TITLE: Why is the identity element of the sandpile group self-similar?
QUESTION [12 upvotes]: 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.  

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)?

REPLY [3 votes]: 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:
          
(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.