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Converting non-periodic tilings with Tile(1, 1) into tilings with three types of pentagons, I

Teruhisa Sugimoto

TL;DR

This work shows how non-periodic tilings generated from the $Tile(1,1)$ monotile can be transformed into tilings composed of three pentagon types without subdividing the original rhombi. Focusing on two base non-periodic tilings, $T_h$ and $T_s$, the authors derive two distinct pentagonal-patterns for each (under the constraint of no rhombus subdivision), using cluster-based substitutions derived from prior work by Smith et al. When optional reflections are allowed, the set of attainable pentagonal tilings expands to countless patterns. The study connects rhombus-based aperiodic tilings to pentagonal tilings via specific conversion schemes and highlights two pattern-series families, including implications for periodic tilings ($T_p$) formed by translation-unit belts. It also outlines open questions on matching conditions for rhombus and pentagon tilings and suggests directions for broader exploration of pattern-generating capabilities of $Tile(1,1)$.

Abstract

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the resulting non-periodic tilings with three types of pentagons exhibit two patterns. Note that, during the conversion process in this manuscript, the rhombus is not subdivided into smaller similar rhombuses.

Converting non-periodic tilings with Tile(1, 1) into tilings with three types of pentagons, I

TL;DR

This work shows how non-periodic tilings generated from the monotile can be transformed into tilings composed of three pentagon types without subdividing the original rhombi. Focusing on two base non-periodic tilings, and , the authors derive two distinct pentagonal-patterns for each (under the constraint of no rhombus subdivision), using cluster-based substitutions derived from prior work by Smith et al. When optional reflections are allowed, the set of attainable pentagonal tilings expands to countless patterns. The study connects rhombus-based aperiodic tilings to pentagonal tilings via specific conversion schemes and highlights two pattern-series families, including implications for periodic tilings () formed by translation-unit belts. It also outlines open questions on matching conditions for rhombus and pentagon tilings and suggests directions for broader exploration of pattern-generating capabilities of .

Abstract

Non-periodic tilings with Tile(1, 1) using the substitution method, as presented by Smith et al. in [2] and [3], can be converted into non-periodic tilings with three types of pentagons. When arbitrary replacements are excluded, the resulting non-periodic tilings with three types of pentagons exhibit two patterns. Note that, during the conversion process in this manuscript, the rhombus is not subdivided into smaller similar rhombuses.
Paper Structure (10 sections, 40 figures)

This paper contains 10 sections, 40 figures.

Table of Contents

  1. Introduction
  2. Preparation
  3. Conversion of non-periodic tiling $T_{h}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  4. First pattern of $T_{h}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  5. Second pattern of $T_{h}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  6. Conversion of non-periodic tiling $T_{s}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  7. First pattern of $T_{s}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  8. Second pattern of $T_{s}$ in the case where the original rhombus is not subdivided into smaller similar rhombuses
  9. Countless patterns of non-periodic tilings that Tile$(1, 1)$ can generate and their conversion
  10. Conclusions

Figures (40)

  • Figure 1-1: Tile$(1, 1)$, and periodic tiling formed by Tile$(1, 1)$.
  • Figure 1-2: Converting non-periodic tiling $T_{s}$ with Tile$(1, 1)$ into tiling consisting of squares, regular hexagons, and rhombuses with an acute angle of $30^ \circ$.
  • Figure 1-3: Non-periodic tiling consisting of squares, regular hexagons, and rhombuses with an acute angle of $30^ \circ$ generated by conversion based on $T_{s}$ with Tile$(1, 1)$.
  • Figure 1-4: Converting non-periodic tiling $T_{h}$ with Tile$(1, 1)$ into tiling consisting of squares, regular hexagons, and rhombuses with an acute angle of $30^ \circ$.
  • Figure 1-5: Non-periodic tiling consisting of squares, regular hexagons, and rhombuses with an acute angle of $30^ \circ$ generated by conversion based on $T_{h}$ with Tile$(1, 1)$.
  • ...and 35 more figures