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.
Recommended Citation
Koca, Nazife O.; Al Qanobi, Amal J.H.; and Koca, Mehmet
(2016)
4D Pyritohedral Symmetry,
Sultan Qaboos University Journal For Science: Vol. 21:
Iss.
2, 150-161.
DOI: https://doi.org/10.24200/squjs.vol21iss2pp150-161
Available at:
https://squjs.squ.edu.om/squjs/vol21/iss2/1