Book contents
- Frontmatter
- Contents
- Preface
- 1 Classical Regular Polytopes
- 2 Regular Polytopes
- 3 Coxeter Groups
- 4 Amalgamation
- 5 Realizations
- 6 Regular Polytopes on Space-Forms
- 7 Mixing
- 8 Twisting
- 9 Unitary Groups and Hermitian Forms
- 10 Locally Toroidal 4-Polytopes: I
- 11 Locally Toroidal 4-Polytopes: II
- 12 Higher Toroidal Polytopes
- 13 Regular Polytopes Related to Linear Groups
- 14 Miscellaneous Classes of Regular Polytopes
- Bibliography
- Indices
- List of Symbols
- Author Index
- Subject Index
3 - Coxeter Groups
Published online by Cambridge University Press: 20 August 2009
- Frontmatter
- Contents
- Preface
- 1 Classical Regular Polytopes
- 2 Regular Polytopes
- 3 Coxeter Groups
- 4 Amalgamation
- 5 Realizations
- 6 Regular Polytopes on Space-Forms
- 7 Mixing
- 8 Twisting
- 9 Unitary Groups and Hermitian Forms
- 10 Locally Toroidal 4-Polytopes: I
- 11 Locally Toroidal 4-Polytopes: II
- 12 Higher Toroidal Polytopes
- 13 Regular Polytopes Related to Linear Groups
- 14 Miscellaneous Classes of Regular Polytopes
- Bibliography
- Indices
- List of Symbols
- Author Index
- Subject Index
Summary
A central role in the theory of abstract regular polytopes is played by an important class of groups known as Coxeter groups. Historically, Coxeter groups made their first appearance as the symmetry groups of the classical regular polytopes and tessellations, or as related reflexion groups [120, Chapter 11]. Since then, they have occurred in many branches of mathematics.
The purpose of the first three sections of this chapter is to review some of the basic properties of Coxeter groups. We shall not give a full exposition of the subject here, but instead focus attention on those results which will be used later. In particular, we shall state most results without proof. For further notes, references, historical remarks, and proofs, the reader is referred to Cohen [81, 82], Coxeter [120], Grove and Benson [195], Hiller [219], Humphreys [222], Koszul [254], and Tits [417].
Then, in Section 3D, we discuss the class of universal regular polytopes {p1, …, pn-1}, which includes the traditional convex regular polytopes as well as the euclidean or hyperbolic regular tessellations. Finally, in Section 3E, we describe how the Brianchon—Gram theorem for angle sums of convex polytopes can be employed to calculate the order of a finite Coxeter group in a purely elementary manner.
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- Abstract Regular Polytopes , pp. 64 - 94Publisher: Cambridge University PressPrint publication year: 2002