4 research outputs found
Medial symmetry type graphs
A -orbit map is a map with its automorphism group partitioning the set of
flags into orbits. Recently -orbit maps were studied by Orbani\' c,
Pellicer and Weiss, for . In this paper we use symmetry type graphs
to extend such study and classify all the types of -orbit maps, as well as
all self-dual, properly and improperly, symmetry type of -orbit maps with
. Moreover, we determine, for small values of , all types of
-orbits maps that are medial maps. Self-dualities constitute an important
tool in this quest
Block Systems of Ranks 3 and 4 Toroidal Hypertopes
This 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