The abelian sandpile group has an identity: a configuration e such that e ⊕ any stable config = that config. The identity element displays an eerie self-similar fractal — an exact, provable fractal arising from pure arithmetic.
Press "Compute Identity" to begin
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Abelian sandpile:
Each cell holds grains. If ≥ 4, it topples: loses 4, each neighbor gains 1. Repeat until stable.
Group identity:
Start from the all-6 configuration, add it to itself (modulo toppling) repeatedly until stable: id = (6·1) ⊕ (6·1) − (6·1)
The result is a fractal with self-similar structure at every scale — entirely determined by arithmetic, no randomness.
Colors: 0=black, 1=dark green, 2=teal, 3=bright green