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Faces of Platonic solids in all dimensions

Faces of Platonic solids in all dimensions This paper considers Platonic solids/polytopes in the real Euclidean space 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, F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non‐crystallographic Coxeter groups H3, H4. The method consists of 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. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Crystallographica Section A Foundations of Crystallography Wiley

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References (15)

Publisher
Wiley
Copyright
Copyright © 2014 Wiley Subscription Services, Inc., A Wiley Company
ISSN
0108-7673
eISSN
1600-5724
DOI
10.1107/S205327331400638X
pmid
25970193
Publisher site
See Article on Publisher Site

Abstract

This paper considers Platonic solids/polytopes in the real Euclidean space 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, F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non‐crystallographic Coxeter groups H3, H4. The method consists of 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. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.

Journal

Acta Crystallographica Section A Foundations of CrystallographyWiley

Published: Jul 1, 2014

Keywords: ; ; ; ;

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