Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes

Mehmet Koca, N. Özdeş Koca

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Emergence of the experimental evidence of E8 in the analysis of one dimensional Ising-model invokes further studies of the Coxeter-Weyl groups generated by reections regarding their applications to polytopes. The Coxeter group W(H4) which describes the Platonic Polytopes 600-cell and 120-cell in 4D singles out in the mass relations of the bound states of the Ising model for it is a maximal subgroup of the Coxeter-Weyl group W(E8). There exists a one-to-one correspondence between the finite subgroups of quaternions and the Coxeter-Weyl groups of rank 4 which facilitates the study of the rank-4 Coxeter-Weyl groups. In this paper we study the systematic classifications of the 3D-polyhedra and 4D-polytopes through their symmetries described by the rank-3 and rank-4 Coxeter-Weyl groups represented by finite groups of quaternions. We also develop a technique on the constructions of the duals of the polyhedra and the polytopes and give a number of examples. Applications of the rank-2 Coxeter groups have been briey mentioned.

Original languageEnglish
Title of host publicationProceedings of the 13th Regional Conference on Mathematical Physics
Pages40-60
Number of pages21
Publication statusPublished - 2013
Event2010 13th Regional Conference on Mathematical Physics - Antalya, Turkey
Duration: Oct 27 2010Oct 31 2010

Other

Other2010 13th Regional Conference on Mathematical Physics
CountryTurkey
CityAntalya
Period10/27/1010/31/10

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Keywords

  • Coxeter group
  • Ising-model
  • Polytopes
  • Quaternions

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics

Cite this

Koca, M., & Koca, N. Ö. (2013). Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes. In Proceedings of the 13th Regional Conference on Mathematical Physics (pp. 40-60)