Sandpile

Per Bak wanted to know why natural phenomena so often follow the same statistics: many small events, occasionally a large one, and no typical size in between. Earthquakes, forest fires, extinction waves. His answer, from 1987, is called self-organised criticality, and the model that came with it is a sandpile.

Drop grains on a single point. At first nothing remarkable happens. Once the pile reaches a certain steepness, the next grain triggers an avalanche — usually one of three squares, sometimes one that runs across the entire field. There is no setting that governs this. The pile tunes itself to the state in which avalanches of every size occur, and stays there.

The word 'abelian' in the name is a mathematical promise: it does not matter in what order you resolve the topplings. Start top left or bottom right, alternate between them — the final state is identical. That is remarkable, because the route there differs completely.

What forms from a single source is moreover not a pile but a figure: a square with triangles inside it, self-similar, and to this day not fully explained. There is a proof that the pattern converges on a fixed limiting shape, but the little triangles themselves still escape any closed description. It is a fractal nobody designed, following from four sentences of legislation.