Sketch · 2026-09-16
Abelian sandpile.
A square lattice where each cell holds a small heap of sand. Any cell that reaches four grains topples one grain onto each of its four neighbours. Grains that fall off the edge are lost. Drop a pile at the centre and the toppling cascades outward until the whole field is stable again. The final picture is the same no matter which unstable cell you resolved first, and that is where the name comes from.
Click anywhere on the lattice to drop a grain at that spot. The simulator relaxes until every cell holds three or fewer grains before the next frame paints. Large drops are done in batches so the page keeps responding.
Note
The rule is short. Pick any cell with four or more grains, subtract four from its count, add one to each of its four axial neighbours. Repeat until nothing is left over threshold. Bak, Tang and Wiesenfeld wrote it down in 1987 as a model for self-organised criticality; Deepak Dhar proved the abelian property soon after.
The abelian property says that the final stable configuration does not depend on the order in which unstable cells were toppled, nor on the order in which grains were added. Any legal schedule reaches the same picture and uses the same total number of topples. That is a strong statement for a system that looks like it should care about timing.
Drop a hundred thousand grains at a single point on a wide enough lattice and the stable pile is a square with visible fractal panels. The pattern's fine structure is still an open research topic; a 2013 result by Levine and Pegden showed the scaling limit is described by a very particular Apollonian family of conics.
Sources
- Bak, P., Tang, C., Wiesenfeld, K. (1987). Self-organized criticality: An explanation of 1/f noise. Physical Review Letters 59(4): 381–384.
- Dhar, D. (1990). Self-organized critical state of sandpile automaton models. Physical Review Letters 64(14): 1613–1616.
- Pegden, W., Smart, C. (2013). Convergence of the Abelian sandpile. Duke Mathematical Journal 162(4): 627–642.