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 \( a_{ij} = 2\cos(\pi / m_{ij}) \) with \( m_{12}=5, m_{23}=3, m_{13}=2 \). - Eigenvalues of H3 Cartan matrix: \( \lambda_1 = 2 + \phi, \lambda_2 = 2, \lambda_3 = 2 - \phi^{-1} \), where \( \phi = (1+\sqrt{5})/2 \approx 1.618 \). - Golden ratio appears: \( \phi, \phi^{-1} \approx 0.618 \). - H4 (4D) extends H3; its Coxeter number is 30, and its Cartan eigenvalues involve \( \phi^2 \approx 2.618 \). - Quaternionic representation: vertices of H4 polytopes (e.g., 600-cell) are orbits under quaternion multiplication, preserving icosahedral symmetry in 3D projection. CONNECTION: - Golden ratio \( \phi \) and its inverse \( \phi^{-1} \) 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