Thought Toys · Exhibit 18

The sandpile that tunes itself

Drop grains of sand one at a time. When a spot gets too steep it topples, spilling onto its neighbours — which can topple too. Most grains do nothing. Then one, no different from the rest, sets off an avalanche that crosses the whole table. The pile isn't tuned to do this; it tunes itself, to the very edge of stability.

The pile — brighter means steeper; amber is one grain from toppling
1 2 3 · ready toppling now
How often each avalanche size happens — on log–log axes

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What you're seeing

Every cell is a little stack of sand. Add a grain and, if a stack reaches four, it topples: it hands one grain to each of its four neighbours and keeps none of the four. That can push a neighbour to four, so the topple spreads — an avalanche. Grains that fall off the table's edge are simply lost. That's the whole machine.

Start from an empty table and rain grains down. At first nothing avalanches — the pile is too flat, and almost every grain just sits. But the average slope creeps up on its own, and once it reaches a certain steepness the pile stops getting any steeper: every further grain that would pile up triggers a slide that carries the excess away. The pile has parked itself at a knife-edge — critical — without anyone setting it there. That's self-organized criticality.

At that edge, avalanches have no typical size. Watch the log–log plot fill in: the points lie on a straight line, which means a slide ten times bigger is just a fixed fraction rarer — there's no "normal" avalanche and no special "huge" one, only the same pattern at every scale. Most grains still do nothing; a rare grain takes down half the table; both come from the identical rule. Flip walls on and the edges stop draining: the table fills until it can't settle, and the scale-free behaviour dies. The same maths is blamed for the sizes of earthquakes, forest fires, and market crashes.

The rule, exactly. On a grid of heights, a site that reaches the threshold topples to its four neighbours: if h ≥ 4: h → h − 4, and each neighbour + 1 Relax until every site is below 4 (the result and the avalanche size don't depend on the order — the pile is Abelian). Open edges lose grains, so the pile reaches a steady critical density (about 2.1 grains per cell). Verified in node (99×99): avalanche sizes ran from 1 to ~64,000 — nearly five orders of magnitude — with the mean (≈815) far above the median (38): a fat tail, not a bump. Counter-examples: a fresh, subcritical pile produces only size-1 avalanches until it self-organizes; and with walls (no draining) the dish just saturates — in both, the scale-free tail is absent.

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