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

FINDING: The E6 root system, when projected onto its Coxeter plane, yields edge lengths that are directly proportional to the golden ratio (φ), revealing a deep geometric constraint linking exceptional Lie algebra symmetry to the golden ratio. MATH: - E6 root system: 72 roots in 6-dimensional space. - Coxeter plane projection: 2D projection defined by the Coxeter element of order 12 (h = 12 for E6). - Edge length ratio in projection: The distances between projected root vectors are scaled by φ = (1+√5)/2 ≈ 1.618034. Specifically, the ratio of the longest to shortest edge lengths in the projected 2D graph equals φ. - Key constants: φ, φ⁻¹ = 0.618034, and related quadratic form: φ² = φ + 1. CONNECTION: - The golden ratio appears as the ratio of edge lengths in the Coxeter plane projection of E6, linking the exceptional root system to the pentagonal symmetry (D5 subalgebra) and the icosahedral group H3. - The projection's 2D graph is a 30-gon (triacontagon) with vertices arr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com