Quaternionic Roots of E₈ Related Coxeter Graphs and Quasicrystals

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.