Faces of platonic solids in all dimensions

TL;DR

This paper classifies and describes Platonic solids and polytopes in all dimensions using Coxeter groups, detailing their faces and dual pairs through a recursive diagram decoration method.

Contribution

It introduces a recursive Coxeter diagram decoration approach to systematically analyze faces of Platonic polytopes in arbitrary dimensions.

Findings

Classification of Platonic polytopes in all dimensions

Recursive method for face analysis using Coxeter diagrams

Identification of conditions for solids to be considered Platonic

Abstract

This paper considers Platonic solids/polytopes in the real Euclidean space R^n of dimension 3 <= n < infinity. 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, F4 and of non-crystallographic Coxeter groups H3, H4. Our method consists in recursively decorating the appropriate Coxeter-Dynkin 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.