Polyhedra obtained from Coxeter groups and quaternions

Mehmet Koca, Mudhahir Al-Ajmi, Ramazan Ko̧

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

Original languageEnglish
Article number113514
JournalJournal of Mathematical Physics
Volume48
Issue number11
DOIs
Publication statusPublished - 2007

Fingerprint

quaternions
Coxeter Group
Weyl Group
Archimedean solid
Quaternion
polyhedrons
Platonic solid
Polyhedron
Semiregular
apexes
diagrams
Diagram
Symmetry
Dynkin Diagram
polytopes
Root System
Polytopes
Polytope
Irreducible Representation
symmetry

ASJC Scopus subject areas

  • Physics and Astronomy(all)
  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Polyhedra obtained from Coxeter groups and quaternions. / Koca, Mehmet; Al-Ajmi, Mudhahir; Ko̧, Ramazan.

In: Journal of Mathematical Physics, Vol. 48, No. 11, 113514, 2007.

Research output: Contribution to journalArticle

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