Finite-state transducers for substitution tilings

TL;DR

This paper introduces finite-state automata-based algorithms for efficiently generating and analyzing substitution tilings, enabling practical tiling patch creation and unique tile addressing systems.

Contribution

It presents novel combinatorial techniques using finite automata for substitution tilings, improving practical generation and analysis methods.

Findings

Efficient algorithms for tiling patch generation

Unambiguous substitution systems for hat tilings

Enhanced analysis capabilities for substitution tilings

Abstract

We present a suite of algorithmic techniques for handling substitution tilings by treating a tile's hierarchy of supertiles in a purely combinatorial fashion using finite state automata. The resulting techniques are very convenient for practical generation of patches of tilings such as hats, Spectres and Penrose tiles, both random and deliberately selected. They also permit some analyses of the represented tiling. A particular product of this process is two substitution systems for the hat tiling which are "unambiguous" in that a single tile address uniquely determines the rest of the plane.