The Five Platonic Solids and their Connection to Root Systems

Abstract

Platonic solids have fascinated humans for thousands of years. In ancient times, they were associated with the elements fire, air, water, earth, and aether. These solids are completely symmetrical three-dimensional polyhedra. In this snapshot, it is first explained that there can only be five such polyhedra in the three-dimensional space. For this purpose, so-called Schläfli symbols and Coxeter graphs are introduced. More precisely, the (linear) Coxeter graphs correspond to the (linear) Schläfli symbols that, in turn, correspond exactly to the regular convex polyhedra. Through this one-to-one relationship, it is possible to classify the regular convex polytopes in any dimension by exploiting the classification of Coxeter graphs.