Sandpile Identity — Abelian Structure

The abelian sandpile group identity is the unique configuration e such that e + f = f for any stable f (with toppling). Constructed by: start with 6 chips everywhere, topple to stability, then subtract 2 from every cell and topple again. The resulting fractal pattern has exact 4-fold symmetry and self-similar structure at all scales — a completely unexpected consequence of simple toppling rules.