Mentors: Dr. Houston Schuerger (UNT Alumni) and Dr. Lior Fishman

Title: Landscapes of the Tetrahedron and Cube: an Exploration of Shortest Paths on Polyhedra

Abstract: We review the established definition of nets and Alexandrov's definition of a star unfolding, leading to the classic result that the shortest path between two points on the surface of a convex polyhedra restricted to the polyhedron is contained as a straight line segment in one of the polyhedra's nets. To do any calculations concerning paths on the surface of a tetrahedron and cube we first define a coordinate system defined on the surface of the surface of a polyhedron. Since the shortest path on the surface of a convex polyhedron is a line segment contained in one of the polyhedron's nets, to calculate the set of points along said paths or the length of said paths we show it is sufficient to do so for each subset of the nets in which said paths can be contained. We introduce the new notion of a landscape which will be a useful tool in determining exactly which subsets of a polyhedron's nets need to be considered for such calculations. We then establish the 5 valid landscapes of the tetrahedron and 15 valid landscapes of the cube and their proofs.