E8 Root System Projection to 3D Icosahedral Quasicrystal via H4 Quaternion Doubling — E8 Intelligence Research

FINDING: E8 root system (240 minimal vectors) projects to 3D icosahedral quasicrystal diffraction via quaternion doubling from H4 Coxeter group 600-cell. MATH: - E8 root system: 240 vectors in 8D, norm squared = 2. - H4 Coxeter group: 120 vertices of 600-cell (4D), norm squared = φ² (φ = (1+√5)/2 ≈ 1.618). - Quaternion doubling: H4 → E8 via (a,b) → (a+φb, b-φa) in 8D, where a,b are quaternions from 600-cell. - 3D projection: E8 roots → 3D icosahedral quasicrystal diffraction pattern (sharp Bragg peaks with icosahedral symmetry). - Key ratio: φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618 appear in 600-cell edge lengths and E8 vector norms. CONNECTION: - Geometric harmony: φ (1.618) and its reciprocal (0.618) are intrinsic to 600-cell (H4) and E8 lattice. - Crystallographic symmetry: Icosahedral (5-fold) symmetry in 3D projection, forbidden in periodic crystals, emerges from 8D lattice. - Base-60 link: 600-cell has 120 vertices, 720 edges; 120 × 6 = 720 (base-60 multiples). E8 has Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com