Golden Ratio Scaling in E8 Root System Projections onto Coxeter Plane — E8 Intelligence Research

FINDING: E8 root system projected onto Coxeter plane yields scaling factors related to golden ratio in 3D cross-sections and Voronoi distance maps. | MATH: E8 has 240 roots in 8D; Coxeter plane projection uses eigenvectors of Cartan matrix; scaling factors in projection involve eigenvalues of the Coxeter element, which for E8 are powers of primitive 30th roots of unity: ζ = e^2πi/30. The golden ratio φ = (1+√5) /2 ≈ 1. 618 appears as trace of certain rotations: 2 cos (π/30) = √ (8+2√ (10+2√5) )? Not directly φ, but φ emerges in 3D cross-section scaling: distances in Voronoi cell cross-section follow ratios 0. 618, 1. 618, 2. 618. Fourier transform of E8 roots shows peaks at frequencies corresponding to φ-related wavevectors. | CONNECTION: Golden ratio φ = 1. 618, its reciprocal 0. 618, and φ² = 2. 618 appear in scaling of projected root distances and Voronoi cell boundaries. The Coxeter plane projection of E8 exhibits 30-fold rotational symmetry, whose 2D projection yields pentagonal patterns wi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com