An introduction to Coxeter polyhedra

Abstract

This paper is an introduction to Coxeter polyhedra in spherical, Euclidean, and hyperbolic geometries. It consists of essentially two parts that could be read independently. In the first we introduce non-obtuse polyhedra in the spherical, Euclidean, and hyperbolic spaces, and prove various fundamental theorems originated from Andreev, Coxeter, and Vinberg. In the second we introduce Coxeter polyhedra and use them to describe regular, semiregular, and uniform polyhedra and tessellations, mostly via the Wythoff construction.