4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices
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arXiv
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| Format: | Preprint |
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2016
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| author | Koca, Mehmet Koca, Nazife Ozdes Al-Qanobi, Amal Juma Hamood |
| author_facet | Koca, Mehmet Koca, Nazife Ozdes Al-Qanobi, Amal Juma Hamood |
| contents | We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4) implying that it is a group relevant to the crystallography as well as quasicrystallographic structures in 4-dimensions. First we review the pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron- pyritohedron and the pseudo icosidodecahedron. In quaternionic representations it finds a natural extension to the 4-dimensions.The related polytopes turn out to be the pseudo snub 24-cell and its dual polytope expressed in terms of a parameter x leading to snub 24-cell and its dual in the limit where the parameter x takes the golden ratio. It turns out that the relevant lattice is the root lattice of W(D4). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1606_02445 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | 4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices Koca, Mehmet Koca, Nazife Ozdes Al-Qanobi, Amal Juma Hamood Mathematical Physics We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4) implying that it is a group relevant to the crystallography as well as quasicrystallographic structures in 4-dimensions. First we review the pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron- pyritohedron and the pseudo icosidodecahedron. In quaternionic representations it finds a natural extension to the 4-dimensions.The related polytopes turn out to be the pseudo snub 24-cell and its dual polytope expressed in terms of a parameter x leading to snub 24-cell and its dual in the limit where the parameter x takes the golden ratio. It turns out that the relevant lattice is the root lattice of W(D4). |
| title | 4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/1606.02445 |