Escher: An Engine for Exploring Hierarchical Combinatorial Tilings
John Bowers and Dakota Lawson

Proceedings of Bridges 2024: Mathematics, Art, Music, Architecture, Culture
Pages 455–458
Short Papers

Abstract

In this paper we present a tool for creating combinatorial tilings defined by finite subdivision rules. Like their better known geometric counterparts (for example Penrose tiles), combinatorial tilings are formed by a finite tile set glued together along their sides. Unlike geometric tilings, however, combinatorial tilings have no inherent geometry and are endowed with geometry after the fact using graph drawing methods. Subdivision tilings are a type of combinatorial tiling that are defined recursively. Such tilings have interesting mathematical properties. Our open source tool, Escher, allows a user to define new tilings, compute properties of the tilings, and produce drawings of them. The Escher engine has both a Python library for exploring tilings using code as well as a simplified language for defining tilings and producing drawings of them that is designed to be accessible to non-programmers.

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