Regular polytope

Abstract: polymake is a software tool designed for the algorithmic treatment of polytopes and polyhedra. We give an overview of the functionality as well as of the structure. This paper can be seen as a first approximation to a polymake handbook.

Abstract: 1 Convex Sets.- 1. The Affine Structure of ?d.- 2. Convex Sets.- 3. The Relative Interior of a Convex Set.- 4. Supporting Hyperplanes and Halfspaces.- 5. The Facial Structure of a Closed Convex Set.- 6. Polarity.- 2 Convex Polytopes.- 7. Polytopes.- 8. Polyhedral Sets.- 9. Polarity of Polytopes and Polyhedral Sets.- 10. Equivalence and Duality of Polytopes.- 11. Vertex-Figures.- 12. Simple and Simplicial Polytopes.- 13. Cyclic Polytopes.- 14. Neighbourly Polytopes.- 15. The Graph of a Polytope.- 3 Combinatorial Theory of Convex Polytopes.- 16. Euler s Relation.- 17. The Dehn-Sommerville Relations.- 18. The Upper Bound Theorem.- 19. The Lower Bound Theorem.- 20. McMullen s Conditions.- Appendix 1 Lattices.- Appendix 2 Graphs.- Appendix 3 Combinatorial Identities.- Bibliographical Comments.- List of Symbols.

Abstract: regular polytopes stand at the end of more than two millennia of geometrical research, which began with regular polygons and polyhedra. They are highly symmetric combinatorial structures with distinctive geometric, algebraic or topological properties; in many ways more fascinating than traditional regular polytopes and tessellations. The rapid development of the subject in the past 20 years has resulted in a rich new theory, featuring an attractive interplay of mathematical areas, including geometry, combinatorics, group theory and topology. regular polytopes and their groups provide an appealing new approach to understanding geometric and combinatorial symmetry. This is the first comprehensive up-to-date account of the subject and its ramifications, and meets a critical need for such a text, because no book has been published in this area of classical and modern discrete geometry since Coxeter's Regular Polytopes (1948) and Regular Complex Polytopes (1974). The book should be of interest to researchers and graduate students in discrete geometry, combinatorics and group theory.

Abstract: H. S. M. Coxeter: Published Works.- I: Polytopes and Honeycombs.- Uniform Tilings with Hollow Tiles.- Spherical Tilings with Transitivity Properties.- Some Isonemal Fabrics on Polyhedral Surfaces.- Convex Bodies which Tile Space.- Geometry of Radix Representations.- Embeddability of Regular Polytopes and Honeycombs in Hypercubes.- The Derivation of Schoenberg's Star-Polytopes from Schoute's Simplex Nets.- The Harmonic Analysis of Skew Polygons as a Source of Outdoor Sculptures.- The Geometry of African Art III. The Smoking Pipes of Begho.- Crystallography and Cremona Transformations.- Cubature Formulae, Polytopes, and Spherical Designs.- Two Quaternionic 4-Polytopes.- Span-Symmetric Generalized Quadrangles.- On Coxeter's Loxodromic Sequences of Tangent Spheres.- II: Extremal Problems.- Elementary Geometry, Then and Now.- Some Researches Inspired by H. S. M. Coxeter.- Some Problems in the Geometry of Convex Bodies.- On an Analog to Minkowski's Lattice Point Theorem.- Intersections of Convex Bodies with Their Translates.- An Extremal Property of Plane Convex Curves- P. Ungar's Conjecture.- III: Geometric Transformations.- Polygons and Polynomials.- Algebraic Surfaces with Hyperelliptic Sections.- On the Circular Transformations of Mobius, Laguerre, and Lie.- The Geometry of Cycles, and Generalized Laguerre Inversion.- Inversive Geometry.- Absolute Polarities and Central Inversions.- Products of Axial Affinities and Products of Central Collineations.- Normal Forms of Isometries.- Finite Geometries with Simple, Semisimple, and Quasisimple Fundamental Groups.- Motions in a Finite Hyperbolic Plane.- IV: Groups and Presentations of Groups.- Generation of Linear Groups.- On Covering Klein's Curve and Generating Projective Groups.- A Local Approach to Buildings.- Representations and Coxeter Graphs.- Coinvariant Theory of a Coxeter Group.- Two-Generator Two-Relation Presentations for Special Linear Groups.- Groups Related to Fa,b,c Involving Fibonacci Numbers.- V: The Combinatorial Side.- Convex Polyhedra.- Non-Hamilton Fundamental Cycle Graphs.- Some Combinatorial Identities.- Binary Views of Ternary Codes.