Faces of Platonic solids in all dimensions

Abstract

Abstract.This paper considers Platonic solids/polytopes in the real Euclidean space R n of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n−1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types An, Bn, Cn, F 4 and of non-crystallographic Coxeter groups H 3 , H 4 .Our method consists in recursively decorating the appropriate CoxeterDynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic.

The polytopes of the H 3 group with 60 vertices and their orbit decompositions

“…It was first introduced by Moody & Patera (1992). In recent years, the method has been used, for example, to describe the faces of Platonic solids (Szajewska, 2014) and root polytopes (Szajewska, 2016) in all dimensions.…”

Faces of root polytopes in all dimensions

“…In Tables 3 and 4 of Champagne et al (1995) faces of four-dimensional polytopes with all symmetry groups were described as an example. Most recently, in Szajewska (2014), faces of Platonic solids in any dimension were described using the method of Champagne et al (1995).…”

“…We use the same rules for decoration of the corresponding connected Coxeter-Dynkin diagrams as used in Champagne et al (1995) and Szajewska (2014). The decoration of the nodes of the corresponding Coxeter-Dynkin diagram, which we use in this paper, coincides with that in Szajewska (2014). This method as it is applies to the root polytopes of the simple Lie algebras of type A n , D n , E 6 ; E 7 and E 8 .…”

“…We consider polytopes of the short roots as a separate case from the polytopes of long roots. Then it can be described again using the same decoration method as in Champagne et al (1995) and Szajewska (2014).…”

“…Root polytopes of the symmetry types A n and C n were studied by Mé szá ros (2009,2011); however those methods are limiting because they cannot be easily extended to root systems of other symmetries. A few root polytopes in dimensions n > 1 are Platonic solids that were described in Szajewska (2014). The root polytopes A 3 ; B 3 ; C 3 ; D 4 ; F 4 and H 4 are special cases of polytopes generated by Coxeter groups in R 3 and R 4 that are found in Tables 2, 3 and 4 in Champagne et al (1995).…”