Affine coxeter group W<sub>a</sub>(A<sub>4</sub>), quaternions, and decagonal quasicrystals

Koca, Mehmet; Koca, Nazife Ozdes; Koc, Ramazan

doi:10.1142/s0219887814500315

Cited by 10 publications

(12 citation statements)

References 31 publications

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“…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…”

Section: Projections Of the Faces Of The Voronoi And Delone Cellsmentioning

confidence: 99%

“…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. The third method is the model set technique initiated by Meyer (Meyer, 1972) and followed by Lagarias (Lagarias, 1996) and developed by Moody (Moody, 1997).…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice are identical rhombuses and equilateral triangles respectively. Their orthogonal projections onto the Coxeter plane display various rhombic and triangular prototiles including thick and thin rhombi of Penrose, Amman-Beenker tiles, Robinson triangles and Danzer triangles to name a few. We point out that the dihedral subgroup of order 2ℎ involving the Coxeter element of order ℎ = + 1 of the Coxeter-Weyl group plays a crucial role for ℎ-fold symmetric tilings of the Coxeter plane. After setting the general scheme we give samples of patches with 4, 5, 6, 7, 8 and 12-fold symmetries. The face centered cubic (f.c.c.) lattice described by the root lattice 3 whose Wigner-Seitz cell is the rhombic dodecahedron projects, as expected, onto a square lattice with an ℎ = 4 fold symmetry.

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Section: Projections Of the Faces Of The Voronoi And Delone Cellsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2019

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“…. ; 4Þ are obtained by using the eigenvectors of the Cartan matrix of WðA 4 Þ and the fifth vector is chosen to be orthogonal to the rest (Koca, Koca & Koc, 2014b):…”

Section: The Coxeter-weyl Group W(b N ) and Its Maximal Subgroupsmentioning

confidence: 99%

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

Koca¹,

Koca²,

Koc³

2015

Acta Cryst Sect A

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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.

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“…Lie algebras derived from the root systems of the crystallographic groups are applied to particle physics as the symmetry of the standard model (Glashow, 1961;Weinberg, 1967;Salam, 1968;Fritzsch et al, 1973) and its embeddings into higher-rank Lie groups (Georgi & Glashow, 1974;Fritzsch & Minkowski, 1975;Gü rsey et al, 1976). As a typical example, let us consider SUð5Þ grand unified theory derived from the a 4 root system that describes the unification of strong and electro-weak interactions where the related polytopes describing the particle content project into two dimensions (Koca et al, 2014). As an application in quasi-crystallography, in a recent article (Koca et al, 2018), we have shown that the facet of the A 4 Voronoi cell is a rhombohedron and its 2-faces project into the Coxeter plane as thick and thin rhombuses of the Penrose tiling.…”

Section: Introductionmentioning

confidence: 99%

Explicit construction of the Voronoi and Delaunay cells ofW(A_n) andW(D_n) lattices and their facets

Koca

Al-Siyabi

et al. 2018

Acta Cryst Sect A

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Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(A) and W(D) are constructed. The face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. The Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is shown that the Delone cells centred at the origin of the root lattice A are the polytopes of the fundamental weights ω, ω,…, ω and the Delone cells of the root lattice D are the polytopes obtained from the weights ω, ω and ω. A simple mechanism explains the tessellation of the root lattice by Delone cells. It is proved that the (n-1)-facet of the Voronoi cell of the root lattice A is an (n-1)-dimensional rhombohedron and similarly the (n-1)-facet of the Voronoi cell of the root lattice D is a dipyramid with a base of an (n-2)-cube. The volume of the Voronoi cell is calculated via its (n-1)-facet which in turn can be obtained from the fundamental simplex. Tessellations of the root lattice with the Voronoi and Delone cells are explained by giving examples from lower dimensions. Similar considerations are also worked out for the weight lattices A* and D*. It is pointed out that the projection of the higher-dimensional root and weight lattices on the Coxeter plane leads to the h-fold aperiodic tiling, where h is the Coxeter number of the Coxeter-Weyl group. Tiles of the Coxeter plane can be obtained by projection of the two-dimensional faces of the Voronoi or Delone cells. Examples are given such as the Penrose-like fivefold symmetric tessellation by the A root lattice and the eightfold symmetric tessellation by the D root lattice.

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Affine coxeter group W_a(A₄), quaternions, and decagonal quasicrystals

Cited by 10 publications

References 31 publications

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

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

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

Explicit construction of the Voronoi and Delaunay cells ofW(A_n) andW(D_n) lattices and their facets

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Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals

Cited by 10 publications

References 31 publications

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

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

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

Explicit construction of the Voronoi and Delaunay cells ofW(An) andW(Dn) lattices and their facets

Contact Info

Product

Resources

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Affine coxeter group W_a(A₄), quaternions, and decagonal quasicrystals

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

Explicit construction of the Voronoi and Delaunay cells ofW(A_n) andW(D_n) lattices and their facets