### 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 |
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Country | Turkey |

City | Antalya |

Period | 10/27/10 → 10/31/10 |

### Keywords

- Coxeter group
- Ising-model
- Polytopes
- Quaternions

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics

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## Cite this

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