Coxeter Group H3: Golden Ratio Eigenvalues Linking Icosahedral Symmetry to Quasicrystals — E8 Intelligence Research

Abstract

FINDING: Coxeter group H3 root system encodes icosahedral symmetry via golden ratio eigenvalues in representation theory, linking discrete geometry to quasicrystalline and fundamental physics structures. MATH: - Coxeter group H3 (icosahedral group) has Coxeter matrix with entries m_{ij} = 5 for the bond between simple reflections generating the golden ratio. - Eigenvalues of the Coxeter element in H3: λ = exp(±iπ/5), exp(±i3π/5), and 1 (trivial). These yield golden ratio φ = (1+√5)/2 ≈ 1.618 and its reciprocal φ⁻¹ ≈ 0.618 via: λ + λ⁻¹ = 2 cos(π/5) = φ, λ + λ⁻¹ = 2 cos(3π/5) = φ⁻¹. - The root system of H3 has 30 roots, with lengths in ratio 1 : φ (short : long). - The Cartan matrix eigenvalues involve φ and φ⁻¹, e.g., largest eigenvalue = φ² ≈ 2.618. CONNECTION: - Golden ratio φ = 1.618, φ⁻¹ = 0.618, φ² = 2.618 appear as eigenvalues and root length ratios — direct geometric harmony. - Icosahedral symmetry (H3) is the largest finite rotation group in 3D, linked to q Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com