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An alternative proof for an aperiodic monotile

Shigeki Akiyama, Yoshiaki Araki

TL;DR

This work presents an elementary alternative proof of the aperiodicity of the Smith Turtle monotile, selected from the Smith–Myers–Kaplan–Goodman-Strass family at $b=\sqrt{3}$. It replaces the original combinatorial substitution with a geometric Golden Hex substitution built from patch-tiles and Golden Sturmian Patches that encode a Sturmian word of slope $\alpha=(5-\sqrt{5})/10$, with central-word structure controlling local combinatorics. A central tool is the Golden Ammann Bar (GAB), whose crossings on a Kagome lattice yield two irrational frequencies, forcing non-periodicity and ruling out any nonzero period. The approach yields a self-contained aperiodicity proof, highlights connections to cut-and-project concepts, and suggests broader applicability to other aperiodic monotiles and related tiling schemes.

Abstract

We give an alternative simple proof that the monotile introduced by Smith, Myers, Kaplan and Goodman-Strass is aperiodic.

An alternative proof for an aperiodic monotile

TL;DR

This work presents an elementary alternative proof of the aperiodicity of the Smith Turtle monotile, selected from the Smith–Myers–Kaplan–Goodman-Strass family at . It replaces the original combinatorial substitution with a geometric Golden Hex substitution built from patch-tiles and Golden Sturmian Patches that encode a Sturmian word of slope , with central-word structure controlling local combinatorics. A central tool is the Golden Ammann Bar (GAB), whose crossings on a Kagome lattice yield two irrational frequencies, forcing non-periodicity and ruling out any nonzero period. The approach yields a self-contained aperiodicity proof, highlights connections to cut-and-project concepts, and suggests broader applicability to other aperiodic monotiles and related tiling schemes.

Abstract

We give an alternative simple proof that the monotile introduced by Smith, Myers, Kaplan and Goodman-Strass is aperiodic.
Paper Structure (6 sections, 8 theorems, 43 equations, 21 figures)

This paper contains 6 sections, 8 theorems, 43 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Sturmian words and central words
  3. Golden Hex substitution
  4. Golden Ammann bar and Aperiodicity
  5. Future works
  6. Appendix

Key Result

Lemma 1

For any palindrome $K\in \{0,1\}^*$, the geometric realization of $K$ and its $\pi$-rotated image $\hbox{o}rigin=c]{180}{K}$ differ only at the two ends by four small kites as in Figure Pal4. In particular, $P(n)$ and $\hbox{o}rigin=c]{180}{P(n)}$ share the same upper and lower boundaries.

Figures (21)

  • Figure 1: Tiling by Smith Turtle
  • Figure 2: Patch-tiles $\mathrm{T}_n, \mathrm{\Pi}_n\ (n=0,1,2,3)$
  • Figure 3: Golden Hex Substitution
  • Figure 4: Geometric realization of $00100010010$. This is a GSP.
  • Figure 5: $\mathrm{T}_n$, $\mathrm{\Pi}_n$ and surrounding GSPs
  • ...and 16 more figures

Theorems & Definitions (17)

  • Lemma 1
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Remark 1
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 7 more