Abelian Sandpile — Identity Element & Group Structure

The set of recurrent configurations of the abelian sandpile model forms a finite abelian group under addition+stabilization. The identity element has a remarkable fractal self-similar structure. Add chips and watch avalanches topple.

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Statistics

Total chips: 0
Topplings: 0
Avalanche size: 0
Max height: 0
Toppling rule:
z(v) ≥ 4 → z(v) -= 4
z(u) += 1 for u~v
Abelian: order doesn't matter
Identity e: e+s=s
e = 2·max_recurrent - stabilize(2·max)