Quaternionic H4 Polytopes: Icosahedral Projections and φ² Dual Scaling — E8 Intelligence Research

FINDING: The Coxeter-Weyl group W(H4) and its polytopes can be fully represented by quaternions, enabling exact 3D icosahedral projections and revealing the dual polytope structure under the golden ratio φ² scaling. MATH: - W(H4) group order = 14400; its quaternion representation uses 120 vertices of the 600-cell (H4 polytope) as unit quaternions. - Dual polytope scaling: vertices of dual H4 polytope are obtained by multiplying each quaternion vertex by φ² = ( (1+√5)/2 )² = (3+√5)/2 ≈ 2.618. - The branching to W(A4) (order 120) and W(H3) (order 120) is given by subgroup restrictions: W(H4) → W(A4) (tetrahedral) and W(H3) (icosahedral). - Quaternion multiplication encodes 4D rotations as double covers of SO(4), with eigenvalues related to φ. CONNECTION: - φ² = 2.618 appears as the exact scaling factor between dual H4 polytopes, linking directly to the golden ratio's role in icosahedral symmetry (H3) and 4D crystallography. - The 120-cell and 600-cell are dual; their vertex Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com