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Convex pentagonal monotiles in the 15 Type families

Teruhisa Sugimoto

TL;DR

This paper systematically reviews the 15 known convex pentagonal monotile (Type) families, detailing how edge-length and angle constraints define tile behavior and how overlaps among families arise. It employs a Venn-diagram framework mapping the intersections of $T_x$ sets to analyze when tiles belong to multiple types and to characterize edge-to-edge versus non-edge-to-edge tilings. Key findings classify tilings by isohedrality: Types 1–5 yield isohedral tilings, Types 6–9 and 11–13 yield 2-isohedral tilings, and Types 10, 14, 15 yield 3-isohedral tilings, with many tilings requiring reflections and others not. The work engages with broader questions about aperiodicity, supports Rao’s claim that all convex pentagonal monotiles lie in these 15 families, and sets up future investigations into tilings outside the standard Type classifications and the necessity of reflections for non-periodic patterns.

Abstract

The properties of convex pentagonal monotiles in the 15 Type families and their tilings are summarized. The Venn diagrams of the 15 Type families are also shown.

Convex pentagonal monotiles in the 15 Type families

TL;DR

This paper systematically reviews the 15 known convex pentagonal monotile (Type) families, detailing how edge-length and angle constraints define tile behavior and how overlaps among families arise. It employs a Venn-diagram framework mapping the intersections of sets to analyze when tiles belong to multiple types and to characterize edge-to-edge versus non-edge-to-edge tilings. Key findings classify tilings by isohedrality: Types 1–5 yield isohedral tilings, Types 6–9 and 11–13 yield 2-isohedral tilings, and Types 10, 14, 15 yield 3-isohedral tilings, with many tilings requiring reflections and others not. The work engages with broader questions about aperiodicity, supports Rao’s claim that all convex pentagonal monotiles lie in these 15 families, and sets up future investigations into tilings outside the standard Type classifications and the necessity of reflections for non-periodic patterns.

Abstract

The properties of convex pentagonal monotiles in the 15 Type families and their tilings are summarized. The Venn diagrams of the 15 Type families are also shown.
Paper Structure (3 sections, 2 theorems, 19 figures)

This paper contains 3 sections, 2 theorems, 19 figures.

Table of Contents

  1. Introduction
  2. Properties of convex pentagonal monotiles and their tilings
  3. Conclusion

Key Result

Theorem 1

If a convex pentagon can generate an edge-to-edge monohedral tiling, it belongs to at least one of the Type 1, 2, or 4--9 families shown in Figure fig01. (In other words, convex pentagonal monotiles that can generate an edge-to-edge tiling are contained in $T_{1}$, $T_{2}$, $T_{4}$, $T_{5}$, $T_{6}$

Figures (19)

  • Figure 1: Fifteen Type families of convex pentagonal monotiles.
  • Figure 2: Examples of edge-to-edge tilings with convex pentagonal monotiles belonging to the Type 1 or Type 2 families.
  • Figure 3: Venn diagram of $T_{x}$.
  • Figure 4: Convex pentagonal monotiles contained in each intersection of the Venn diagram shown in Figure \ref{['fig03']}.
  • Figure 5: Convex pentagonal monotile belonging to both the Type 1 and Type 7 families, the representative tilings of Type 1 and of Type 7 with the convex pentagon. (In this figure, the convex pentagons corresponding to the reflected tiles are marked with an asterisk "*.")
  • ...and 14 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2