Hex structures on singular Euclidean surfaces with conical singularities

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.;Singular Euclidean surfaces arise in several contexts within mathematics: (1) Masur and Tabachnikov studied them from the viewpoint of billiard flows on Euclidean polygons ([MT02]); (2) Troyanov analyzed them in the context of Riemann surfaces and their parameter spaces (Teichmuller and Moduli spaces) ([Tro07]); (3) Rivin investigated their relations with volumes in hyperbolic geometry ([ Riv94]); (4) They arise in the study of holomorphic quadratic differentials: in this case all the cone angles are multiples of pi.;In this thesis I describe a parameter space for the moduli space of singular Euclidean structures on a sphere with 4 cone points, provided that the cone angles are integer multiples of 2p3 but less than 2pi. This classification involves finding a canonical decomposition of a singular Euclidean surface into its Voronoi cells. There are exactly two cone points of cone angle 4pi/3 and to each of these is assigned the Voronoi cell centered at this point, which consists of all points in the surface that are closer to it than they are to the other cone point of angle 4p3 . Using combinatorial, geometric and topological techniques I analyze all possibilities for the Voronoi cells case by case, finding geometric parameters that characterize the Voronoi cells of singular Euclidean structures. This classification of Voronoi cells enable us to find a parameter space for the moduli space of singular Euclidean structures on a sphere with 4 cone points (with some numeric restriction on the cone angles).