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Sturmian lattices and Aperiodic tile sets

Shigeki Akiyama, Tadahisa Hamada, Katsuki Ito

TL;DR

The paper presents an explicit algorithm to construct aperiodic tile sets from Sturmian words with quadratic slopes, using Sturmian lattices and BD equivalence as the central toolset. For any quadratic irrational slope $\alpha$, the expansion constant $\lambda$ is a unit in the real quadratic field $\mathbb{Q}(\alpha)$, yielding self-similar Sturmian lattices and infinitely many essentially different aperiodic tile sets realized via cross BD correspondences. A key geometric principle is that a quadratic curve and a line intersect in at most two points, which underpins the aperiodicity proof, while cabinet cells and Ammann bars encode the Sturmian structure in a tile-combinatorial framework. The authors provide bounds on tile counts in terms of expansion data, develop a practical construction workflow, and illustrate several explicit examples (including recoveries of Smith Turtle), suggesting a pathway to aperiodic monotiles guided by Sturmian dynamics.

Abstract

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce infinitely many aperiodic tile sets whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is ``Sturmian lattices''; an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. We shall give a classification of Sturmian lattices. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction.

Sturmian lattices and Aperiodic tile sets

TL;DR

The paper presents an explicit algorithm to construct aperiodic tile sets from Sturmian words with quadratic slopes, using Sturmian lattices and BD equivalence as the central toolset. For any quadratic irrational slope , the expansion constant is a unit in the real quadratic field , yielding self-similar Sturmian lattices and infinitely many essentially different aperiodic tile sets realized via cross BD correspondences. A key geometric principle is that a quadratic curve and a line intersect in at most two points, which underpins the aperiodicity proof, while cabinet cells and Ammann bars encode the Sturmian structure in a tile-combinatorial framework. The authors provide bounds on tile counts in terms of expansion data, develop a practical construction workflow, and illustrate several explicit examples (including recoveries of Smith Turtle), suggesting a pathway to aperiodic monotiles guided by Sturmian dynamics.

Abstract

We give an explicit algorithm to construct aperiodic tile sets based on Sturmian words of quadratic slopes. The method works for any quadratic irrational slope, and we can produce infinitely many aperiodic tile sets whose underlying scaling constant is a unit of any real quadratic field. There are two key ingredients in our construction. The first one is ``Sturmian lattices''; an interesting grid structure generated by Sturmian words that emerged in an aperiodic monotile called Smith Turtle. We shall give a classification of Sturmian lattices. The second is the bounded displacement equivalence of Delone sets, which plays a central role in this construction.

Paper Structure

This paper contains 30 sections, 40 theorems, 312 equations, 50 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Balanced words
  3. Sturmian lattices
  4. Geometric representation
  5. Symbolic representation
  6. Classification of Sturmian lattices
  7. Irrational case
  8. Rational case
  9. Other cases
  10. Super Sturmian lattices
  11. Negative super Sturmian lattices
  12. Positive super Sturmian lattices
  13. Expansion constants of quadratic Sturmian lattices
  14. Aperiodic tile sets
  15. Bounded Displacement equivalence
  16. ...and 15 more sections

Key Result

Lemma 1

Let $w\in \mathcal{A}^{\mathbb Z}$ be a $1$-balanced word of a rational slope $\alpha$ and $c$ the central word of slope $\alpha$. Then $w$ must be one of the following three forms:

Figures (50)

  • Figure 1: Ammann bars on Smith Turtle
  • Figure 2: Sturmian lattice
  • Figure 3: Two geometric forms for a triple $(\mathfrak{A}, \mathfrak{B}, \mathfrak{C})$ of bi-infinite sequences
  • Figure 4: Examples of periodic Sturmian lattices
  • Figure 5: Example of a $3$-color Sturmian lattice
  • ...and 45 more figures

Theorems & Definitions (88)

  • Lemma 1: Markoff
  • proof
  • Definition 1
  • Example 1
  • Example 2
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 2
  • proof
  • ...and 78 more