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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.

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