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: m₈₉ = 5 for the bond between simple roots of lengths ratio φ (since 2 cos (π/5) = φ). - Cartan matrix entry: a₈₉ = -2 cos (π/m₈₉) → 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