### Abstract

In this paper, we construct the quasi regular polyhedra and their duals, which are the generalizations of the Archimedean and Catalan solids, respectively. This work is an extension of two previous papers of ours, which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups, W(A_{3}), W(B_{3}) and W (H_{3}). When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups, which necessarily leads to the quaternionic representation of the vertices. We note the fact that the C_{60} molecule can best be represented by a truncated icosahedron, where the hexagonal faces are not regular. Rather, they are isogonal hexagons with single and double bonds of the carbon atoms represented by the alternating edge lengths of isogonal hexagons.

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

Pages (from-to) | 53-67 |

Number of pages | 15 |

Journal | African Review of Physics |

Volume | 6 |

Publication status | Published - 2011 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

**Quasi regular polyhedra and their duals with coxeter symmetries represented by quaternions -II.** / Koca, Mehmet; Ajmi, Mudhahir Al; Al-Shidhani, Saleh.

Research output: Contribution to journal › Article

*African Review of Physics*, vol. 6, pp. 53-67.

}

TY - JOUR

T1 - Quasi regular polyhedra and their duals with coxeter symmetries represented by quaternions -II

AU - Koca, Mehmet

AU - Ajmi, Mudhahir Al

AU - Al-Shidhani, Saleh

PY - 2011

Y1 - 2011

N2 - In this paper, we construct the quasi regular polyhedra and their duals, which are the generalizations of the Archimedean and Catalan solids, respectively. This work is an extension of two previous papers of ours, which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups, W(A3), W(B3) and W (H3). When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups, which necessarily leads to the quaternionic representation of the vertices. We note the fact that the C60 molecule can best be represented by a truncated icosahedron, where the hexagonal faces are not regular. Rather, they are isogonal hexagons with single and double bonds of the carbon atoms represented by the alternating edge lengths of isogonal hexagons.

AB - In this paper, we construct the quasi regular polyhedra and their duals, which are the generalizations of the Archimedean and Catalan solids, respectively. This work is an extension of two previous papers of ours, which were based on the Archimedean and Catalan solids obtained as the orbits of the Coxeter groups, W(A3), W(B3) and W (H3). When these groups act on an arbitrary vector in 3D Euclidean space they generate the orbits corresponding to the quasi regular polyhedra. Special choices of the vectors lead to the platonic and Archimedean solids. In general, the faces of the quasi regular polyhedra consist of the equilateral triangles, squares, regular pentagons as well as rectangles, isogonal hexagons, isogonal octagons, and isogonal decagons depending on the choice of the Coxeter groups of interest. We follow the quaternionic representation of the group elements of the Coxeter groups, which necessarily leads to the quaternionic representation of the vertices. We note the fact that the C60 molecule can best be represented by a truncated icosahedron, where the hexagonal faces are not regular. Rather, they are isogonal hexagons with single and double bonds of the carbon atoms represented by the alternating edge lengths of isogonal hexagons.

UR - http://www.scopus.com/inward/record.url?scp=84860489456&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84860489456&partnerID=8YFLogxK

M3 - Article

VL - 6

SP - 53

EP - 67

JO - African Review of Physics

JF - African Review of Physics

SN - 2223-6589

ER -