Technique · Simulations
Sandpile
The abelian sandpile: a huge pile collapses into a fractal of nested patterns, then a drip keeps setting off avalanches.
Sandpile is a simulation that evolves step by step. Every piece on Synapse Drip is made from a recipe that layers techniques like this one, and you can change any of it in the Studio.
Good to know
- In 1996 a team in Oslo ran experiments with rice piles and found avalanches of all sizes, a closer match to the sandpile model than ordinary sand, whose grains roll and bounce too much.
- The final pattern from a single huge pile has a fractal structure; in 2013 Wesley Pegden and Charles Smart proved that, as the pile grows, the pattern converges toward a definite limit.
- Because the order of topplings never changes the result, the model is called abelian, and the stable piles on a grid form a mathematical group.
- The sandpile model was introduced in 1987 by Per Bak, Chao Tang and Kurt Wiesenfeld as the simplest example of self-organised criticality, a system that drives itself to the edge of instability.
- Avalanches of every size occur in it, with many small ones and few large ones; real earthquakes follow a similar rule, known as the Gutenberg-Richter law, though real sand does not.
The mathematics
- The first pile holds about 17,401 grains; a pile settles to an average of about 2.1 grains a cell, so it ends up covering about 8,286 cells, a disc-like shape whose radius is 0.34 of the picture's height.
- A cell topples when it holds four or more, sending one grain to each of its four neighbours: if g is the number of grains, the new count is g minus four times the whole number of fours in it, plus what its neighbours send.
- The pattern is exactly the same however the topplings are ordered, and no random numbers are used at all: it is a drawing of integer arithmetic.
In art
- Every cell ends with 0, 1, 2 or 3 grains, so the picture is a four-colour image with exact symmetry: a mandala that nobody drew.
- The nested diamonds, squares and flowers of the settled pile recall the tile floors of mosaics and the patterns of kaleidoscopes, though they come from nothing but counting.
More ways to use it
Related techniques
Reaction-diffusion · Cellular automata · Grain growth · Slime mould · Flocking boids · Attractor cloud · Elementary automaton · Synchrony