Euclidean Space

The main purpose of this chapter is to discuss groups generated by reflexions, concentrating here on the finite and discrete infinite groups in euclidean spaces. While establishing notation and conventions, there are surveys of the algebraic and metrical properties of euclidean spaces, and a treatment of the main features of convex sets that are appealed to subsequently. The classification of the finite and discrete infinite reflexion groups in euclidean spaces is a core feature; the initial part of the treatment is novel. There is then a brief description of subgroup relationships among these groups. Certain angle-sum relations for polytopes and cones are employed to find the orders of the finite Coxeter groups by purely elementary geometric methods; these are established here for polyhedral sets in general. The lower-dimensional spaces are somewhat special. The finite rotation groups in three dimensions are classified, and are shown to be subgroups of reflexion groups. Finally, there is an introduction to quaternions, which provide an alternative approach to finite orthogonal groups in 4-dimensional space; these are needed to describe certain regular polyhedra in that space.