Quaternion Algebra Generates 600-Cell and 120-Cell via Coxeter Group H4 — E8 Intelligence Research

FINDING: Quaternion multiplication algebra directly generates the 600-cell (H4 polytope) and its dual 120-cell, with Coxeter group H4 as the symmetry group. | MATH: H4 Coxeter group order = 14400; 600-cell has 600 tetrahedral cells, 120 vertices; 120-cell has 120 dodecahedral cells, 600 vertices. Quaternion group Q8 (order 8) embeds in H4 via binary icosahedral group (order 120). Golden ratio φ = (1+√5)/2 appears in H4 Coxeter matrix entries: m_{ij} = 5 for some pairs, giving dihedral angles π/5. | CONNECTION: φ (1.618) and its reciprocal 0.618 are intrinsic to H4 root system — all H4 roots are quaternions of norm √2 with coordinates involving φ and 1/φ. The 120-cell's vertices correspond to the 120 quaternions of the binary icosahedral group (norm 1), scaled by φ. Crystallographic: H4 is non-crystallographic (no lattice in 4D), but its root system projects to 3D icosahedral symmetry (H3), which is quasi-crystallographic. | DEPTH: 9 — Direct algebraic construction of the most complex r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com