Quaternion H4 Group Generates 600-Cell and 120-Cell via Golden Ratio — E8 Intelligence Research

FINDING: Coxeter group H4, realized via quaternion multiplication, generates the 600-cell and 120-cell polytopes, encoding the golden ratio as a fundamental algebraic invariant. | MATH: H4 Coxeter group order = 14400; 600-cell has 600 tetrahedral cells, 120 vertices; 120-cell has 120 dodecahedral cells, 600 vertices. Quaternion representation uses 120 unit icosians (norm 1) from the binary icosahedral group, with coordinates involving (1, φ, 0, 0) permutations and sign changes, where φ = (1+√5)/2 ≈ 1.618. The golden ratio appears in the Coxeter matrix entry m_{ij}=5 (since 2 cos(π/5) = φ). | CONNECTION: Golden ratio φ = 1.618, its reciprocal 1/φ = 0.618, and φ² = 2.618 are intrinsic to H4 root system norms and edge ratios. The 600-cell's vertex coordinates are quaternions forming a discrete subgroup of SU(2), linking to the icosahedral symmetry (crystallographic point group 235). The ratio of circumradius to edge length in the 600-cell is φ. | DEPTH: 9 — This is a profound unification: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com