Koca, Nazife O.; Koca, Mehmet; Al-Sawafic, Muna Quasicrystals from higher dimensional lattices. (English) Zbl 1324.52016 Symmetry Cult. Sci. 25, No. 3, 233-260 (2014). Summary: We introduce a general technique and apply it to the projections of the lattices described by the affine Coxeter-Weyl groups \(W_a(A_4), W_a(F_4), W_a(B_6)\) and \(W_a(B_6)\) onto the Euclidean plane. The dihedral subgroups \(D_5\) and \(D_{12}\) of these groups ,respectively, implying the importance of the Coxeter number, play the crucial role in the symmetry of the projected set of points, edges, etc. We define two generators \(R_1\) and \(R_2\) which act as generator reflections in certain planes orthogonal to the projection one where the product \(R_1R_2\) describes the rotation-like elements of the dihedral group. The canonical projections (strip projections) of the lattices determine the nature of the quasicrystallographic structures with 5-fold and 12-fold symmetries. We note that the projections of the lattices \(W_a(F_4), W_a(B_6)\) describe 12-fold symmetric quasicrystal structure, e.g., of the \(\mathrm{Ni}\)-\(\mathrm{C}_r\) particles in crystallography. MSC: 52C23 Quasicrystals and aperiodic tilings in discrete geometry 20F55 Reflection and Coxeter groups (group-theoretic aspects) 20H15 Other geometric groups, including crystallographic groups 82D25 Statistical mechanics of crystals Keywords:lattices; Coxeter-Weyl groups; strip projections; quasicrystallography PDFBibTeX XMLCite \textit{N. O. Koca} et al., Symmetry Cult. Sci. 25, No. 3, 233--260 (2014; Zbl 1324.52016)