The Sandpile
Up close, it is one rule you can watch: a grain of sand lands, a stack that reaches four topples, and its grains hop to the four neighbours, sometimes setting off a chain. Zoom out to ten thousand stacks and nothing is tuned, yet the board fills to a little over two grains a stack and stays there, balanced where one grain can move nothing or sweep the whole board. Pour a million grains onto one square instead and the same rule draws a mandala.
In 1987 three physicists at Brookhaven — Per Bak, Chao Tang and Kurt Wiesenfeld — wrote down a toy that would not stay a toy. A square board. Each cell holds a few grains of sand. Drop grains one at a time at random, and whenever a cell holds four, it topples: one grain to each of its four neighbours, and grains pushed off the edge are gone. A topple can tip a neighbour over four, which topples in turn, and so on. That is the whole model.
Watch it fill. At first the grains just pile up and nothing much happens. Then the avalanches start, and keep getting bigger, until the board reaches a little over two grains a cell and stops filling. From then on it hovers there by itself. Every grain that lands is paid for, on average, by grains carried off the edge, and the pile sits permanently on the brink. Most grains do nothing; a few set off a cascade that crosses the board. Nobody chose that level and there is no dial for it. The pile finds it.
The avalanche sizes are the surprise. Plot how often each size happens on logarithmic axes and the points fall on a straight line: a power law. There is no typical avalanche, only small ones that are common and big ones that are rarer in a fixed proportion all the way up, until the board runs out of room. Physicists knew this shape from systems tuned carefully to a critical point, like a magnet held at exactly its Curie temperature. Bak's claim was that many natural systems tune themselves there, and he called it self-organised criticality. Earthquake sizes follow the same kind of line (the Gutenberg–Richter law), and so do forest fires and solar flares. Word frequencies in a book do too; the Lab's Zipf's law experiment shows that one.
Change one thing and the same rule makes art. Pour every grain onto the centre cell and let it settle: the result is a mandala of nested triangles and patches with the symmetry of the square. Mathematicians proved only in 2013 that the picture has a limiting shape at all, and tied its patches to Apollonian circle packings in 2016. The rule also hides an algebra. The order of topplings never changes the outcome, which makes the settled piles a group, and that group's do-nothing element, its identity, is a fractal of its own.
Nine ways to put a row in order, played as music — and on 1,024 values, a gap of nearly thirty to one you can hear.