Click or tap the grid to drop grains · Space pauses · A auto-drops · C builds a centre pile · I computes the identity · P changes colours · keyboard? Tab to the grid, arrows move, Enter drops
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01 · DROP
GRAINS PER CLICK
Auto drops one grain on a random cell and waits for the avalanche to finish before the next — the slow driving of the original model.
02 · PATTERNS
Grains that topple off the edge are gone for good. A centre pile of at most grid² grains stays clear of it: 4,096 on the 64 grid, 16,384 on 128, 65,536 on 256.
03 · COLOUR
04 · AVALANCHES
- AVALANCHES
- 0
- LARGEST
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- LAST
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- LOG–LOG SLOPE
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Only single-grain drops on a settled pile are logged. Size = number of topplings; area = cells that toppled. Bins double in width and both axes are logarithmic, so a power law shows up as a straight line. The slope is a least-squares fit to your run, not a published exponent.
05 · THE SCIENCE
Every cell holds grains. A cell with four or more is unstable and topples: it sends one grain to each of its four neighbours. Grains pushed past the edge fall off — the edge is the sink. One toppling can tip its neighbours over, so a single grain can trigger an avalanche.
The final settled pile never depends on the order of the topplings, which is why Dhar’s formulation (1990) is called the abelian sandpile. This page relies on it: big loads settle in whole-grid sweeps, single grains cell by cell, and both give the same answer.
Bak, Tang and Wiesenfeld introduced the model in 1987 as an example of self-organized criticality: fed one grain at a time, the pile drives itself to a barely stable state where avalanche sizes have a power-law tail — small ones are common and huge ones keep happening. Levine and Propp’s short explainer walks through it.
CENTRE PILE drops N grains on one cell and lets them settle; bigger piles look like scaled-up smaller ones — observed, but not yet proved (Levine and Propp). IDENTITY computes the identity element of this grid’s sandpile group with the recipe I = (σ − σ°)°: σ puts 6 grains on every cell and ° means “topple until stable” (Holroyd et al.). Add I to a recurrent pile, let it settle, and you get the same pile back.
SOURCES
- P. Bak, C. Tang & K. Wiesenfeld, “Self-organized criticality: An explanation of the 1/f noise”, Physical Review Letters 59, 381 (1987). doi:10.1103/PhysRevLett.59.381
- D. Dhar, “Self-organized critical state of sandpile automaton models”, Physical Review Letters 64, 1613 (1990). doi:10.1103/PhysRevLett.64.1613
- L. Levine & J. Propp, “What is … a sandpile?”, Notices of the AMS 57(8), 976–979 (2010). ams.org (PDF) · author copy (PDF)
- A. E. Holroyd, L. Levine, K. Mészáros, Y. Peres, J. Propp & D. B. Wilson, “Chip-firing and rotor-routing on directed graphs” (2008), §2 — identity recipe I = (σ − σ°)°; §6 — the identity’s apparent fractal structure. arXiv:0801.3306