Quasicrystals from higher dimensions
Author
Al-Sawafiyah, Muna Khalifa Aamer
English abstract
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 of the octahedral symmetry of the cubic lattices which have extensive applications in material science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices. Also we introduce a general technique and apply it to the projections of the lattices described by the affine Coxeter-Weyl groups W.(A4) and Wa(Bs). The dihedral subgroup Dg of the W.(AA), implying the importance of the Coxeter number plays the crucial role in the symmetry of the projected set of points. We define two generators R1 and Ry which act as reflections generators in certain planes. The canonical projections (strip projections) of the lattices determine the nature of the quasi-crystallographic structures with 5-fold and 10-fold symmetries.
