11 research outputs found
Problems on Polytopes, Their Groups, and Realizations
The paper gives a collection of open problems on abstract polytopes that were
either presented at the Polytopes Day in Calgary or motivated by discussions at
the preceding Workshop on Convex and Abstract Polytopes at the Banff
International Research Station in May 2005.Comment: 25 pages (Periodica Mathematica Hungarica, Special Issue on Discrete
Geometry, to appear
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
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
Surface realization with the intersection edge functional
Deciding realizability of a given polyhedral map on a (compact, connected)
surface belongs to the hard problems in discrete geometry, from the
theoretical, the algorithmic, and the practical point of view.
In this paper, we present a heuristic algorithm for the realization of
simplicial maps, based on the intersection edge functional. The heuristic was
used to find geometric realizations in R^3 for all vertex-minimal
triangulations of the orientable surfaces of genus g=3 and g=4. Moreover, for
the first time, examples of simplicial polyhedra in R^3 of genus 5 with 12
vertices were obtained.Comment: 22 pages, 11 figures, various minor revisions, to appear in
Experimental Mathematic
Map operations and k-orbit maps
AbstractA k-orbit map is a map with k flag-orbits under the action of its automorphism group. We give a basic theory of k-orbit maps and classify them up to k⩽4. “Hurwitz-like” upper bounds for the cardinality of the automorphism groups of 2-orbit and 3-orbit maps on surfaces are given. Furthermore, we consider effects of operations like medial and truncation on k-orbit maps and use them in classifying 2-orbit and 3-orbit maps on surfaces of small genus
Internal and external duality in abstract polytopes
We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type for and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type for even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here