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

TL;DR

This paper offers a simplified, geometric proof of a classic result on plane tilings with equilateral convex polygons, building on Rao's computer-assisted classification of pentagon types, and explores potential links to quasicrystals.

Contribution

It provides a concise geometric proof of a known tiling theorem, assuming Rao's classification, and discusses possible connections to quasicrystals.

Findings

Confirmation of the completeness of the fifteen pentagon types for tiling the plane

A simplified geometric proof of the tiling theorem

Discussion of potential links between tilings and quasicrystals

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.