Regular Complex Polytopes

  title={Regular Complex Polytopes},
  author={H. S. M. Coxeter},
Frontispiece Preface to the second edition Preface to the first edition 1. Regular polygons 2. Regular polyhedra 3. Polyhedral kaleidoscopes 4. Real four-space and the unitary plane 5. Frieze patterns 6. The geometry of quaternions 7. The binary polyhedral groups 8. Unitary space 9. The unitary plane, using quaternions 10. The complete enumeration of finite reflection groups in the unitary plane 11. Regular complex polygons and Cayley diagrams 12. Regular complex polytopes defined and described… 
Beautiful Math, Part 6: Visualizing 4D Regular Polytopes Using the Kaleidoscope Principle
This article presents three interesting methods to visualize 4-RPs using a fundamental region algorithm.
Incidence-polytopes with toroidal cells
This work uses hyperbolic geometry to derive certain types of incidence-polytopes whose cells are isomorphic to maps of type {4, 4}, {6, 3}, or {3, 6} on a torus, and gives a criterion on the finiteness in terms of groups of 2 × 2 matrices.
Symmetry of Polytopes and Polyhedra
  • E. Schulte
  • Mathematics
    Handbook of Discrete and Computational Geometry, 2nd Ed.
  • 2004
Many symmetric figures discussed in earlier sections could be treated (and their structure clarified) in this more general framework.
Regular Polygonal Complexes in Space, II
A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces
Definitions in Polytopes
Multi-shell clusters represent complex structures, the study of which needs rigorous definitions in graph theory, geometry, set theory, etc. Within this chapter, main definitions for polyhedra,
Polygons, polyhedra, polytopes
The polytopes are, by definition, the convex envelopes of finite sets of points of an affine space. When this space is of dimension 2 (a plane), we speak of polygons; if the dimension is 3, we speak
Stereographic Visualization of 5-Dimensional Regular Polytopes
Based on the generalized stereographic projection, this paper establishes visualization methods for 5D RPs, which can preserve symmetries and convey general metric and topological data.
Wythoffian Skeletal Polyhedra in Ordinary Space, I
The blueprint for the construction is described and the Wythoffians for distinguished classes of regular polyhedra are treated, which are vertex-transitive and often feature vertex configurations with an attractive mix of different face shapes.
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly