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Translational tilings: structured or wild?

Rachel Greenfeld

Abstract

The study of the structure of translational tilings has captivated mathematicians, scientists, and the general public for centuries and continues to thrive at the crossroads of analysis, combinatorics, dynamics, logic, number theory, and geometry. This vibrant field seeks to uncover the delicate divide between rigid structures and unpredictable, ``wild'' behaviors that arise when sets fill space by translations without gaps or overlaps. We provide an overview of this study and recent developments, highlighting its multidisciplinary nature and offering a glimpse into the process behind the results.

Translational tilings: structured or wild?

Abstract

The study of the structure of translational tilings has captivated mathematicians, scientists, and the general public for centuries and continues to thrive at the crossroads of analysis, combinatorics, dynamics, logic, number theory, and geometry. This vibrant field seeks to uncover the delicate divide between rigid structures and unpredictable, ``wild'' behaviors that arise when sets fill space by translations without gaps or overlaps. We provide an overview of this study and recent developments, highlighting its multidisciplinary nature and offering a glimpse into the process behind the results.

Paper Structure

This paper contains 25 sections, 5 theorems, 37 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Notation and preliminary definitions.
  3. The domino problem: encoding wild behavior
  4. Translational tilings with multiple tiles.
  5. Tiling language.
  6. Discrete translational monotilings
  7. Elementary properties.
  8. Different approaches and tools.
  9. Dilation-invariance.
  10. Periodicity of monotilings.
  11. The periodic tiling conjecture.
  12. Reversible structure theorem
  13. Structured monotilings: periodicity in low dimensions
  14. The structure of one-dimensional monotilings.
  15. The structure of two-dimensional monotilings.
  16. ...and 10 more sections

Key Result

Lemma 3.5

Let $F$ be such that $\operatorname{Tile}(F;G)$ is nonempty. For sufficiently divisibleFor instance, we can choose the number $q$ to be $p|F|$, where $p$ is the exponent of the finite Abelian group $G_0$ such that $G=\mathbb{Z}^d\times G_0$.$q$ we have where $rF=\{rf\colon f\in F\}$ for an integer $r$.

Figures (10)

  • Figure 2.1: On the left, a set of 13 Wang square tiles (over five colors). On the right, a piece of a Wang tiling with the given squares.
  • Figure 3.1: On the left, a discrete square tile $F=\{0,1\}^2$ (where a point $x$ in $\mathbb{Z}^2$ is illustrated by a continuous unit square $x+[0,1]^2$). In the middle, a piece of a tiling by $F$ of type $A_a$ with columns shifted independently, while on the right is a tiling $A^a$, whose rows are shifted independently.
  • Figure 3.2: On the left, the discrete disconnected tile $F=\{0,2\}\times \{0,1\}$ (where a point $x$ in $\mathbb{Z}^2$ is illustrated by the continuous unit square $x+[0,1]^2$). On the right, a piece of a tiling $A_a^b$ by $F$ with both columns (grey) and rows (green and red) shifted independently.
  • Figure 3.3: On the bottom, the square tile $F=\{0,1\}^2$ (where a point $x$ in $\mathbb{Z}^2$ is represented as $x+[0,1]^2$) and a piece of a tiling $A\in \operatorname{Tile}(F;\mathbb{Z}^2)$. For this $F$ we can let $q=4$. On the top is the dilation $rF=\{0,5\}^2$ of $F$ by the dilation factor $r=5$ with $r=1\ (\mathrm{mod}\ q)$, and we see that $A$ also form a tiling of $\mathbb{Z}^2$ by $rF$.
  • Figure 5.1: An illustration of the set $F$ in Example \ref{['ex:multi']} (where a point $x$ in $\mathbb{Z}^2$ is represented by $x+[0,1]^2$).
  • ...and 5 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Example 3.1: Singleton tile
  • Example 3.2: Non-tiling example
  • Example 3.3: Discrete square tile
  • Example 3.4: Disconnected tile
  • Lemma 3.5: Dilation lemma
  • Remark 3.6
  • Definition 3.7: Periodicity and aperiodicity
  • Conjecture 3.8: Discrete periodic tiling conjecture
  • ...and 17 more