The moduli space of hex spheres

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.