Abelian Sandpile Group: Identity Element

Self-similar fractal structure from algebraic identity
Computing identity element...
Max (4·3ᴺ)
Identity
Random stable

Sandpile Group

The set of stable sandpile configurations on a graph forms an abelian group under "addition then toppling". The identity element has a beautiful fractal structure.

Identity: e + e = e (after toppling) Addition: (a ⊕ b)ᵢ = stabilize(aᵢ + bᵢ) Toppling rule: if hᵢ ≥ 4: hᵢ -= 4 each neighbor: +1

The identity element emerges from computing e = stabilize(3·max_stable) — it exhibits remarkable self-similar geometry with 2-adic and 3-adic symmetry patterns.

Grid size N×N Group order ≈ det(Laplacian) Fractal dim ≈ log(4)/log(2) = 2
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