Weiss, Asia Ivic2018-11-212018-11-212018-08-142018-11-21http://hdl.handle.net/10315/35556This dissertation deals with abstract combinatorial structure of toroidal polytopes and toroidal hypertopes. Abstract polytopes are objects satisfying the main combinatorial properties of a classical (geometric) polytope. A regular toroidal polytope is an abstract polytope which can be constructed from the string affine Coxeter groups. A hypertope is a generalization of an abstract polytope, and a regular toroidal hypertope is a hypertope which can be constructed from any affine Coxeter group. In this thesis we classify the rank 4 regular toroidal hypertopes. We also seek to find all block systems on a set of (hyper)faces of toroidal polytopes and hypertopes of ranks 3 and 4 as well as the regular and chiral toroidal polytopes of ranks 3. A block system of a set X under the action of a group G is a partition of X which is invariant under the action of G.enAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.MathematicsElectronic Thesis or Dissertation2018-11-21RegularityChiralityToroidsThin GeometriesHypermapsAbstract PolytopesBlock SystemsSystems of ImprimitivityDiscrete GeometryPolytopesRegular PolytopesToroidal Polytopes