Abelian Sandpile

DEFINITION

z ≥ 4 → z −= 4, one grain to each neighbour
grains leave at the boundary
topple order does not matter — the pile is abelian

NOTES

Drop sand on one spot forever. Mostly nothing happens; sometimes a single grain sets off an avalanche that reorganises half the pile — and the avalanche sizes follow a power law with no preferred scale, which Bak, Tang and Wiesenfeld proposed as the origin of 1/f noise everywhere. The concentric fractal the centre-fed pile settles into looks designed; it is pure arithmetic. Abelian means the topple order is provably irrelevant, so the plate may defer work across frames without changing the result by one grain.

PROVENANCE

Origin

P. Bak, C. Tang & K. Wiesenfeld, "Self-Organized Criticality: An Explanation of 1/f Noise", Physical Review Letters 59, 1987

Standing

Public domain — a toppling rule

Constants

The pile reseeds the moment a grain first falls off the edge

Source

doi:10.1103/PhysRevLett.59.381