## 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 |
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Pages (from-to) | 421-435 |

Number of pages | 15 |

Journal | Turkish Journal of Physics |

Volume | 22 |

Issue number | 5 |

Publication status | Published - 1998 |

## ASJC Scopus subject areas

- Physics and Astronomy(all)