### Abstract

The lattice matching of two sets of quaternionic roots of F_{4} leads to quaternionic roots of E_{8} which has a decomposition H_{4} + σ H_{4} where the Coxeter graph H_{4} is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H_{3} which has a natural extension to H_{3} + σ H_{3} describing the root system of the Lie algebra D_{6}. It is noted that there exist three lattices in 6-dimensions whose point group W(D_{6}) admits the icosahedral symmetry H_{3} as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H_{2} of order 10 is the Weyl group W(A_{4}) where H_{2} + σ H_{2} constitute the root system of the Lie algebra A_{4}. The relevance of these systems to quasicrystals are discussed.

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

Pages (from-to) | 421-435 |

Number of pages | 15 |

Journal | Turkish Journal of Physics |

Volume | 22 |

Issue number | 5 |

Publication status | Published - 1998 |

### Fingerprint

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

_{8}Related Coxeter Graphs and Quasicrystals.

*Turkish Journal of Physics*,

*22*(5), 421-435.

**Quaternionic Roots of E _{8} Related Coxeter Graphs and Quasicrystals.** / Koca, Mehmet; Koca, Nazife Özdeş; Koç, Ramazan.

Research output: Contribution to journal › Article

_{8}Related Coxeter Graphs and Quasicrystals',

*Turkish Journal of Physics*, vol. 22, no. 5, pp. 421-435.

_{8}Related Coxeter Graphs and Quasicrystals. Turkish Journal of Physics. 1998;22(5):421-435.

}

TY - JOUR

T1 - Quaternionic Roots of E8 Related Coxeter Graphs and Quasicrystals

AU - Koca, Mehmet

AU - Koca, Nazife Özdeş

AU - Koç, Ramazan

PY - 1998

Y1 - 1998

N2 - The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + σ H4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + σ H3 describing the root system of the Lie algebra D6. It is noted that there exist three lattices in 6-dimensions whose point group W(D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W(A4) where H2 + σ H2 constitute the root system of the Lie algebra A4. The relevance of these systems to quasicrystals are discussed.

AB - The lattice matching of two sets of quaternionic roots of F4 leads to quaternionic roots of E8 which has a decomposition H4 + σ H4 where the Coxeter graph H4 is represented by the 120 quaternionic elements of the binary icosahedral group. The 30 pure imaginary quaternions constitute the roots of H3 which has a natural extension to H3 + σ H3 describing the root system of the Lie algebra D6. It is noted that there exist three lattices in 6-dimensions whose point group W(D6) admits the icosahedral symmetry H3 as a subgroup, the roots of which describe the mid-points of the edges of an icosahedron. A natural extension of the Coxeter group H2 of order 10 is the Weyl group W(A4) where H2 + σ H2 constitute the root system of the Lie algebra A4. The relevance of these systems to quasicrystals are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0039207672&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039207672&partnerID=8YFLogxK

M3 - Article

VL - 22

SP - 421

EP - 435

JO - Turkish Journal of Physics

JF - Turkish Journal of Physics

SN - 1300-0101

IS - 5

ER -