⚡ Attractors
🌱 Emergence
⚛ Quantum
🕸 Networks
🏆 #9800

Strange Attractors — Chaos in Phase Space

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Strange attractors: bounded, non-periodic, fractal invariant sets.

Lorenz (1963): First discovered strange attractor. σ=10, ρ=28, β=8/3 gives the iconic butterfly.
Lyapunov exponent λ₁ ≈ 0.906, dim ≈ 2.06.

Rössler: Simplest 3D continuous chaotic flow. Single funnel structure.

Thomas: Cyclic symmetry dx/dt = sin(y)−bx, fully symmetric under cyclic permutation of axes.

Conway's Game of Life — Emergence & Complexity

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Generation: 0 | Alive: 0

Rules: B3/S23 — Cell born with 3 neighbors, survives with 2-3.
Universal Turing machine: glider guns compute arbitrary logic.
Activity heatmap shows temporal complexity patterns.

Quantum Double-Slit Interference

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1.0 5
Path integral formulation
ψ = ψ₁ + ψ₂ = A(e^(ikr₁)/r₁ + e^(ikr₂)/r₂)
I = |ψ|² ∝ cos²(πd·sinθ/λ) × sinc²(πa·sinθ/λ)

Fringe spacing: Δy = λL/d
First minimum: a·sinθ = λ
Fringe spacing: — mm

Barabási-Albert Growing Network

2 N=0
Barabási-Albert model
P(attach to i) ∝ k_i (preferential attachment)
Degree distribution: P(k) ~ k^(-3) for m=1
γ = 2m/(m+1) + 1 → 3 as m→∞
"Rich get richer" / Matthew effect
γ estimated: —
Click the canvas for fireworks