Quaternion Construction and 3D Projection of H4 Polytopes — E8 Intelligence Research

FINDING: The Coxeter-Weyl group W(H4) and its polytopes are representable by quaternions, enabling explicit construction of H4 polytopes and their duals, with 3D projections preserving icosahedral symmetry. MATH: - Group W(H4) order = 14400; vertices of H4 polytopes (e.g., 600-cell, 120-cell) are orbits under quaternion multiplication. - Dual polytopes constructed via reciprocal radii or polar reciprocation in 4D. - Projection to 3D uses quaternion map preserving icosahedral group W(H3) (order 120). - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in coordinates of H4 vertices (e.g., 600-cell vertices involve φ, 1, 0). - Scaling factor φ² = φ+1 ≈ 2.618 emerges in dual scaling (e.g., edge lengths of dual polytopes relate by φ²). CONNECTION: - φ² = 2.618 is the square of the golden ratio, linking directly to icosahedral symmetry (H3) and 4D 600-cell/120-cell duality. - Quaternion representation ties to 3D rotation groups (SO(3)) and crystallographic root systems (H4 is non-c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com