@inproceedings{04fe7bf352c94e958ab3ab14a32c802a,

title = "Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes",

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.",

keywords = "Coxeter group, Ising-model, Polytopes, Quaternions",

author = "Mehmet Koca and Koca, {N. {\"O}zde{\c s}}",

year = "2013",

language = "English",

isbn = "9789814417525",

series = "Proceedings of the 13th Regional Conference on Mathematical Physics",

pages = "40--60",

booktitle = "Proceedings of the 13th Regional Conference on Mathematical Physics",

note = "2010 13th Regional Conference on Mathematical Physics ; Conference date: 27-10-2010 Through 31-10-2010",

}