978 research outputs found
Regular Polyhedra of Index Two, II
A polyhedron in Euclidean 3-space is called a regular polyhedron of index 2
if it is combinatorially regular and its geometric symmetry group has index 2
in its combinatorial automorphism group; thus its automorphism group is
flag-transitive but its symmetry group has two flag orbits. The present paper
completes the classification of finite regular polyhedra of index 2 in 3-space.
In particular, this paper enumerates the regular polyhedra of index 2 with
vertices on one orbit under the symmetry group. There are ten such polyhedra.Comment: 33 pages; 5 figures; to appear in "Contributions to Algebra and
Geometry
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
Polygonal Complexes and Graphs for Crystallographic Groups
The paper surveys highlights of the ongoing program to classify discrete
polyhedral structures in Euclidean 3-space by distinguished transitivity
properties of their symmetry groups, focussing in particular on various aspects
of the classification of regular polygonal complexes, chiral polyhedra, and
more generally, two-orbit polyhedra.Comment: 21 pages; In: Symmetry and Rigidity, (eds. R.Connelly, A.Ivic Weiss
and W.Whiteley), Fields Institute Communications, to appea
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