E8 Root System's Coxeter Plane Encodes Golden Ratio and 5-Fold Symmetry — E8 Intelligence Research

FINDING: E8 root system's Coxeter plane projection yields eigenvalues that encode the golden ratio φ = (1+√5)/2, linking the 240-root polytope to 5-fold symmetry and quasicrystalline order. | MATH: The Coxeter plane for E8 is defined by eigenvectors of the Cartan matrix's Coxeter element. Eigenvalues are primitive 30th roots of unity: exp(2πi k/30) for k coprime to 30 (1,7,11,13,17,19,23,29). The sum of two specific eigenvalues gives φ: e2πi/30 + e2πi 29/30 = 2 cos(π/15) ≈ 1.956, but more directly, the golden ratio appears in the ratio of root lengths and in the projection's 2D coordinates: coordinates of the 120 vertices of the 600-cell (a 4D polytope related to E8's 3D shadow) involve φ = (1+√5)/2, with its reciprocal φ⁻¹ = φ-1 ≈ 0.618. The E8 lattice's Voronoi cell cross-section in the Coxeter plane reveals distances matching φ-scaled rings. | CONNECTION: The golden ratio φ = 1.618… and its reciprocal 0.618… emerge directly from E8's Coxeter plane projection, which exhibits 5- Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com