FINDING: E8 root system's 3D Coxeter plane projection reveals 30 pairs of vertices arranged in pentagonal symmetry, directly encoding the golden ratio. MATH: - E8 has 240 roots, 6720 edges, Coxeter number 30. - Projection onto Coxeter plane yields 30 concentric rings of 8 vertices each, forming 30 pairs of regular pentagons. - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in edge length ratios of the projected pentagons. - Key constants: φ, 1/φ = 0.618, φ² = 2.618, φ⁻² = 0.382. - Coxeter plane eigenvectors correspond to eigenvalues of the Cartan matrix; for E8, these involve φ and √5. CONNECTION: - Pentagonal symmetry (order 5) is crystallographically forbidden in periodic 3D lattices but appears in E8's 2D projection due to its 8D non-crystallographic structure. - Golden ratio emerges from the 30 pairs of roots: each pair's angular separation in the Coxeter plane is 2π/5 or multiples thereof. - The ratio of radii of successive pentagonal rings approximates φ. - This li Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Andrew Stewart Caldin (Mon,) studied this question.