### Abstract

We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I _{2}(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group D_{h} of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A_{4} onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine W_{a}(A _{4}) can be decomposed into two orthogonal spaces whose point group is the dihedral group D_{5} which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group W _{a}(A_{3}), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group W_{a}(A_{n}), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of W_{a}(A_{4}) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

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

Article number | 1450031 |

Journal | International Journal of Geometric Methods in Modern Physics |

Volume | 11 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2014 |

### Fingerprint

### Keywords

- affine Coxeter groups
- quasicrystal
- Quaternions

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

_{a}(A

_{4}), quaternions, and decagonal quasicrystals.

*International Journal of Geometric Methods in Modern Physics*,

*11*(4), [1450031]. https://doi.org/10.1142/S0219887814500315

**Affine coxeter group W _{a}(A_{4}), quaternions, and decagonal quasicrystals.** / Koca, Mehmet; Koca, Nazife Ozdes; Koc, Ramazan.

Research output: Contribution to journal › Article

_{a}(A

_{4}), quaternions, and decagonal quasicrystals',

*International Journal of Geometric Methods in Modern Physics*, vol. 11, no. 4, 1450031. https://doi.org/10.1142/S0219887814500315

}

TY - JOUR

T1 - Affine coxeter group Wa(A4), quaternions, and decagonal quasicrystals

AU - Koca, Mehmet

AU - Koca, Nazife Ozdes

AU - Koc, Ramazan

PY - 2014

Y1 - 2014

N2 - We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I 2(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A 4) can be decomposed into two orthogonal spaces whose point group is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group W a(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

AB - We introduce a technique of projection onto the Coxeter plane of an arbitrary higher-dimensional lattice described by the affine Coxeter group. The Coxeter plane is determined by the simple roots of the Coxeter graph I 2(h) where h is the Coxeter number of the Coxeter group W(G) which embeds the dihedral group Dh of order 2h as a maximal subgroup. As a simple application, we demonstrate projections of the root and weight lattices of A4 onto the Coxeter plane using the strip (canonical) projection method. We show that the crystal spaces of the affine Wa(A 4) can be decomposed into two orthogonal spaces whose point group is the dihedral group D5 which acts in both spaces faithfully. The strip projections of the root and weight lattices can be taken as models for the decagonal quasicrystals. The paper also revises the quaternionic descriptions of the root and weight lattices, described by the affine Coxeter group W a(A3), which correspond to the face centered cubic (fcc) lattice and body centered cubic (bcc) lattice respectively. Extensions of these lattices to higher dimensions lead to the root and weight lattices of the group Wa(An), n ≥ 4. We also note that the projection of the Voronoi cell of the root lattice of Wa(A4) describes a framework of nested decagram growing with the power of the golden ratio recently discovered in the Islamic arts.

KW - affine Coxeter groups

KW - quasicrystal

KW - Quaternions

UR - http://www.scopus.com/inward/record.url?scp=84899116454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84899116454&partnerID=8YFLogxK

U2 - 10.1142/S0219887814500315

DO - 10.1142/S0219887814500315

M3 - Article

VL - 11

JO - International Journal of Geometric Methods in Modern Physics

JF - International Journal of Geometric Methods in Modern Physics

SN - 0219-8878

IS - 4

M1 - 1450031

ER -