Quaternionic representation of pyritohedral symmetry, related polyhedra and lattices
مؤلف
, Aida YousufAl-Mukhainiyah
الملخص الإنجليزي
ABSTRACT
One of the most efficient tools in studying crystallography symmetry is to use quaternionic representation for the root system of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The cubic systems FCC (face centered cubic), BCC (body centered cubic) and SC (simple cubic) lattices can be constructed from the root and the weight lattices of the affine extended Coxeter groups W.(D) and W.(Bz) respectively.
The rank-3 Coxeter-Weyl groups W (D3) and W(B.) Aut(D2) describing the point. tetrahedral symmetry and the octahedral symmetry of the cubic lattices have extensive applications in material science. The constructions of the vertices of the Wigner-Seitz cells have been presented also in terms of quaternionic imaginary units. Reflection planes of the Coxeter-Dynkin diagrams are identified with certain planes of the unit cube,
The pyritohedral symmetry takes a simpler form in terms of quaternionic representation, We use Dz to construct the vertices of a family of polyhedra relevant to the cubic lattices in particular; constructions of the pseudoicosahedron and its dual pyritohedron are explicitly worked out. Many candidates of a parameter x give pseudoicosahedra and its dual pyritohedra as a structure in simple cubic lattice.
When x is the ratios of consecutive Fibonacci numbers approach to the golden ratio 1, we obtain a family of pseudoicosahedron approaching to icosahedron. The dual of this family of polyhedra gives a pyritohedron approaching to dodecahedron. The pseudoicosahedron and its dual pyritohedron play an essential role in understanding the crystallographic structures with the pyritohedral symmetry.
