Deterministic Abelian Sandpile and square-triangle tilings

TL;DR

This paper explores the deterministic Abelian Sandpile Model on various lattices, revealing complex self-similar patterns like Sierpinski structures, and introduces a unifying mechanism related to deterministic surfaces in higher-dimensional lattices.

Contribution

It uncovers a universal mechanism behind diverse Sierpinski structures in the Abelian Sandpile Model, linking them to deterministic surfaces in a 4D lattice.

Findings

Identification of Sierpinski structures across different lattices

Proposal of a unifying mechanism for these patterns

Connection to deterministic surfaces in higher dimensions

Abstract

The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome,...), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and involve different mechanisms. As will be shown elsewhere, all these structures fall under one roof, and are in fact different projections of a unique mechanism pertinent to a family of deterministic surfaces in a 4-dimensional lattice. This short note gives a description of this surface, and of the combinatorics associated to its construction.