In the Bak-Tang-Wiesenfeld (1987) sandpile model, each cell holds an integer number of grains. When a cell reaches 4 grains, it topples: loses 4 grains and each of its 4 neighbors gains 1. This can trigger cascading avalanches.
The model self-organizes to a critical state — avalanche sizes follow a power law P(s) ~ s−τ with τ ≈ 1.2 (2D). The "abelian" property means the final configuration is independent of the order topplings occur. The set of stable sandpiles forms an abelian group under addition modulo topplings, with a remarkable fractal identity element — click "Show Identity" to see it.