User:Mosher/Draft of intro

A Coxeter group is a mathematical group defined by a presentation which says that every generator has order two, and which gives the order of the product of every pair of generators. Coxeter groups are also called abstract reflection groups. Many (but not all) Coxeter groups act discretely by isometries on Euclidean or hyperbolic space of some dimension, generated by reflections across hyperplanes. On the other hand, all Coxeter groups act discretely by affine transformations on Euclidean space of some dimension, generated by affine reflections across hyperplanes.

Coxeter groups are ubiquitous in mathematics and geometry. The finite Coxeter groups are precisely the finite Euclidean reflection groups. Consequently, the symmetry groups of all the regular polytopes are finite Coxeter groups. The Weyl groups of root systems are also all special cases of finite Coxeter groups. A triangle group is a special kind of Coxeter group, generated by reflections in the sides of a triangle in the sphere, the Euclidean plane, or the hyperbolic plane.

Coxeter groups are named for the geometer H. S. M. Coxeter.