Truncation symmetry type graphs

There are operations that transform a map M (an embedding of a graph on a surface) into another map in the same surface, modifying its structure and consequently its set of flags F(M). For instance, by truncating all the vertices of a map M, each flag in F(M) is divided into three flags of the truncated map. Orbanic, Pellicer and Weiss studied the truncation of k-orbit maps for k < 4. They introduced the notion of T-compatible maps in order to give a necessary condition for a truncation of a k-orbit map to be either k-, 3k/2- or 3k-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of k-orbit maps for k < 8 and k=9

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

Hereditary Polytopes

Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary, but the other polytopes in this class are interesting, have possible applications in modeling of structures, and have not been previously investigated. This paper establishes the basic theory of hereditary polytopes, focussing on the analysis and construction of hereditary polytopes with highly symmetric faces.Comment: Discrete Geometry and Applications (eds. R.Connelly and A.Ivic Weiss), Fields Institute Communications, (23 pp, to appear

Symmetry-type graphs of Platonic and Archimedean solids

A recently developed theory of flag-graphs and $k$-orbit maps classifies maps according to their symmetry-type graphs. We propose a similar classification for polyhedra showing that Platonic and Archimedean solids with the same vertex pattern have isomorphic symmetry-type graphs and introducing some tools for the determination of symmetry-type graphs of any polyhedron

Realizing finite edge-transitive orientable maps

J.E. Graver and M.E. Watkins, Memoirs Am. Math. Soc. 126 (601) (1997) established that the automorphism group of an edge-transitive, locally finite map manifests one of exactly 14 algebraically consistent combinations (called types) of the kinds of stabilizers of its edges, its vertices, its faces, and its Petrie walks. Exactly eight of these types are realized by infinite, locally finite maps in the plane. H.S.M. Coxeter (Regular Polytopes, 2nd ed., McMillan, New York, 1963) had previously observed that the nine finite edge-transitive planar maps realize three of the eight planar types. In the present work, we show that for each of the 14 types and each integer n ≥ 11 such that n ≡ 3, 11 (mod 12), there exist finite, orientable, edge-transitive maps whose various stabilizers conform to the given type and whose automorphism groups are (abstractly) isomorphic to the symmetric group Sym(n). Exactly seven of these types (not a subset of the planar eight) are shown to admit infinite families of finite, edge-transitive maps on the torus, and their automorphism groups are determined explicitly. Thus all finite, edge-transitive toroidal maps are classified according to this schema. Finally, it is shown that exactly one of the 14 types can be realized as an abelian group of an edge-transitive map, namely, as ℤn × ℤ2 where n ≡ 2 (mod 4)