### 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(H_{4}) 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 |
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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 |

### Fingerprint

### Keywords

- Coxeter group
- Ising-model
- Polytopes
- Quaternions

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics

### Cite this

*Proceedings of the 13th Regional Conference on Mathematical Physics*(pp. 40-60)

**Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes.** / Koca, Mehmet; Koca, N. Özdeş.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 13th Regional Conference on Mathematical Physics.*pp. 40-60, 2010 13th Regional Conference on Mathematical Physics, Antalya, Turkey, 10/27/10.

}

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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UR - http://www.scopus.com/inward/citedby.url?scp=84892736411&partnerID=8YFLogxK

M3 - Conference contribution

SN - 9789814417525

SP - 40

EP - 60

BT - Proceedings of the 13th Regional Conference on Mathematical Physics

ER -