Abelian Sandpile — Identity Element

The abelian sandpile group (Dhar 1990): stable configurations (height 0-3) on an N×N grid form a group under addition (add then topple until stable). Sites with ≥4 grains topple: each neighbor gets 1 grain; boundary grains are lost. The identity element is computed by the burning algorithm: start from 2·(max stable) + max stable = 3·max stable, then subtract max stable twice. The fractal self-similar identity configuration (above) is a celebrated result of sandpile mathematics. Colors: 0=black, 1=dark blue, 2=gold, 3=white. Click Compute Identity to see the fractal identity.