39 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
Polyhedra, Complexes, Nets and Symmetry
Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are
finite or infinite 3-periodic structures with interesting geometric,
combinatorial, and algebraic properties. They can be viewed as finite or
infinite 3-periodic graphs (nets) equipped with additional structure imposed by
the faces, allowed to be skew, zig-zag, or helical. A polyhedron or complex is
"regular" if its geometric symmetry group is transitive on the flags (incident
vertex-edge-face triples). There are 48 regular polyhedra (18 finite polyhedra
and 30 infinite apeirohedra), as well as 25 regular polygonal complexes, all
infinite, which are not polyhedra. Their edge graphs are nets well-known to
crystallographers, and we identify them explicitly. There also are 6 infinite
families of "chiral" apeirohedra, which have two orbits on the flags such that
adjacent flags lie in different orbits.Comment: Acta Crystallographica Section A (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
Regular incidence-polytopes with Euclidean or toroidal faces and vertex-figures
AbstractRegular incidence-polytopes are combinatorial generalizations of regular polyhedra. Certain group-theoretical constructions lead to many new regular incidence-polytopes whose faces and vertex-figures are combinatorially isomorphic to classical Euclideanly regular polytopes or regular maps on the torus
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