### Abstract

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in n-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter-Weyl groups W_{a}(F_{4}), W_{a}(B_{6}) and W_{a}(E_{6}). These groups share the same Coxeter number h = 12 with different Coxeter exponents. The dihedral subgroup D_{12} of the Coxeter groups can be obtained by defining two generators R_{1} and R_{2} as the products of generators of the Coxeter-Weyl groups. The reflection generators R_{1} and R_{2} operate in the Coxeter planes where the Coxeter element R_{1}R_{2} of the Coxeter-Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the lattices W_{a}(F_{4}) and W_{a}(B_{6}) are compatible with some experimental results.

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

Pages (from-to) | 605-615 |

Number of pages | 11 |

Journal | Acta Crystallographica Section A: Foundations and Advances |

Volume | 70 |

DOIs | |

Publication status | Published - Nov 1 2014 |

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### Keywords

- Coxeter-Weyl groups
- cut-and-project technique
- lattices
- quasicrystallography
- strip projection

### ASJC Scopus subject areas

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

### Cite this

**Twelvefold symmetric quasicrystallography from the lattices F _{4}, B_{6} and E_{6} .** / Koca, Nazife O.; Koca, Mehmet; Koc, Ramazan.

Research output: Contribution to journal › Article

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*Acta Crystallographica Section A: Foundations and Advances*, vol. 70, pp. 605-615. https://doi.org/10.1107/S2053273314015812

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TY - JOUR

T1 - Twelvefold symmetric quasicrystallography from the lattices F4, B6 and E6

AU - Koca, Nazife O.

AU - Koca, Mehmet

AU - Koc, Ramazan

PY - 2014/11/1

Y1 - 2014/11/1

N2 - One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in n-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6). These groups share the same Coxeter number h = 12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter-Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) are compatible with some experimental results.

AB - One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter-Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors in n-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter-Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter-Weyl groups Wa(F4), Wa(B6) and Wa(E6). These groups share the same Coxeter number h = 12 with different Coxeter exponents. The dihedral subgroup D12 of the Coxeter groups can be obtained by defining two generators R1 and R2 as the products of generators of the Coxeter-Weyl groups. The reflection generators R1 and R2 operate in the Coxeter planes where the Coxeter element R1R2 of the Coxeter-Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the lattices Wa(F4) and Wa(B6) are compatible with some experimental results.

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=84992736427&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84992736427&partnerID=8YFLogxK

U2 - 10.1107/S2053273314015812

DO - 10.1107/S2053273314015812

M3 - Article

VL - 70

SP - 605

EP - 615

JO - Acta Crystallographica Section A: Foundations and Advances

JF - Acta Crystallographica Section A: Foundations and Advances

SN - 0108-7673

ER -