4 research outputs found

    Medial symmetry type graphs

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    A kk-orbit map is a map with its automorphism group partitioning the set of flags into kk orbits. Recently kk-orbit maps were studied by Orbani\' c, Pellicer and Weiss, for k≤4k \leq 4. In this paper we use symmetry type graphs to extend such study and classify all the types of 55-orbit maps, as well as all self-dual, properly and improperly, symmetry type of kk-orbit maps with k≤7k\leq 7. Moreover, we determine, for small values of kk, all types of kk-orbits maps that are medial maps. Self-dualities constitute an important tool in this quest

    Block Systems of Ranks 3 and 4 Toroidal Hypertopes

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    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
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