FINDING: E8 lattice projection to 3D icosahedral quasicrystal via quaternion orientational order parameter classification of solidification phase transitions. MATH: - E8 root system: 240 vertices, 6720 edges; 8D lattice with Coxeter group symmetry E8 (order 696729600). - Projection to 3D icosahedral quasicrystal uses cut-and-project method; quaternion order parameter Q ∈ S³ / binary icosahedral group (order 120). - Key constants: golden ratio φ = (1+√5) /2 ≈ 1. 618, its inverse φ⁻¹ ≈ 0. 618, and φ² ≈ 2. 618 appear in icosahedral symmetry (e. g. , vertex coordinates of icosahedron involve φ). - Phase transition classification: Landau-type free energy expansion in quaternion order parameter; symmetry breaking from SO (3) to icosahedral group Ih (order 120). CONNECTION: - Icosahedral symmetry directly encodes φ (0. 618, 1. 618, 2. 618) in its geometry (e. g. , edge/radius ratios). - E8 lattice's 240 roots correspond to 240 vertices of the 4₂₁ polytope; 3D quasicrystal diffraction pat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Tue,) studied this question.