### Abstract

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

Original language | English |
---|---|

Article number | 113514 |

Journal | Journal of Mathematical Physics |

Volume | 48 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*48*(11), [113514]. https://doi.org/10.1063/1.2809467

**Polyhedra obtained from Coxeter groups and quaternions.** / Koca, Mehmet; Al-Ajmi, Mudhahir; Ko̧, Ramazan.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 48, no. 11, 113514. https://doi.org/10.1063/1.2809467

}

TY - JOUR

T1 - Polyhedra obtained from Coxeter groups and quaternions

AU - Koca, Mehmet

AU - Al-Ajmi, Mudhahir

AU - Ko̧, Ramazan

PY - 2007

Y1 - 2007

N2 - We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

AB - We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

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U2 - 10.1063/1.2809467

DO - 10.1063/1.2809467

M3 - Article

VL - 48

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 11

M1 - 113514

ER -