### Abstract

A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa (Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) 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.

Original language | English |
---|---|

Pages (from-to) | 175-185 |

Number of pages | 11 |

Journal | Acta Crystallographica Section A: Foundations and Advances |

Volume | 71 |

DOIs | |

Publication status | Published - Mar 1 2015 |

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### Keywords

- aperiodic tilings
- Coxeter-Weyl groups
- cut-and-project technique
- Lattices
- quasicrystallography
- strip projection

### ASJC Scopus subject areas

- Physical and Theoretical Chemistry
- Materials Science(all)
- Structural Biology
- Inorganic Chemistry
- Condensed Matter Physics
- Biochemistry

### Cite this

**Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices B _{n}.** / Koca, Mehmet; Ozdes Koca, Nazife; Koc, Ramazan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Group-theoretical analysis of aperiodic tilings from projections of higher-dimensional lattices Bn

AU - Koca, Mehmet

AU - Ozdes Koca, Nazife

AU - Koc, Ramazan

PY - 2015/3/1

Y1 - 2015/3/1

N2 - A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa (Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) 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.

AB - A group-theoretical discussion on the hypercubic lattice described by the affine Coxeter-Weyl group Wa (Bn) is presented. When the lattice is projected onto the Coxeter plane it is noted that the maximal dihedral subgroup Dh of W(Bn) 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.

KW - aperiodic tilings

KW - Coxeter-Weyl groups

KW - cut-and-project technique

KW - Lattices

KW - quasicrystallography

KW - strip projection

UR - http://www.scopus.com/inward/record.url?scp=84924024510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84924024510&partnerID=8YFLogxK

U2 - 10.1107/S2053273314025492

DO - 10.1107/S2053273314025492

M3 - Article

AN - SCOPUS:84924024510

VL - 71

SP - 175

EP - 185

JO - Acta Crystallographica Section A: Foundations and Advances

JF - Acta Crystallographica Section A: Foundations and Advances

SN - 0108-7673

ER -