Abelian Sandpile
Stack grains until they spill, then watch the spill spread
Stack grains on a grid. Any pile holding four or more spills one grain to each side, which can set off a chain of further spills.
How it works
In plain English, before the notation
Each square holds a number of grains. A square holding four or more is unstable: it gives away four grains, one to each of its four orthogonal neighbours. That can push a neighbour over four, which spills in turn, and so on. Drop a large pile on one square and the collapse takes thousands of steps and leaves behind a fixed pattern with clear repeating structure at several scales.
Collapse one large pile
- Open Presets and load "Single Pile, 80,000 Grains" — all of them stacked on one square.
- Press Play. The collapse takes several thousand steps; the final shape is worth waiting for.
- Once it has settled, use Drop Sand with the +80k setting to add another pile off-centre and watch the two interact.
Starting configurations
Loads straight into the simulatorTry any rule
The catalogue covers a few dozen rules. Here you can run any of the 262,144 two-state grid rules, or any of the 256 one-dimensional rules, including ones nobody has written up.
Well-known rules
Where it came from
Per Bak, Chao Tang and Kurt Wiesenfeld introduced the model in 1987 at Brookhaven National Laboratory as the founding example of self-organised criticality: the idea that some systems drive themselves to a critical point without anyone tuning a parameter to get them there.
Deepak Dhar showed in 1990 that the model has an exact algebraic structure — the stable configurations that can be reached repeatedly form a finite abelian group. That result is why the model is studied by mathematicians as well as physicists, and it is where the "abelian" in the name comes from.
The rule, precisely
What each cell looks at
The 4 cells sharing an edge — north, south, east and west (the von Neumann neighbourhood)
What a cell can be
A non-negative integer per cell; stable configurations use only 0, 1, 2 and 3
The update
if h(x,y) >= 4 then h(x,y) -= 4 and each of the 4 orthogonal neighbours gains 1
The model is called abelian because the order of toppling does not matter: if several squares are unstable, toppling them in any order gives exactly the same final configuration.
Structure at every scale
The settled pattern is far more organised than the rule suggests it should be:
- A single large pile settles into a shape with distinct triangular regions and repeating structure at several scales.
- The scaling limit of that shape has been characterised mathematically, and is related to Apollonian circle packings — a result of Levine, Pegden and Smart.
- Adding one grain to a settled pile can cause no toppling at all, or a cascade across a large part of the grid.
- The reachable stable configurations form a finite abelian group. Its identity element is itself a striking pattern, computed rather than designed.
Algebraically rich, and harder than it looks
The abelian group structure gives the model exact algebraic identities, and sandpile dynamics have been shown capable of carrying logic.
Where the idea of self-organised criticality came from
The model was proposed to explain why so many natural systems produce events with no characteristic size — many small ones, occasionally a very large one, with no separate mechanism for the large events. That framing has been influential and also contested: the two-dimensional sandpile’s own avalanche statistics turn out to be more complicated than a single clean power law. Its lasting contribution is the mechanism, not the fit.
Things to try
- Use the +4 Drop Sand setting on a settled pile. Most single additions do nothing visible; occasionally one sets off a large cascade. That contrast is the whole point of the model.
- Drop two large piles apart from each other and let them collide. The result is not the sum of the two settled shapes.
- Because the edges are open, a pile near the boundary settles much faster than one in the middle.
Frequently Asked Questions
References
The original Physical Review Letters paper that introduced the model and the term.
The abelian group structure and its connection to the grid Laplacian.
Good summary of Dhar’s group result and the work on the scaling limit.