The moduli space of hex spheres

TL;DR

This paper characterizes the moduli space of unit-area hex spheres, showing it is homeomorphic to Voronoi polygons, and derives geometric properties and transformations of these spheres.

Contribution

It establishes a homeomorphism between the moduli space of hex spheres and Voronoi polygons, revealing their geometric structure and properties.

Findings

Moduli space of hex spheres is homeomorphic to Voronoi polygons.

Each hex sphere contains an embedded Euclidean annulus.

Hex spheres can be realized as tetrahedral boundaries in 3D.

Abstract

A hex sphere is a singular Euclidean sphere with four cone points whose cone angles are (integer) multiples of $\frac{2\pi}{3}$ but less than $2\pi$. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the the space of similarity classes of Voronoi polygons in the Euclidean plane. This result gives us as a corollary that each unit-area hex sphere $M$ satisfies the following properties: (1) it has an embedded (open Euclidean) annulus that is disjoint from the singular locus of $M$; (2) it embeds isometrically in the 3-dimensional Euclidean space as the boundary of a tetrahedron; and (3) there is a simple closed geodesic $\gamma$ in $M$ such that a fractional Dehn twist along $\gamma$ converts $M$ to the double of a parallelogram.