The decoration of a Coxeter—Dynkin diagram and the Schläfli symbol as two methods to describe polytopes generated by finite reflection groups

Abstract

The aim of this paper is to present two methods of describing uniform convex polytopes in three dimensions, namely, the Schläfli symbol and the method of decorating a Coxeter–Dynkin diagram. The reflection-generated polytopes are important objects, both in an abstract and in a physical context. In mathematics such polytopes are orbits of finite reflections groups, while in chemistry, biology and physics they serve as physical models for molecules, proteins, viruses and other three-dimensional structures. For this reason it is important to be able to determine the structure of polytopes and polyhedra, and to have a uniform notation that describes them. The two techniques are explained and demonstrated on the examples of polytopes generated by the finite Coxeter groups A3 and H3.