Graduation Year


Document Type




Degree Granting Department


Major Professor

Gregory L. McColm


Polyhedra, K-skeletons, Complexes, Crystal Nets, Geometric Group Theory, Graphs


We present a formal description of `Face Fundamental Transversals' on the faces of the Complexes of polyhedra (meaning threedimensional polytopes). A Complex of a polyhedron is the collection of the vertex points of the polyhedron, line segment edges and polygonal faces of the polyhedron. We will prove that for the faces of any 3-dimensional complex of a polyhedron under face adjacency relations, that a `Face Fundamental Transversal' exists, and it is a union of the connected orbits of faces that are intersected exactly once. While exploring the problem of finding a face fundamental transversal, we have found a partial result for edges that are incident to faces in a face fundamental transversal. Therefore we will present this partial result, as The Edge Transversal Proposition 1. We will also discuss a few conjectures that arose out this proposition. In order to reach our approaches we will first discuss some history of polyhedra, group theory, and incorporate a little crystallography, as this will appeal to various audiences.