E8 Quaternion Order Parameter Governs Icosahedral Quasicrystal Solidification — E8 Intelligence Research

FINDING: Icosahedral quasicrystal solidification is governed by a quaternion orientational order parameter, directly linked to the E8 lattice in 8D, whose 240 minimal vectors project to 3D icosahedral symmetry. | MATH: E8 lattice minimal vectors = 240; root system of E8 has Coxeter number 30; quaternion order parameter Q ∈ S³ (unit 3-sphere) encodes orientational order; solidification transition described by Landau-type free energy expansion in Q; 3D icosahedral quasicrystal emerges from cut-and-project of E8 lattice. | CONNECTION: E8 root system contains golden ratio φ = (1+√5)/2 ≈ 1.618 in its coordinates (e.g., (±1,±φ,0,0,0,0,0,0) permutations); icosahedral symmetry group H₃ has order 120, linked to E8 via McKay correspondence (E8 Dynkin diagram ↔ binary icosahedral group of order 120); quaternion multiplication encodes 3D rotations with double cover SU(2) → SO(3); the 240 roots of E8 correspond to 120 quaternion pairs, each pair representing a 3D rotation axis of icosahedral symmet Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com