Electrons fired one-by-one through a double slit build up an interference pattern — wave and particle simultaneously. Adjust slit separation and watch the fringes shift. The detection probability |ψ|² creates the bands even though each electron passes through "one slit."
xₙ₊₁ = r·xₙ(1−xₙ). As r increases from 1→4 the system period-doubles repeatedly (Feigenbaum δ≈4.669…), then dissolves into chaos. Hover to inspect. The infinitely deep self-similar structure is a fractal of period-doubling cascades.
Four local rules → gliders, oscillators, spaceships, self-replicators. A universal Turing machine can be built from these rules. Life demonstrates how global complexity emerges from purely local interactions with zero top-down design.
A Möbius strip has χ=0, one edge, one face — non-orientable. The torus has χ=0 with two edges, orientable. Euler's formula V−E+F=χ is a topological invariant: unchanged by any continuous deformation. Rotate to explore the geometry.
dx/dt=σ(y−x), dy/dt=x(ρ−z)−y, dz/dt=xy−βz. With σ=10, ρ=28, β=8/3 the system never repeats yet never diverges — a strange attractor. Two trajectories starting ε apart diverge exponentially (butterfly effect). Lyapunov exponent λ₁≈0.906.