Sandpile toppling on Penrose tilings: identity and isotropic dynamics

TL;DR

This paper investigates sandpile dynamics on Penrose tilings and other grids, revealing near-isotropic shapes and stability properties, with a publicly available simulator for further exploration.

Contribution

It introduces the first study of sandpile behavior on aperiodic Penrose tilings, highlighting identity elements and near-circular stabilization shapes.

Findings

Surprising circular shapes emerge in sandpile stabilization.

Shapes are close to, but not perfect, circles.

Numerical comparison shows near-isotropic dynamics across tilings.

Abstract

We present experiments of sandpiles on grids (square, triangular, hexagonal) and Penrose tilings. The challenging part is to program such simulator; and our javacript code is available online, ready to play! We first present some identity elements of the sandpile group on these aperiodic structures, and then study the stabilization of the maximum stable configuration plus the identity, which lets a surprising circular shape appear. Roundness measurements reveal that the shapes are not approaching perfect circles, though they are close to be. We compare numerically this almost isotropic dynamical phenomenon on various tilings.