Golden Ratio in E8 Coxeter Eigenvalues Links Symmetry to Quasicrystals — E8 Intelligence Research

FINDING: Coxeter element eigenvalues in E8 yield the golden ratio as a fundamental geometric constant, linking root system symmetries to pentagonal tiling and quasicrystalline order. MATH: For Coxeter group E8, the Coxeter element has eigenvalues \( e2π i m_i / h \) where \( h = 30 \) is the Coxeter number. The set includes \( e2π i · 6/30 = e2π i/5 \) and \( e2π i · 12/30 = e2π i · 2/5 \), whose real parts are \( cos(72^∘) = (√5-1)/4 ≈ 0.309 \) and \( cos(144^∘) = -(√5+1)/4 ≈ -0.809 \). The golden ratio \( φ = (1+√5)/2 ≈ 1.618 \) appears via \( 2cos(36^∘) = φ \) and \( 2cos(72^∘) = 1/φ ≈ 0.618 \). The eigenvalue spectrum includes \( φ \) and \( 1/φ \) as algebraic integers in the cyclotomic field \( Q(ζ_5) \). CONNECTION: The golden ratio emerges from the E8 root system's 30-fold rotational symmetry (Coxeter element order 30), which projects to 5-fold symm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com