Quasicrystals are obtained as the projections of higher dimensional cubic lattices. This paper aims to show that the affine dihedral subgroup Wa(I2(h)) of the affine group Wa(Bn), h = 2n being the Coxeter number, offers a different perspective to -fold symmetric quasicrystallography. The affine group Wa(I2(h)) is const …
Affine subgroups having the same Coxeter number with the affine Coxeter groups W(An), W(Dn) and W(En) are constructed by graph folding techniques. The affine groups W(Cn) and W(Bn) are obtained from the Coxeter groups W(A2n-1) and W(D2n-2), respectively. The affine groups W(E6), W(D6) and W(E8) lead to the affine group …
We analyze the group structures of two groups of order 1344 which are respectively non-split and split extensions of the elementary Abelian group of order 8 by its automorphism group 𝑃𝑆𝐿 2 (7).Two groups have the same number of conjugacy classes and the set of dimensions of irreducible representations is equal.The grou …
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 rel …
The pyritohedron consisting of twelve identical but non regular pentagonal faces and its dual pseudoicosahedron that possess the pyritohedral (Th) symmetry play an essential role in understanding the crystallographic structures with the pyritohedral symmetry. The pyritohedral symmetry takes a simpler form in terms of q …
3-dimensional convex uniform polyhedra have been projected onto their corresponding Coxeter planes defined by the simple roots of the Coxeter diagram ,
We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine extended Coxeter groups W(A3) and W(B3)=Aut(A3). It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral …
There are two chiral Archimedean polyhedra, the snub cube and snub dodecahedron together with their dual Catalan solids, pentagonal icositetrahedron and pentagonal hexacontahedron. In this paper we construct the chiral polyhedra and their dual solids in a systematic way. We use the proper rotational subgroups of the Co …