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

We consider the problem of determining the length of the shortest paths between points on the surfaces of tetrahedra and cubes. Our approach parallels the concept of Alexandrov's star unfolding but focuses on specific polyhedra and uses their symmetries to develop coordinate based formulae. We do so by defining a coordinate system on the surfaces of these polyhedra. Subsequently, we identify relevant regions within each polyhedron's nets and develop formulae which take as inputs the coordinates of the points and produce as an output the distance between the two points on the polyhedron being discussed.