Chip-firing on a grid: drop chips, fire when height≥4. The abelian sandpile model (Bak-Tang-Wiesenfeld 1987). The set of stable configurations forms an abelian group. Toppling waves create fractal patterns.
Abelian sandpile (Dhar 1990): chip-firing is abelian — order of firings doesn't matter, only the final stable state. Recurrent configurations form a group under addition. Identity element has fractal structure. Toppling waves have fractal dimension ≈ 1.21. The sandpile drives itself to criticality without fine-tuning — self-organized criticality. Avalanche sizes follow power law P(s)∝s^{-τ} with τ=1 in 2D.