Grains poured on one square of a grid, where any square holding four topples and gives one to each neighbour. They settle into a fractal mandala, the same whatever order the squares topple in.
Bak, Tang & Wiesenfeld, 1987 · Deepak Dhar, 1990
Pour grains of sand, one at a time, onto the middle square of a grid. Whenever a square holds four grains it topples, passing one grain to each of its four neighbours, and they may topple in turn, so one grain can set off an avalanche of any size. Per Bak, Chao Tang and Kurt Wiesenfeld proposed the model in 1987 as the simplest system that tunes itself to the edge of collapse, which they called self-organised criticality.
Deepak Dhar then showed that the order of the topplings makes no difference to where the grains end up, and that the stable piles on a grid form a group. When everything has settled each square holds 0, 1, 2 or 3 grains. Here full squares are left as paper and the emptier a square the darker its ink, and the pattern that appears is a fractal of nested squares and lattices that nobody has fully explained.
Choose Identity for that group's identity on a square grid with lost edges: the one stable pile that changes nothing when added to any other, a strange carpet of its own. In motion the grains are poured in and the pile grows outward from the centre.
Behind the mathematics
The sandpile model of self-organised criticality, 1987
Danish physicist who with Chao Tang and Kurt Wiesenfeld proposed the sandpile model in 1987, the founding example of self-organised criticality: systems that drive themselves to the edge of avalanches. His book How Nature Works carried the idea to earthquakes, extinctions and markets.
The abelian property and the sandpile group, 1990
Indian physicist at the Tata Institute of Fundamental Research who in 1990 showed that the sandpile's topplings can happen in any order with the same result, and that its stable piles form a group, which made the model exactly solvable. In 2022 he became the first Indian to receive the Boltzmann Medal.
