I've been reading about the Abelian Sandpile Model and noticed the identity element of the sandpile group on the square has self-similar components.

enter image description here

The sandpile group of the 198x198 square of height 4 above is a finite abelian group. However, the sandpile corresponding to elements of this group can be fractal. Where is the complexity coming from?

Also, how does can you find the identity element (say, with a computer)?


Since the time this question was asked, there have been some important breakthroughs on the convergence of the scaling limit of the sandpile model (and its identity element), which may explain some of the self-similarity you note.

See especially the papers: Pegden and Smart - Convergence of the Abelian sandpile, Levine, Pegden, and Smart - Apollonian structure in the Abelian sandpile, and Levine, Pegden, and Smart - The Apollonian structure of integer superharmonic matrices.

  • $\begingroup$ Could you please fix the first link to go to the abstract rather than directly to the PDF? Thank you! $\endgroup$ – Harry Altman Nov 25 '13 at 23:37
  • $\begingroup$ @HarryAltman: fixed. $\endgroup$ – Sam Hopkins Nov 25 '13 at 23:39
  • $\begingroup$ Added much later: I'm actually not sure that these papers address convergence of the identity element on a finite two-dimensional grid. Their main focus is on the stabilization of a large number of chips placed at the origin of the infinite two-dimensional grid. At any rate, fractal patterns emergence in both settings and I believe they are at least very closely morally related. $\endgroup$ – Sam Hopkins Jun 15 '18 at 21:30

As per your second question, the following algorithm allows one to compute the identity element.

Let $c$ denote the maximal stable configuration; i.e. $c = \sum_{v\in V}(d(v)-1) v$ This is always recurrent. Let $a^{\circ}$ denote the stabilization of a configuration $a$. Then this will give you the identity $e$:

$e =(2c - (2c)^\circ)^\circ$

If you are interested, check out this applet for doing a lot of this stuff (and it's pretty, too): http://people.reed.edu/~davidp/sand/program/program.html


Another way to compute the identity is to do it iteratively: start with an empty grid (all cells contain 0 chips), then iterate the following:

  • perform a reverse firing from the edges: add 2 chips to each corner cell and one chip to each edge cell
  • if the resulting configuration is not stable, i.e. some cells contain at least 4 chips, then fire these cells until you obtain a stable configuration (some chips will disappear from the grid at corner and edge cells).

Once you obtain twice in a row the same stable configuration, you have the identity. During my PhD I have made a video of the stable configurations that you obtain at each step of this algorithm:

          ASM identity

(many initial steps are missing to speed up the video).

You can see how the self-similar patterns evolve at each step and the formation of the center square.


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