On monohedral tilings of a regular polygon

TL;DR

This paper characterizes monohedral tilings of regular polygons with up to three tiles, extending previous results from squares and circles to all regular n-gons for n ≥ 5.

Contribution

It generalizes the classification of monohedral tilings to all regular polygons with at most three tiles, unifying prior specific cases.

Findings

Characterization of monohedral tilings for regular polygons with up to three tiles.

Extension of previous results from squares and circles to all regular n-gons.

Provides a unified framework connecting tilings of different regular polygons.

Abstract

A tiling of a topological disc by topological discs is called monohedral if all tiles are congruent. Maltby (J. Combin. Theory Ser. A 66: 40-52, 1994) characterized the monohedral tilings of a square by three topological discs. Kurusa, L\'angi and V\'\i gh (Mediterr. J. Math. 17: article number 156, 2020) characterized the monohedral tilings of a circular disc by three topological discs. The aim of this note is to connect these two results by characterizing the monohedral tilings of any regular $n$-gon with at most three tiles for any $n \geq 5$.