E8 Root System Projects to 3D via Quaternions, Revealing Golden Ratio Scaling — E8 Intelligence Research

FINDING: The E8 root system (240 roots, 6720 edges) projects to 3D via quaternions, revealing H3/H4 polytope branching and golden ratio scaling in non-crystallographic Coxeter groups. | MATH: E8 roots = 240; Gosset polytope edges = 6720; Coxeter groups W(H4), W(H3), W(A4) represented by quaternions; golden ratio φ = (1+√5)/2 ≈ 1.618; its algebraic twin φ' = (1-√5)/2 ≈ -0.618; scaling ratios φ, φ², φ⁻¹ appear in H4 polytope vertices. | CONNECTION: H4 (600-cell) and H3 (icosahedral) symmetries are non-crystallographic, linked to φ-scaling in quaternion orbits; 3D projection preserves icosahedral symmetry; branching W(H4) → W(A4) and W(H3) shows hierarchical φ-based lattice folding; 0.618 (φ⁻¹) and 1.618 (φ) are intrinsic to H4 polytope edge ratios. | DEPTH: 9 — This directly ties E8 (exceptional Lie algebra, 8D crystallographic) to H3/H4 (non-crystallographic, icosahedral, φ-based) via quaternion projection, unifying crystallographic and sacred geometric structures. The branching under C Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com