Abelian Sandpile

Abelian sandpile model (Bak-Tang-Wiesenfeld 1987): each cell holds ≤3 grains; if a cell reaches 4 grains it topples — distributing one grain to each neighbor. The "abelian" property: the final configuration is independent of the order of topplings. Recurrent configurations form a group under addition mod topplings (the "sandpile group"), isomorphic to the cokernel of the graph Laplacian. The identity element has fractal geometry with self-similar structure at all scales. Avalanche sizes follow a power law P(s) ∝ s⁻ᵗ with τ ≈ 1.2 — the hallmark of self-organized criticality. Click on the grid to add grains.