A Riemann surface is a two dimensional space (like a sphere or the surface of a donut) with a notion of angle between tangent vectors. The amount by which angles have to be distorted to get from one Riemann surface to another defines a distance on the space of all Riemann surfaces of a given topological type, called the Teichmuller metric. The most efficient way to get from one Riemann surface to another in this metric can be described in terms of flat structures. A flat structure is a way to assemble the surface from polygons in the plane by gluing pairs of parallel edges via isometries. One can deform a flat structure by stretching it in a fixed direction. The geodesics (or straight lines) in the Teichmuller metric are given precisely by these stretch paths. Despite this concrete description, the geometry of this metric is still poorly understood. For example, it was discovered only recently that balls in this metric are not necessarily convex, at least when the topology of the surface is sufficiently complicated. The goal of the project is to study the low complexity cases that remain, starting with the sphere with 5 points removed. The problem is amenable to numerical computations.