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Quasicrystals from higher dimensional lattices. (English) Zbl 1324.52016

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
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