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Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

Arun Maiti

Abstract

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.

Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

Abstract

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.

Paper Structure

This paper contains 6 sections, 5 theorems, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Tiles with Arbitrarily Large Heesch Numbers
  3. Aperiodicity in Hyperbolic Homogeneous Tilings
  4. Weakly Aperiodic Vertex Types
  5. Aperiodic Convex Monotiles
  6. Concluding remarks

Key Result

Theorem 1.1

For any given positive integer $n$, there exists a cyclic tuple $\mathfrak{k}_n$ with Heesch number $n$.

Figures (4)

  • Figure 1.1: Schematic of the layer-by-layer growth of a $[4, 5, 4, 5]$ tiling centered around $X_0$
  • Figure 2.1: Neighborhoods of type $F_2$ around odd faces
  • Figure 2.2: Completing a partial neighborhood of a 10-gon
  • Figure 2.3: Sequence of enforced neighborhoods

Theorems & Definitions (10)

  • Theorem 1.1
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Remark 3.1