4D Pyritohedral Symmetry
We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 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 th...
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| Format: | Article |
| Language: | English |
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Sultan Qaboos University
2016-12-01
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| Series: | Sultan Qaboos University Journal for Science |
| Subjects: | |
| Online Access: | https://journals.squ.edu.om/index.php/squjs/article/view/469 |
| _version_ | 1819142598438158336 |
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| author | Nazife O. Koca Amal J.H. Al Qanobi Mehmet Koca |
| author_facet | Nazife O. Koca Amal J.H. Al Qanobi Mehmet Koca |
| author_sort | Nazife O. Koca |
| collection | DOAJ |
| description | We describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 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 crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes. |
| first_indexed | 2024-12-22T12:12:53Z |
| format | Article |
| id | doaj.art-ce5c8f2ca0ba4b489aa88287a98942c6 |
| institution | Directory Open Access Journal |
| issn | 1027-524X 2414-536X |
| language | English |
| last_indexed | 2024-12-22T12:12:53Z |
| publishDate | 2016-12-01 |
| publisher | Sultan Qaboos University |
| record_format | Article |
| series | Sultan Qaboos University Journal for Science |
| spelling | doaj.art-ce5c8f2ca0ba4b489aa88287a98942c62022-12-21T18:26:15ZengSultan Qaboos UniversitySultan Qaboos University Journal for Science1027-524X2414-536X2016-12-0121215016110.24200/squjs.vol21iss2pp150-1614644D Pyritohedral SymmetryNazife O. Koca0Amal J.H. Al Qanobi1Mehmet Koca2Department of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of OmanDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of OmanDepartment of Physics, College of Science, Sultan Qaboos University, P.O. Box 36, Al-Khoud 123, Muscat, Sultanate of OmanWe describe an extension of the pyritohedral symmetry in 3D to 4-dimensional Euclidean space and construct the group elements of the 4D pyritohedral group of order 576 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 crystallographic as well as quasicrystallographic structures in 4-dimensions. We derive the vertices of the 24 pseudoicosahedra, 24 tetrahedra and the 96 triangular pyramids forming the facets of the pseudo snub 24-cell. It turns out that the relevant lattice is the root lattice of W (D4). The vertices of the dual polytope of the pseudo snub 24-cell consists of the union of three sets: 24-cell, another 24-cell and a new pseudo snub 24-cell. We also derive a new representation for the symmetry group of the pseudo snub 24-cell and the corresponding vertices of the polytopes.https://journals.squ.edu.om/index.php/squjs/article/view/469PseudoicosahedronPyritohedronLatticeCoxeter groups and Quaternions. |
| spellingShingle | Nazife O. Koca Amal J.H. Al Qanobi Mehmet Koca 4D Pyritohedral Symmetry Sultan Qaboos University Journal for Science Pseudoicosahedron Pyritohedron Lattice Coxeter groups and Quaternions. |
| title | 4D Pyritohedral Symmetry |
| title_full | 4D Pyritohedral Symmetry |
| title_fullStr | 4D Pyritohedral Symmetry |
| title_full_unstemmed | 4D Pyritohedral Symmetry |
| title_short | 4D Pyritohedral Symmetry |
| title_sort | 4d pyritohedral symmetry |
| topic | Pseudoicosahedron Pyritohedron Lattice Coxeter groups and Quaternions. |
| url | https://journals.squ.edu.om/index.php/squjs/article/view/469 |
| work_keys_str_mv | AT nazifeokoca 4dpyritohedralsymmetry AT amaljhalqanobi 4dpyritohedralsymmetry AT mehmetkoca 4dpyritohedralsymmetry |