Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn

Acta Crystallographica. Section A, Foundations and Advances
Mehmet KocaRamazan Koc

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.

References

Oct 15, 1985·Physical Review. B, Condensed Matter·V Elser
Apr 15, 1987·Physical Review. B, Condensed Matter·D S RokhsarD C Wright
Jun 24, 1985·Physical Review Letters·M Duneau, A Katz
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Aug 31, 1987·Physical Review Letters·N WangK H Kuo
Aug 18, 2012·Nano Letters·Steven M LubinTeri W Odom
Apr 25, 2008·Science and Technology of Advanced Materials·An Pang Tsai

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Citations

Sep 6, 2018·Acta Crystallographica. Section A, Foundations and Advances·Mehmet KocaRamazan Koc
Jan 6, 2021·Acta Crystallographica. Section A, Foundations and Advances·Alexander TalisValentin Kraposhin
Mar 2, 2021·Acta Crystallographica. Section A, Foundations and Advances·Nazife Ozdes KocaAbeer Al-Siyabi

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