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 \( e^{2\pi i m_i / h} \) where \( h = 30 \) is the Coxeter number. The set includes \( e^{2\pi i \cdot 6/30} = e^{2\pi i/5} \) and \( e^{2\pi i \cdot 12/30} = e^{2\pi i \cdot 2/5} \), whose real parts are \( \cos(72^\circ) = (\sqrt{5}-1)/4 \approx 0.309 \) and \( \cos(144^\circ) = -(\sqrt{5}+1)/4 \approx -0.809 \). The golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) appears via \( 2\cos(36^\circ) = \phi \) and \( 2\cos(72^\circ) = 1/\phi \approx 0.618 \). The eigenvalue spectrum includes \( \phi \) and \( 1/\phi \) as algebraic integers in the cyclotomic field \( \mathbb{Q}(\zeta_5) \). 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