E6 Root System's Coxeter Plane Projection Reveals Golden Ratio φ — E8 Intelligence Research

FINDING: E6 root system Coxeter plane projection yields edge lengths in golden ratio φ, linking exceptional Lie algebra E6 to pentagonal symmetry and the golden ratio. MATH: - E6 root system: 72 roots in 6D, rank 6, Coxeter number h = 12. - Coxeter plane projection: 2D projection using Coxeter element of order h. - Edge lengths in projection: ratios of distances between root projections yield φ = (1+√5)/2 ≈ 1.618, and its reciprocal 1/φ ≈ 0.618. - Specifically, the squared distances between certain projected roots are in ratio φ² : 1 : 1/φ², i.e., (φ², 1, φ⁻²). - φ² = φ + 1 ≈ 2.618, φ⁻² = 2 - φ ≈ 0.382. CONNECTION: - Golden ratio φ appears naturally from the Coxeter plane projection of E6, a crystallographic root system of exceptional type. - The projection exhibits pentagonal (D5) and decagonal symmetry, directly linking E6 to the golden ratio and fivefold symmetry, which is forbidden in periodic 3D lattices but appears in quasicrystals. - The ratios 0.382, 0.618, 1. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com