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

2025/06/23 by Bernhard Klaaßen, Klaassen, Bernhard
Materials Science · Mathematics · #52C20 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG) #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2506.18473

openalex publication_date 2025/06/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Tegula -- exploring a galaxy of two-dimensional periodic tilings
• Convex pentagons that admit i-block transitive tilings
• Rotationally Symmetric Tilings with Convex Pentagons and Hexagons

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