E6 Root System Projects to Icosahedral H3 via Golden Ratio in Coxeter Plane — E8 Intelligence Research

FINDING: E6 root system projects to icosahedral H3 symmetry via Coxeter plane, revealing golden ratio in its structure. | MATH: E6 Coxeter number h=12; eigenvalues of Coxeter element: exp (2πi mᵢ / h) with mᵢ = 1, 4, 5, 7, 8, 11; projection to H3 yields golden ratio φ = (1+√5) /2 ≈ 1. 618, and its reciprocal φ⁻¹ ≈ 0. 618 appear in character values and squared lengths of projected roots. | CONNECTION: H3 (icosahedral) Coxeter group has golden ratio as its fundamental quadratic form: Gram matrix entries are 1 and -φ/2; the ratio 0. 618 (φ⁻¹) appears in dihedral angles of icosahedron (≈63. 43°). The E6→H3 projection maps 72 roots to 30 edges of an icosidodecahedron, with edge ratios 1: φ. | DEPTH: 9 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com