Realizations of Regular Abstract Polyhedra of Types {3,6} and {6,3}

Abstract

This paper classifies and gives methods for computing the irreducible realizations of the abstract polyhedra corresponding to regular maps of type {3, 6} and {6, 3}. A complete list of irreducible realizations is given for polyhedra of type {3, 6}.

Realizations of Regular Toroidal Maps of Type {4,4}

“…As yet, we lack a more insightful proof of this. (The same situation occurs for maps of type {3, 6}, as explained in [1]. )…”

“…If these realizations were congruent, corresponding translations g j x g k y would have equal traces, as described in(6). Taking ( j, k) = (1, 0) and(1,1), and recalling that 0 < m j < j < b/2, we soon find that m 1 = m 2 and 1 = 2 . Cases (ii)-(vii) follow similarly.…”

“…Now while an abstract approach clarifies the general properties and construction of polytopes, it is nevertheless still interesting and useful to consider concrete realizations, for example, as symmetric objects in Euclidean space (see [6]). For example, in [1], Burgiel and Stanton describe the pure realizations of the finite, regular toroidal maps of type {3, 6}, essentially by examining the action of the group on a unitary space whose basis is identified with the vertex set of the map.…”

Problems on polytopes, their groups, and realizations

“…The fine structure of the realization cone is only known for a small number of polytopes, including the regular convex polytopes, with the exception of the 120-cell {5, 3, 3} and 600-cell {3, 3, 5}, and the regular toroids of rank 3 (see [42, Section 5B] and [6,45,46]).…”

Realizations of Regular Toroidal Maps

“…In this paper, we investigate the pure realizations of finite regular toroidal polyhedra (or maps) of type {3, 6} and {6, 3}. Burgiel and Stanton have described elsewhere the pure realizations of these maps, essentially by examining the action of the automorphism group on a unitary space whose basis is identified with the vertex set of the map [1]. Here we take a somewhat different approach, which allows us to explicitly describe real representations of the group.…”