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 language | English |
|---|---|
| Title of host publication | Proceedings of the 13th Regional Conference on Mathematical Physics |
| Pages | 40-60 |
| Number of pages | 21 |
| Publication status | Published - 2013 |
| Event | 2010 13th Regional Conference on Mathematical Physics - Antalya, Turkey Duration: Oct 27 2010 → Oct 31 2010 |
Other
| Other | 2010 13th Regional Conference on Mathematical Physics |
|---|---|
| Country | Turkey |
| City | Antalya |
| Period | 10/27/10 → 10/31/10 |
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Keywords
- Coxeter group
- Ising-model
- Polytopes
- Quaternions
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
Cite this
Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes. / Koca, Mehmet; Koca, N. Özdeş.
Proceedings of the 13th Regional Conference on Mathematical Physics. 2013. p. 40-60.Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
TY - GEN
T1 - Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes
AU - Koca, Mehmet
AU - Koca, N. Özdeş
PY - 2013
Y1 - 2013
N2 - 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.
AB - 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.
KW - Coxeter group
KW - Ising-model
KW - Polytopes
KW - Quaternions
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M3 - Conference contribution
SN - 9789814417525
SP - 40
EP - 60
BT - Proceedings of the 13th Regional Conference on Mathematical Physics
ER -