### Abstract

The vertices of the four-dimensional polytope {3, 3, 5} and its dual {5, 3, 3} admitting the symmetry of the non-crystallographic Coxeter group W(H _{4}) of order 14 400 are represented in terms of quaternions with unit norm where the polytope {3, 3, 5} is represented by the elements of the binaryicosahedral group of quaternions of order 120. We projected the polytopes to three-dimensional Euclidean space where the quaternionic vertices are the orbits of the Coxeter group W(H_{3}), icosahedral group with inversion, where W(H_{3}) × Z_{2} is one of the maximal subgroups of the Coxeter group W(H_{4}). The orbits of the icosahedral group W(H _{3}) in the polytope {3, 3, 5} are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons, dodecahedrons and one icosidodecahedron in three dimensions. The 15 orbits of the icosahedral group W(H_{3}) in the polytope {5, 3, 3} represent the dodecahedrons, icosidodecahedrons, small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry. One of the convex solids with 60 vertices is very similar to the truncated icosahedron (soccer ball) but with two different edge lengths which can be taken as a realistic model of the C _{60} molecule at extreme temperature and pressure.

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

Article number | 013 |

Pages (from-to) | 14047-14054 |

Number of pages | 8 |

Journal | Journal of Physics A: Mathematical and General |

Volume | 39 |

Issue number | 45 |

DOIs | |

Publication status | Published - Nov 10 2006 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Physics A: Mathematical and General*,

*39*(45), 14047-14054. [013]. https://doi.org/10.1088/0305-4470/39/45/013

**Quaternionic representation of the Coxeter group W(H4) and the polyhedra.** / Koca, Mehmet; Al-Ajmi, Mudhahir; Koç, Ramazan.

Research output: Contribution to journal › Article

*Journal of Physics A: Mathematical and General*, vol. 39, no. 45, 013, pp. 14047-14054. https://doi.org/10.1088/0305-4470/39/45/013

}

TY - JOUR

T1 - Quaternionic representation of the Coxeter group W(H4) and the polyhedra

AU - Koca, Mehmet

AU - Al-Ajmi, Mudhahir

AU - Koç, Ramazan

PY - 2006/11/10

Y1 - 2006/11/10

N2 - The vertices of the four-dimensional polytope {3, 3, 5} and its dual {5, 3, 3} admitting the symmetry of the non-crystallographic Coxeter group W(H 4) of order 14 400 are represented in terms of quaternions with unit norm where the polytope {3, 3, 5} is represented by the elements of the binaryicosahedral group of quaternions of order 120. We projected the polytopes to three-dimensional Euclidean space where the quaternionic vertices are the orbits of the Coxeter group W(H3), icosahedral group with inversion, where W(H3) × Z2 is one of the maximal subgroups of the Coxeter group W(H4). The orbits of the icosahedral group W(H 3) in the polytope {3, 3, 5} are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons, dodecahedrons and one icosidodecahedron in three dimensions. The 15 orbits of the icosahedral group W(H3) in the polytope {5, 3, 3} represent the dodecahedrons, icosidodecahedrons, small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry. One of the convex solids with 60 vertices is very similar to the truncated icosahedron (soccer ball) but with two different edge lengths which can be taken as a realistic model of the C 60 molecule at extreme temperature and pressure.

AB - The vertices of the four-dimensional polytope {3, 3, 5} and its dual {5, 3, 3} admitting the symmetry of the non-crystallographic Coxeter group W(H 4) of order 14 400 are represented in terms of quaternions with unit norm where the polytope {3, 3, 5} is represented by the elements of the binaryicosahedral group of quaternions of order 120. We projected the polytopes to three-dimensional Euclidean space where the quaternionic vertices are the orbits of the Coxeter group W(H3), icosahedral group with inversion, where W(H3) × Z2 is one of the maximal subgroups of the Coxeter group W(H4). The orbits of the icosahedral group W(H 3) in the polytope {3, 3, 5} are the conjugacy classes of the binary icosahedral group and represent a number of icosahedrons, dodecahedrons and one icosidodecahedron in three dimensions. The 15 orbits of the icosahedral group W(H3) in the polytope {5, 3, 3} represent the dodecahedrons, icosidodecahedrons, small rhombicosidodecahedrons and some convex solids possessing the icosahedral symmetry. One of the convex solids with 60 vertices is very similar to the truncated icosahedron (soccer ball) but with two different edge lengths which can be taken as a realistic model of the C 60 molecule at extreme temperature and pressure.

UR - http://www.scopus.com/inward/record.url?scp=33947268117&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33947268117&partnerID=8YFLogxK

U2 - 10.1088/0305-4470/39/45/013

DO - 10.1088/0305-4470/39/45/013

M3 - Article

VL - 39

SP - 14047

EP - 14054

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 45

M1 - 013

ER -