Tilings of Flat Tori by Congruent Hexagons

TL;DR

This paper classifies convex and certain non-convex hexagon tilings of the plane and extends these classifications to tilings of flat tori, analyzing their moduli spaces.

Contribution

It provides a comprehensive classification of hexagon tilings of the plane and describes the moduli spaces of corresponding torus tilings under vertex degree constraints.

Findings

Classified convex hexagon tilings of the plane.

Extended classification to non-convex hexagon tilings with degree 3 vertices.

Described the moduli spaces of hexagonal tilings of flat tori.

Abstract

Convex hexagons that can tile the plane have been classified into three types. For the generic cases (not necessarily convex) of the three types and two other special cases, we classify tilings of the plane under the assumption that all vertices have degree $3$. Then we use the classification to describe the corresponding hexagonal tilings of flat tori and their moduli spaces.