H₃ Coxeter Group: Golden Ratio Algebra for Icosahedral Quasicrystals — E8 Intelligence Research

FINDING: Coxeter group H₃ is the symmetry group of the icosahedron/dodecahedron, directly encoding the golden ratio φ in its Cartan matrix and root system, providing the algebraic foundation for quasicrystalline order. MATH: - H₃ Coxeter matrix: m_{ij} = 5 for the bond between simple roots of lengths ratio φ (since 2 cos(π/5) = φ). - Cartan matrix entry: a_{ij} = -2 cos(π/m_{ij}) → for m=5: a = -2 cos(36°) = -φ ≈ -1.618. - Eigenvalues of H₃ Cartan matrix: {4 cos²(π/5), 4 cos²(π/3), 4 cos²(π/2)} = {φ², 1, 0} where φ² = φ+1 ≈ 2.618. - Root system: 30 roots, 15 positive, norm ratio √φ (long/short). - Coxeter number h = 10; eigenvalues of Coxeter element: exp(2πi k/h) for k ∈ {1,3,5,7,9} → golden ratio appears in the trace: Tr = 2 cos(π/5) + 2 cos(3π/5) = φ - 1/φ = 1. CONNECTION: - φ = 1.618, 1/φ = 0.618, φ² = 2.618, φ⁻² = 0.382 — all appear in H₃ eigenvalues, root lengths, and projection matrices. - Quasicrystal: H₃ is the point group of icosahedral quasicrystals (e.g., Al Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com