-free, grid, and complete networks. Consensus time scales differ dramatically with topology — N² vs N/ln N.
Spectral Zeta: Can You Hear the Shape?
Visualize Laplacian eigenmodes of drums. Weyl's law, isospectral pairs, and the spectral zeta ζ(s)=Σλₙ⁻ˢ encoding geometry — Kac's famous 1966 question.
Geodesic Billiards: Hadamard Surface Chaos
Sinai billiard, Bunimovich stadium, and triangle tables demonstrate hyperbolic chaos. Negative curvature drives exponential trajectory divergence — Lyapunov exponents in real time.
Neutral Theory: Ecological Drift & Speciation
Hubbell's zero-sum community dynamics. Stochastic birth-death with immigration rate m and speciation ν. Watch the Fisher log-series species abundance distribution emerge.
Abelian Sandpile: Recurrent States
Drop grains, watch avalanches cascade, and compute the identity element of the sandpile group. Self-organized criticality with fractal recurrent configurations.
Casimir Force: Quantum Vacuum Plates
Conducting plates attract via quantum vacuum fluctuations. Visualize allowed modes, compute F ∝ d⁻⁴, and explore thermal corrections to the Casimir effect.
Stochastic Resonance: Bistable Well
Noise enhances weak signal detection in a double-well potential. Watch optimal noise drive coherent switching between wells — the counterintuitive power of randomness.
Network Motif: C1 Feed-Forward Loop
The most common gene regulatory motif. Compare coherent (delay generator) vs incoherent (pulse generator) FFLs — real transcription factor network dynamics.
Vortex Pairs: Kosterlitz-Thouless Transition
XY model on a lattice: watch vortex-antivortex pairs bind below T_KT and unbind above it. The topological phase transition that earned the 2016 Nobel Prize.
Minimal Surfaces: Plateau's Problem
Catenoid, helicoid, Enneper, Scherk, and Costa surfaces — all with zero mean curvature H=0. Rotate and explore the Weierstrass-Enneper representation in 3D.
Chaotic Billiard: Bunimovich Stadium
Bunimovich's 1974 proof: add semicircles to a rectangle and the billiard becomes fully chaotic. Compare nearby trajectories diverging exponentially — with Poincaré section.
TASEP: Shock Profiles & Phase Diagram
Totally asymmetric exclusion process with open boundaries. Tune injection/extraction rates α, β to navigate the LD/HD/MC phase diagram and watch density shocks form.
Epidemic Superspreaders: Heterogeneous Networks
SIR on a scale-free network with tunable degree heterogeneity. High-degree hubs act as superspreaders — R₀ = β⟨k²⟩/γ⟨k⟩ diverges as power-law exponent γ→2.
Berry Phase & Holonomy
Parallel transport on the Bloch sphere reveals geometric phase γ_B = −½Ω. Watch the quantum state accumulate phase purely from curvature as a parameter loop closes.
Kondo Resonance
Abrikosov-Suhl resonance from magnetic impurity screening in metals — DOS, resistivity, AC conductivity
Adaptive Dynamics PIP
Pairwise invasibility plots, evolutionary branching, and the canonical equation of adaptive dynamics
Bravais Lattice & Brillouin Zone
Interactive 2D Bravais lattice explorer with reciprocal space construction and first Brillouin zone
Double Well Tunneling
Quantum tunneling splitting in symmetric/asymmetric double wells — Numerov eigenstates, coherent oscillations
Epidemic Contact Tracing
SEIQR model on heterogeneous networks with contact tracing and isolation interventions
Drude Model Conductivity
Classical electron gas, scattering dynamics, AC conductivity and Lorentzian frequency response
Oscillatory Shear Rheology
Maxwell/Kelvin-Voigt/Burgers models — Lissajous-Bowditch figures, G' and G'' spectra, tan delta
Bayesian Belief Network
DAG belief propagation — evidence updates via Bayes' theorem, likelihood ratios, posterior distributions
Sine-Gordon Kink Collision
Kink/antikink collisions and breather formation in the sine-Gordon field — topological charge, energy
Cilia Metachronal Wave
Hydrodynamically coupled ciliary oscillators forming symplectic/antiplectic metachronal waves
Erdős–Rényi Phase Transition
Giant component emergence in random graphs: mean-field percolation at ⟨k⟩=1
Anderson Impurity & Kondo Effect
Spectral function and NRG energy flow for a correlated quantum impurity in a metal
Ricci Flow — Surface Smoothing
Hamilton's curvature-driven metric evolution on 2-manifolds; Gaussian curvature coloring
Delay-Coordinate Embedding
Takens' theorem: reconstruct strange attractors from scalar time series via delay vectors
Active Gel — Toner-Tu Flocking
Self-propelled particles with alignment noise; polar order parameter and flocking transition
Rayleigh–Bénard Convection
Thermal instability rolls: buoyancy vs diffusion at Ra_c=1708, Nusselt number scaling
Information Geometry & Natural Gradient
Fisher-Rao metric on statistical manifolds; Amari's natural gradient vs Euclidean paths
Collective Memory & Cascade
Social imitation threshold dynamics on networks: Watts cascade model and opinion avalanches
Josephson AC Effect
RSJ model: washboard potential, I-V characteristic, and AC Josephson frequency ω_J=2eV/ℏ
Spectral Flow & Chiral Anomaly
Dirac fermion energy levels flow through zero as gauge flux winds; index theorem visualization
Protein Coevolution DCA
Direct coupling analysis extracts residue-residue contacts from multiple sequence alignments via inverse Boltzmann statistics
Rossby Wave Baroclinic Instability
Quasi-geostrophic potential vorticity dynamics: westward-propagating Rossby waves and baroclinic instability on a β-plane
Quantum Scars — Stadium Billiard
Semiclassical wavefunctions enhanced along unstable periodic orbits in a classically chaotic stadium billiard
Sandstone Pore Network Drainage
Capillary-controlled invasion percolation: non-wetting fluid displacing wetting fluid through heterogeneous pore throats
Wilson-Cowan Neural Field
Spatially extended excitatory-inhibitory neural population dynamics with Gaussian connectivity kernels — waves and patterns
Mutualistic Network Nestedness
Plant-pollinator bipartite networks: nested interaction structure, NODF metric, and cascade extinction dynamics
Granular Force Chains
Hertzian contact mechanics in compressed polydisperse packings — heterogeneous force chain networks carry the stress
Polariton Rabi Splitting
Exciton-photon strong coupling in semiconductor microcavities: anticrossing dispersion, Hopfield coefficients, and BEC condensation
Forest Fire SOC (Drossel-Schwabl)
Self-organized criticality without parameter tuning: power-law fire size distributions from simple growth and lightning rules
Quantum Discord & Separability
Quantum correlations beyond entanglement: discord, concurrence, Werner states, and the separability boundary in bipartite systems
KT Vortex Defects
XY model vortex-antivortex pairs, Kosterlitz-Thouless transition, Monte Carlo spin dynamics
Stochastic Resonance Neuron
FitzHugh-Nagumo neuron fires subthreshold signals with optimal noise — the resonance sweet spot
Sandpile Burning Bijection
Abelian sandpile burning algorithm reveals bijection between recurrent configs and spanning trees
Hamiltonian Chaos: Twist Map
Chirikov standard map Poincaré sections — KAM tori, resonance islands, and the onset of chaos
Louvain Community Detection
Greedy modularity maximization finds network communities — resolution parameter controls granularity
Smoluchowski Coagulation
Cluster aggregation with product kernel undergoes gelation — infinite cluster forms at finite time
MBE Epitaxial Growth
Ehrlich-Schwoebel barrier drives mound instability during molecular beam epitaxy crystal growth
CFT Minimal Models
Kac tables, conformal dimensions, and fusion rules for minimal models M(p,q) including Ising CFT
Flashing Ratchet Motor
Asymmetric potential switching on/off rectifies thermal noise into directed particle motion
Moody Diagram
Darcy-Weisbach friction factor vs Reynolds number — laminar, transitional, turbulent, rough pipes