A primer on substitution tilings of the Euclidean plane

TL;DR

This paper introduces the theory of substitution tilings of the Euclidean plane, covering geometric and combinatorial classes, highlighting recent developments, and discussing their connections and research questions.

Contribution

It provides an overview of substitution tilings, introduces new insights into combinatorial substitutions, and discusses the relationship between geometric and combinatorial classes.

Findings

Geometric substitution tilings include well-known examples like Penrose tilings.

Combinatorial substitutions are a developing area with emerging research.

Connections between geometric and combinatorial substitutions are explored.

Abstract

This paper is intended to provide an introduction to the theory of substitution tilings. For our purposes, tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined, and some of what we present here is new. We give numerous examples, mention selected major results, discuss connections between the two classes of substitutions, include current research perspectives and questions, and provide an extensive bibliography. Although the author attempts to fairly represent the as a whole, the paper is not an exhaustive survey, and she apologizes for any important omissions.