Golden Ratio Eigenvalues Link H3 Icosahedral Group to H4 and Quaternions — E8 Intelligence Research

FINDING: Icosahedral symmetry group H3 is non-crystallographic, simple, and its Cartan matrix eigenvalues involve the golden ratio, linking to H4 and quaternionic representations. MATH: - H3 Coxeter group: Cartan matrix entries \( aᵢⱼ = 2cos(π / mᵢⱼ) \) with \( m₁₂=5, m₂₃=3, m₁₃=2 \). - Eigenvalues of H3 Cartan matrix: \( λ_1 = 2 + φ, λ_2 = 2, λ_3 = 2 - φ⁻¹ \), where \( φ = (1+√5)/2 ≈ 1.618 \). - Golden ratio appears: \( φ, φ⁻¹ ≈ 0.618 \). - H4 (4D) extends H3; its Coxeter number is 30, and its Cartan eigenvalues involve \( φ^2 ≈ 2.618 \). - Quaternionic representation: vertices of H4 polytopes (e.g., 600-cell) are orbits under quaternion multiplication, preserving icosahedral symmetry in 3D projection. CONNECTION: - Golden ratio \( φ \) and its inverse \( φ⁻¹ \) are fundamental to H3 and H4, linking to pentagonal and icosahedral symmetry. - Non-crystallographic: no lattice in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com