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Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling

James Smith

Abstract

These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling. A simple recursive substitution system is then introduced to show the existence of a non-periodic tiling without the need for computer-aided verification. A new cut-and-project style construction linking the Turtle tiling with 1-dimensional Fibonacci words provides a second proof of non-periodicity, and an alternative demonstration that the Turtle can tile the plane. Deforming the Turtle into the Hat tile then provides a third proof for non-periodicity by considering the effect on the lattice underlying the Rhombille tiling. Finally, attention turns to the Spectre tile. In collaboration with Erhard Künzel and Yoshiaki Araki, we present two new substitution rules for generating Spectre tilings. This pair of conjugate rules show that the aperiodic monotile tilings can be considered as a 2-dimensional analog to Sturmian words.

Turtles, Hats and Spectres: Aperiodic structures on a Rhombic tiling

Abstract

These notes derive aperiodic monotiles (arXiv:2303.10798) from a set of rhombuses with matching rules. This dual construction is used to simplify the proof of aperiodicity by considering the tiling as a colouring game on a Rhombille tiling. A simple recursive substitution system is then introduced to show the existence of a non-periodic tiling without the need for computer-aided verification. A new cut-and-project style construction linking the Turtle tiling with 1-dimensional Fibonacci words provides a second proof of non-periodicity, and an alternative demonstration that the Turtle can tile the plane. Deforming the Turtle into the Hat tile then provides a third proof for non-periodicity by considering the effect on the lattice underlying the Rhombille tiling. Finally, attention turns to the Spectre tile. In collaboration with Erhard Künzel and Yoshiaki Araki, we present two new substitution rules for generating Spectre tilings. This pair of conjugate rules show that the aperiodic monotile tilings can be considered as a 2-dimensional analog to Sturmian words.
Paper Structure (19 sections, 1 theorem, 21 equations, 55 figures, 2 tables)

This paper contains 19 sections, 1 theorem, 21 equations, 55 figures, 2 tables.

Table of Contents

  1. Initial observation
  2. Matching rules
  3. The Turtle
  4. Tiling properties
  5. First proof of non-periodicity: via Ammann Bars
  6. Tilability
  7. Hexagonal clusters and the H7/H8 substitution rules.
  8. Bricks and Mortar substitution system
  9. Proof of the substitution rules
  10. Fibonacci cube construction
  11. Second proof of non-periodicity: via projection
  12. The Hat
  13. Alternative proofs of aperiodicity
  14. Third proof of non-periodicity (sketch): morphing Turtles into Hats
  15. The Spectre
  16. ...and 4 more sections

Key Result

Proposition 7

The Turtle tile is not periodic

Figures (55)

  • Figure 2.1: A Rhomb tile set
  • Figure 2.2: A Hex and Rhomb tile set
  • Figure 2.3: An aperiodic tile set
  • Figure 2.4: Enforcing matching rules to remove the 'hole' tile
  • Figure 2.5: The aperiodic monotile's family
  • ...and 50 more figures

Theorems & Definitions (10)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • Proposition 7
  • proof
  • proof
  • proof