FINDING: SU (2) double cover of SO (3) maps quaternion orientational order to icosahedral quasicrystal solidification via E8 lattice in 8D. MATH: SU (2) ≅ Spin (3) double covers SO (3) with kernel ±1; quaternion order parameter Q ∈ S³; E8 root lattice in 8D projects to 3D icosahedral quasicrystal; quaternion orientational order parameter classifies isotropic→icosahedral phase transition. CONNECTION: Icosahedral symmetry (H3 Coxeter group) embeds in E8; golden ratio φ = (1+√5) /2 ≈ 1. 618 appears in E8 Cartan matrix and 600-cell polytope coordinates; 600-cell has 120 vertices, 720 edges, 1200 faces, 600 tetrahedral cells — all scaled by φ; quaternion double cover of SO (3) yields spinorial representation of icosahedral group (binary icosahedral group of order 120). DEPTH: 9 — Directly links quaternion spinors, SU (2) topology, quasicrystal order, and E8 exceptional lattice, unifying geometric harmony (φ, icosahedral symmetry) with condensed matter physics and high-dimensional root system Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Mon,) studied this question.