Quaternion Order Parameter Links Icosahedral Quasicrystals to E8 Lattice — E8 Intelligence Research

FINDING: Solidification of icosahedral quasicrystals is classified by a quaternion orientational order parameter, linking the 3D quasicrystal to the 8D E8 lattice via projection. MATH: - E8 lattice: 240 minimal vectors (roots) in 8D, with 8D coordinates (±1,±1,0,0,0,0,0,0) and permutations, plus (±1/2,±1/2,±1/2,±1/2,±1/2,±1/2,±1/2,±1/2) with even number of minus signs. - Quaternion order parameter: q ∈ ℍ (4D), with norm |q| = 1 (unit quaternion), representing orientational degrees of freedom. - Icosahedral symmetry group H3 (order 120) is a subgroup of SO(3), and its double cover is the binary icosahedral group (order 240), which is isomorphic to the E8 root system's Weyl group modulo center. - Projection: 3D icosahedral quasicrystal vertices are obtained by projecting a slice of the E8 lattice onto 3D, using a cut-and-project method with a 3D irrational subspace (golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the projection matrix). - Key ratio: φ = 1.618, and its inverse φ⁻¹ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com