The Moduli Space of Hex Spheres
College of Liberal Arts and Sciences
A surface is called singular Euclidean if it can be obtained from a finite disjoint collection of Euclidean triangles by identifying pairs of edges by Euclidean isometries. The surface is locally isometric to the Euclidean plane except at finitely many points, at which it is locally modeled on Euclidean cones. These singular points are called the cone points. For each cone point there is a cone angle, which is the sum of the angles of the triangles that are incident to the cone point. A hex sphere is a singular Euclidean sphere with 4 cones whose cone angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere, we consider its Voronoi decomposition centered at the two cone points with greatest cone angles. This decomposes the hex sphere into two cells, the Voronoi cells, which intersect along a graph. By cutting a Voronoi cell along a special shortest geodesic, the Voronoi cell becomes a polygon on the Euclidean plane. This polygon will be referred to as a Voronoi polygon. We prove that the Moduli space of hex spheres of unit area is homeomorphic to the space of similarity classes of Voronoi polygons in the Euclidean plane.
International Conference Japan-Mexico on Topology and its Applications
Cruz-Cota, Aldo-Hilario, "The Moduli Space of Hex Spheres" (2010). Faculty Scholarly Dissemination Grants. Paper 46.
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