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Finite-state transducers for substitution tilings

Simon Tatham

TL;DR

The work develops a comprehensive, geometry-free framework that leverages finite-state automata to manipulate substitution tilings via combinatorial tile addresses. It constructs recognisers and deterministic transducers for neighbor relations, and introduces refinement techniques to handle ambiguity, enabling practical tiling generation and analysis across Penrose, Ammann-Beenker, Spectre, and Hat tilings. Key contributions include a complete description of combinatorial substitution data, algorithms for neighbor computation, and demonstrations of unambiguous transducers along with methods to detect and resolve ambiguity. The approach yields efficient, scalable tools for generating patches, locating addresses, and analyzing tilings, with concrete results on multiple famous tilings and discussion of infinite boundaries and singular patterns.

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.

Finite-state transducers for substitution tilings

TL;DR

The work develops a comprehensive, geometry-free framework that leverages finite-state automata to manipulate substitution tilings via combinatorial tile addresses. It constructs recognisers and deterministic transducers for neighbor relations, and introduces refinement techniques to handle ambiguity, enabling practical tiling generation and analysis across Penrose, Ammann-Beenker, Spectre, and Hat tilings. Key contributions include a complete description of combinatorial substitution data, algorithms for neighbor computation, and demonstrations of unambiguous transducers along with methods to detect and resolve ambiguity. The approach yields efficient, scalable tools for generating patches, locating addresses, and analyzing tilings, with concrete results on multiple famous tilings and discussion of infinite boundaries and singular patterns.

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.

Paper Structure

This paper contains 35 sections, 2 theorems, 20 equations, 53 figures, 9 algorithms.

Table of Contents

  1. Introduction
  2. Combinatorial description of a substitution system
  3. Sub-tiles, sub-edges, and adjacency map
  4. Spurs
  5. Edges are not always polygon edges
  6. Layers
  7. Complete description of a combinatorial system
  8. Geometric 'source code'
  9. Addresses of tiles and edges
  10. Relation to regular languages
  11. Algorithms for handling addresses
  12. Computing neighbours recursively
  13. Computing neighbours using a transducer
  14. Recogniser for neighbour edge pairs
  15. Deterministic transducer
  16. ...and 20 more sections

Key Result

Theorem 1

In a substitution system without spurs, $\mathcal{N}$ is a regular language.

Figures (53)

  • Figure 1: Substitution system for Penrose P2 tiling using Robinson triangles
  • Figure 2: Adjacency map for the type-$A$ Robinson triangle
  • Figure 3: Substitution system for Penrose P2 whole tiles
  • Figure 4: The long edge of a hat monotile can meet one hat or two
  • Figure 5: Substitution system for Penrose P3 whole tiles
  • ...and 48 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Conjecture 1
  • Theorem 2
  • proof
  • Conjecture 2