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 \ (a₈₉ = 2 (/ m₈₉) \) with \ (m₁₂=5, m₂₃=3 \). Eigenvalues: \ (₁ = 4 ² (/5) = ² 2. 618 \), \ (₂ = 4 ² (/3) = 1 \), \ (₃ = 4 ² (/2) = 0 \). - H4 Coxeter group: Cartan matrix eigenvalues: \ (², ², 1, 0 \) (double φ²). - Golden ratio: \ (= (1+5) /2 1. 618 \), \ (^-1 0. 618 \), \ (² 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