Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues

  title={Regular Skew Polyhedra in Three and Four Dimension, and their Topological Analogues},
  author={H. S. M. Coxeter},
  journal={Proceedings of The London Mathematical Society},
  • H. Coxeter
  • Published 1938
  • Mathematics
  • Proceedings of The London Mathematical Society
Petrie-Coxeter maps revisited.
This paper presents a technique for constructing new chiral or regular polyhedra (or maps) from self-dual abstract chiral polytopes of rank 4. From improperly selfdual chiral polytopes we derive
Geometric Realizations of Cyclically Branched Coverings over Punctured Spheres.
In classical differential geometry, a central question has been whether abstract surfaces with given geometric features can be realized as surfaces in Euclidean space. Inspired by the rich theory of
Convex-Faced Combinatorially Regular Polyhedra of Small Genus
It is strongly conjecture that in addition to those eight regular maps of genus g known to have polyhedral realizations, there are no other regular maps, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3.
The {4, 5} Isogonal Sponges on the Cubic Lattice
A notation for labeling three-dimensional isogonal polyhedra is introduced and it is shown how this notation can be combinatorially used to find all of the isogsonal polyhedral shapes that can be created given a specific vertex star configuration.
Polyhedral Surfaces, Polytopes, and Projections
The author thanks his beloved wife for her love, support and patience during all the ups and downs during the preparation of his PhD thesis.
The Regular Maps on a Surface of Genus Three
  • F. A. Sherk
  • Mathematics
    Canadian Journal of Mathematics
  • 1959
A considerable volume of research on the theory of regular maps is now in existence. Systematic enumerations of regular maps on the surfaces of genus 1 and 2 were begun by Brahana (1; 2) and
The five Platonic solids have nice algebraic, geometric, and combinatorial properties, so mathematicians have often tried to find analogues, which have some, but not all, properties in common with
On Laves' Graph Of Girth Ten
  • H. Coxeter
  • Mathematics
    Canadian Journal of Mathematics
  • 1955
1. Introduction. This note shows how a certain infinite graph of degree three, discovered by Laves in connection with crystal structure, can be inscribed (in sixteen ways, all alike) in an infinite
On the Rational Bredon Cohomology of Equivariant Configuration Spaces
. Bredon cohomology is a cohomology theory that applies to topological spaces equipped with the group actions. For any group G , given a real linear representation V , the configuration space of V has