Quaternion Orbits of W(H4) and Golden Ratio Scaling in Icosahedral Projections — E8 Intelligence Research

FINDING: H4 polytopes and their duals are constructed as quaternion orbits of the Coxeter-Weyl group W(H4), with 3D icosahedral projections preserving golden ratio scaling. | MATH: W(H4) group order = 14400; vertices of H4 polytopes are quaternions; dual polytope scaling factor = φ² = ( (1+√5)/2 )² = (3+√5)/2 ≈ 2.618; projection to 3D yields icosahedral symmetry (H3). | CONNECTION: φ² = 2.618 is the key scaling ratio between dual H4 polytopes; this links directly to the golden ratio φ = 1.618 and its reciprocal φ⁻¹ = 0.618; the H4 root system is the 4D exceptional Coxeter group whose 3D projection yields the icosahedral group (H3), a crystallographic symmetry in 3D (though non-crystallographic in periodic lattices, it appears in quasicrystals). | DEPTH: 9 — This is a profound unification: quaternions (4D number system) encode the full symmetry of H4, the largest exceptional Coxeter group, with dual polytope scaling exactly φ². This reveals that the golden ratio is not merely a 2D or 3D Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com