How many times does each site topple during relaxation?
The odometer function u(v) records how many times site v topples during the relaxation of a sandpile. It satisfies the discrete Laplacian equation Δu = ρ_final − ρ_initial where ρ is the chip configuration. The odometer reveals the fractal geometry of the abelian sandpile: its level sets form intricate self-similar boundaries with 8-fold symmetry.