Abelian Sandpile — Recurrent States & Group Identity

Chip-firing on Z² with sink: the identity element has fractal self-similar structure
Current sandpile state (0-3 grains)
Group identity element (fractal)
40
50
8
Topplings: 0
Total grains: 0
Stable: YES
Avalanche size: 0
The abelian sandpile group (Dhar 1990): recurrent stable configurations form an abelian group under addition+stabilization. The identity element is the unique recurrent config c such that c+c stabilizes to c. Its fractal structure (Levine-Peres 2007) converges to a limit shape with exact 4-fold symmetry.