An introduction to Coxeter polyhedra

TL;DR

This paper introduces Coxeter polyhedra across different geometries, covering fundamental theorems and their applications to regular and uniform tessellations using the Wythoff construction.

Contribution

It provides an accessible introduction to Coxeter polyhedra, including key theorems and their use in describing various symmetric polyhedra and tessellations.

Findings

Proves fundamental theorems from Coxeter, Vinberg, and Andreev.

Describes regular, semiregular, and uniform polyhedra using Coxeter groups.

Explains the Wythoff construction for generating tessellations.

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.