Vertex-Transitive Polyhedra in Three-Space

Vertex-transitive polyhedra of positive genus are natural generalizations of the well-known convex uniform polyhedra. Examples of non-spherical vertex-transitive polyhedra were first examined by Grunbaum and Shephard in a 1984 survey article on Polyhedra with transitivity properties. Besides two infinite families of toroidal vertex-transitive polyhedra, only five examples of genus g ≥ 2 were given. To this day it is unclear whether all such polyhedra of genus two and higher have been found. Recent progress has been made by Gevay, Schulte, and Wills (to appear in Advances in Geometry), not only in providing the newest example (there are now seven), but in restricting the symmetry groups to the rotation groups of the Platonic solids. Yet the principal question of complete enumeration of the combinatorial types is still open. In this thesis, the case of tetrahedral symmetry is settled - the unique example is already known. For the case of octahedral and icosahedral symmetry, plausible avenues of resolving the problem are sketched, yet a complete solution remains beyond the scope of this work.