Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional latticesB_n

Abstract

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group W(a)(B(n)) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup D(h) of W(B(n)) with h = 2n representing the Coxeter number describes the h-fold symmetric aperiodic tilings. Higher-dimensional cubic lattices are explicitly constructed for n = 4, 5, 6. Their rank-3 Coxeter subgroups and maximal dihedral subgroups are identified. It is explicitly shown that when their Voronoi cells are decomposed under the respective rank-3 subgroups W(A(3)), W(H(2)) × W(A(1)) and W(H(3)) one obtains the rhombic dodecahedron, rhombic icosahedron and rhombic triacontahedron, respectively. Projection of the lattice B(4) onto the Coxeter plane represents a model for quasicrystal structure with eightfold symmetry. The B(5) lattice is used to describe both fivefold and tenfold symmetries. The lattice B(6) can describe aperiodic tilings with 12-fold symmetry as well as a three-dimensional icosahedral symmetry depending on the choice of subspace of projections. The novel structures from the projected sets of lattice points are compatible with the available experimental data.

Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

“…See for a review the paper "Modelling of quasicrystals" by Kramer [12] and references therein. Similar work has also been carried out in the reference [13]. Kramer and Andrle [14] have investigated Danzer tiles from the wavelet point of view and their relations with the lattice D 6 .…”

Prototiles and Tilings from Voronoi and Delone Cells of the Root Lattice An

“…The same technique is extended to an arbitrary cubic lattice (Whittaker & Whittaker, 1987). It is not a surprise that we obtain the same tiling from the 4 lattice because Voronoi cell of 4 is obtained as the projection of the Voronoi cell of the cubic lattice 5 (Koca et al, 2015) (see also Appendix A for further details). The Coxeter-Weyl group ( 5 ) is of order 2 5 .…”

“…Here ℎ = + 1 is the Coxeter number. Going to the real space, the Coxeter element acts on the plane ∥ spanned by the unit vectors In (Koca, et al, 2014(Koca, et al, , 2015 we have introduced an equivalent definition of the Coxeter plane through the eigenvalues and eigenvectors of the Cartan matrix of the root system of the lattice . The eigenvalues and eigenvectors of the Cartan matrix of the group can be written as…”

“…In particular, Baake et al (Baake et al, 1990) exemplified the projection of the 4 lattice. The group theoretical treatments of the projection of ndimensional cubic lattices have been worked out by the references (Whittaker & Whittaker, 1987;Koca et al, 2015) and a general projection technique on the basis of dihedral subgroup of the root lattices has been proposed by Boyle & Steinhardt (Boyle & Steinhardt, 2016). In a recent article (Koca et al, 2018a) we pointed out that the 2dimensional facets of the 4 Voronoi cell projects onto thick and thin rhombuses of the Penrose tilings.…”

4D Pyritohedral Symmetry

“…attices in higher dimensions described by the affine Coxeter groups, when projected into lower dimensions, may represent the quasicrystal structures [1][2][3][4][5]. It is known that the 4 A lattice projects into the aperiodic lattice with 5fold symmetry [1].…”