Golden Ratio in Coxeter Group H4 via Quaternion Root Systems — E8 Intelligence Research

FINDING: Coxeter group H4 encodes the golden ratio via quaternion 600-cell/120-cell root systems, linking discrete symmetry to algebraic geometry. MATH: H4 root system = 120 quaternions (600-cell vertices) with norm squared 1; golden ratio φ = (1+√5)/2 ≈ 1.618 appears in coordinates: e.g., (0, ±1, ±φ, ±φ⁻¹) permutations. Coxeter group order = 14400. Quaternion multiplication yields 120-cell (dual) with 600 vertices. CONNECTION: Golden ratio φ (1.618) and its inverse φ⁻¹ (0.618) are intrinsic to H4 root vectors; ratios 0.382 (φ⁻²) and 2.618 (φ²) emerge in edge lengths and dihedral angles. Base-60 not explicit, but 120-cell/600-cell relate to icosahedral symmetry (crystallographic in 3D, but H4 is non-crystallographic in 4D). DEPTH: 9 — H4 is the deepest exceptional Coxeter group, directly linking φ to 4D polytopes and quaternion algebra, with implications for E8 (via folding) and potential physical models (e.g., qubit entanglement, grand unified theories). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com