Faces of Platonic solids in all dimensions

A polytope is called Platonic if it has faces of one kind in every dimension, 0 (vertices), 1 (edges), 2 (faces), …. The three-dimensional Platonic solids/polytopes have been known since antiquity: tetrahedron, octahedron, cube, dodecahedron and icosahedron. Platonic solids in any dimension n < ∞ were identified in the 20th century. In this paper a simple recursive method of decoration of the Coxeter–Dynkin diagram is used to describe the faces of all types and dimensions.