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Old problem revisited: Which equilateral convex polygons tile the plane?

Bernhard Klaassen

TL;DR

This work leverages Rao's reported completeness of the 15 tiling pentagon types to provide a streamlined geometric characterization of all equilateral strictly convex plane tilings: any such tile must be a triangle or quadrangle, a pentagon with two inner angles summing to $\pi$ (or the special pentagon $P7$), or a hexagon with a triple of inner angles summing to $2\pi$ wherein two angles share an edge. The proof proceeds by constructing representative cases (notably $P8$ for type 8) and systematically excluding types 5 and 6, then verifying that only types 1, 2, 4, 7, 8 (and certain hexagons) satisfy the tiling criteria. The analysis also details possible symmetry groups for periodic tilings and discusses nonperiodic equilateral tilings with potential quasicrystal connections, while proposing the Delaney–Dress framework as an independent avenue for verification. Overall, the paper provides a concise, geometry-led consolidation of the equilateral tiling landscape under the assumption of Rao’s result, highlighting both classical and novel tiling structures and their broader implications.

Abstract

We present a simplified proof of a forty-year-old result concerning the tiling of the plane with equilateral convex polygons. Our approach is based on a theorem by M. Rao, who used an exhaustive computer search to confirm the completeness of the well-known list of fifteen pentagon types. Assuming the validity of Rao's result, we provide a concise and mainly geometric proof of a tiling theorem originally due to Hirschhorn and Hunt. Finally, a possible connection to quasicrystals is sketched.

Old problem revisited: Which equilateral convex polygons tile the plane?

TL;DR

This work leverages Rao's reported completeness of the 15 tiling pentagon types to provide a streamlined geometric characterization of all equilateral strictly convex plane tilings: any such tile must be a triangle or quadrangle, a pentagon with two inner angles summing to (or the special pentagon ), or a hexagon with a triple of inner angles summing to wherein two angles share an edge. The proof proceeds by constructing representative cases (notably for type 8) and systematically excluding types 5 and 6, then verifying that only types 1, 2, 4, 7, 8 (and certain hexagons) satisfy the tiling criteria. The analysis also details possible symmetry groups for periodic tilings and discusses nonperiodic equilateral tilings with potential quasicrystal connections, while proposing the Delaney–Dress framework as an independent avenue for verification. Overall, the paper provides a concise, geometry-led consolidation of the equilateral tiling landscape under the assumption of Rao’s result, highlighting both classical and novel tiling structures and their broader implications.

Abstract

We present a simplified proof of a forty-year-old result concerning the tiling of the plane with equilateral convex polygons. Our approach is based on a theorem by M. Rao, who used an exhaustive computer search to confirm the completeness of the well-known list of fifteen pentagon types. Assuming the validity of Rao's result, we provide a concise and mainly geometric proof of a tiling theorem originally due to Hirschhorn and Hunt. Finally, a possible connection to quasicrystals is sketched.

Paper Structure

This paper contains 9 sections, 4 theorems, 6 figures.

Table of Contents

  1. Introduction
  2. Proofs of lemmas and proposition
  3. Analysis of the resulting equilateral tilings
  4. Type 5
  5. Type 6
  6. Symmetry groups
  7. Nonperiodic equilateral tilings and possible applications
  8. Alternative approach
  9. Conclusion

Key Result

Proposition 1.1

An equilateral strictly convex polygon P tiles the plane if and only if one of the following holds:

Figures (6)

  • Figure 1: Existence of a type $8$ pentagon
  • Figure 2: Variation of vertex $C$ as $C'$
  • Figure 3: Constructing pentagon type $9$ (left) and $7$ (right)
  • Figure 4: Constructing pentagon type $5$ (left) and $6$ (right)
  • Figure 5: Nonperiodic tiling scheme with equilateral hexagons for any rotational symmetry
  • ...and 1 more figures

Theorems & Definitions (7)

  • Proposition 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof