E8 Projects to 600-Cell with Golden Ratio Edge-to-Radius — E8 Intelligence Research

FINDING: E8 (240 roots, Gosset 4₂₁) admits a projection to the H4/600-cell whose edge-to-radius ratio is the golden ratio φ⁻¹ ≈ 0.618, linking the largest exceptional Lie algebra to icosahedral symmetry. | MATH: E8 root system: 240 roots, |Φ(E8)|=240, Weyl group order |W(E8)| = 696729600 = 2¹⁴·3⁵·5²·7. Kissing number τ₈=240. 600-cell: 120 vertices, 720 edges, edge/radius = 1/φ = (√5−1)/2 ≈ 0.618034. H4 Coxeter group order = 14400 = 120². E8 theta series: Θ_E8(q) = 1 + 240·Σ σ₃(n)qⁿ = E₄ (Eisenstein series of weight 4); 240 = coefficient, modular form identity Θ_E8 = E₄. | CONNECTION: φ = (1+√5)/2 = 1.618034; φ⁻¹ = 0.618034; φ² = 2.618034. The 600-cell is the 4D icosahedral analogue; its vertex figure is an icosahedron (12 vertices, φ-coordinates). E8 ⊃ H4 via folding of the E8 Coxeter diagram (E8 → H4 by identifying the 8-node diagram under the 5-fold symmetry of the affine extension). Root counts: 240 = 2·120 (E8 roots ↔ 600-cell vertices ×2, ±). 6720 = number of E8 edges (4₂₁ polytop Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com