Abstract
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.
Keywords
Publication details
- DOI
- 10.24200/squjs.vol21iss2pp150-161
- Journal
- Sultan Qaboos University Journal for Science, 21(2), 150
- Publisher
- Sultan Qaboos University
- Open access
- Gold open access
- License
- CC BY 4.0
Cite this article
APA 7
Koca, N. Ö., Al Qanobi, A. J., & Koca, M. (2016). 4D Pyritohedral Symmetry. Sultan Qaboos University Journal for Science, 21(2), 150. https://doi.org/10.24200/squjs.vol21iss2pp150-161
MLA 9
Koca, Nazife Özdeş, et al. "4D Pyritohedral Symmetry." Sultan Qaboos University Journal for Science, vol. 21, no. 2, 2016, pp. 150. https://doi.org/10.24200/squjs.vol21iss2pp150-161.
Chicago (author–date)
Koca, Nazife Özdeş, Amal J.H. Al Qanobi, and Mehmet Koca. 2016. "4D Pyritohedral Symmetry." Sultan Qaboos University Journal for Science 21 (2): 150. https://doi.org/10.24200/squjs.vol21iss2pp150-161.
Harvard
Koca, N. Ö., Al Qanobi, A. J. and Koca, M. (2016) '4D Pyritohedral Symmetry', Sultan Qaboos University Journal for Science, 21(2), pp. 150. doi:10.24200/squjs.vol21iss2pp150-161.
Vancouver
Koca NÖ, Al Qanobi AJ, Koca M. 4D Pyritohedral Symmetry. Sultan Qaboos University Journal for Science. 2016;21(2):150. doi:10.24200/squjs.vol21iss2pp150-161
IEEE
N. Ö. Koca, A. J. Al Qanobi, and M. Koca, "4D Pyritohedral Symmetry," Sultan Qaboos University Journal for Science, vol. 21, no. 2, pp. 150, 2016, doi: 10.24200/squjs.vol21iss2pp150-161.