Golden Ratio Unifies H3/H4 Coxeter Groups, Finite Simple Groups, and Quaternionic Polytopes — E8 Intelligence Research

FINDING: Icosahedral symmetry (H3) and its 4D extension (H4) are non-crystallographic Coxeter groups whose Cartan matrix eigenvalues and root system ratios are governed by the golden ratio φ, linking finite simple groups to quaternionic polytopes and 3D/4D geometric harmony. MATH: - H3 Coxeter group: Cartan matrix entries \( a_{ij} = 2 \cos(\pi / m_{ij}) \) with \( m_{12}=5, m_{23}=3 \). Eigenvalues: \( \lambda_1 = 4 \cos^2(\pi/5) = \phi^2 \approx 2.618 \), \( \lambda_2 = 4 \cos^2(\pi/3) = 1 \), \( \lambda_3 = 4 \cos^2(\pi/2) = 0 \). - H4 Coxeter group: Cartan matrix eigenvalues: \( \phi^2, \phi^2, 1, 0 \) (double φ²). - Golden ratio: \( \phi = (1+\sqrt{5})/2 \approx 1.618 \), \( \phi^{-1} \approx 0.618 \), \( \phi^2 \approx 2.618 \). - Quaternionic representation: H4 polytopes (e.g., 600-cell, 120-cell) vertices as quaternion orbits under W(H4), projected to 3D preserving icosahedral symmetry. CONNECTION: - Ratios 0.618, 1.618, 2.618 appear directly in Cartan eigenvalues a Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com