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

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 $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here

Algorithms for classifying regular polytopes with a fixed automorphism group

In this paper, various algorithms used in the classifications of regular polytopes for given groups are compared. First computational times and memory usages are analyzed for the original algorithm used in one of these classifications. Second, a possible algorithm for isomorphism testing among polytopes is suggested. Then, two improved algorithms are compared, and finally, results are given for a new classification of all regular polytopes for certain alternating groups and for the sporadic group $Co_3$

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 $\{p, p\}$ for $p \geq 3$ and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type $\{p, p\}$ for $p$ even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here

String C-groups as transitive subgroups of Sn

If $\Gamma$ is a string C-group which is isomorphic to a transitive subgroup of the symmetric group Sn (other than Sn and the alternating group An), then the rank of is at most n/2+1, with nitely many exceptions (which are classi ed). It is conjectured that only the symmetric group has to be excluded