Metallic Ratios, Coxeter Groups, and E8 Symmetry in Expander Graphs — E8 Intelligence Research

FINDING: Hyperbolic Coxeter groups yield highly regular expander graphs; metallic ratios (golden, silver, bronze) form a family linked to root system discriminants; E8 root system projects to 3D with golden-ratio-based symmetries. | MATH: Metallic ratios defined by recurrence \( \phi_n = n + 1/\phi_n \), giving golden ratio \( \phi_1 = (1+\sqrt{5})/2 \approx 1.618 \), silver ratio \( \phi_2 = 1+\sqrt{2} \approx 2.414 \), bronze ratio \( \phi_3 = (3+\sqrt{13})/2 \approx 3.303 \). Coxeter group discriminant for hyperbolic type relates to these ratios. E8 root system has 240 roots; its Cartan matrix determinant = 1; Coxeter number = 30; eigenvalues involve golden ratio. | CONNECTION: Golden ratio (1.618) and its reciprocal (0.618) appear in E8 root system projections and in hyperbolic Coxeter group growth rates. The metallic ratios correspond to eigenvalues of certain Coxeter matrices. Base-60 not directly present, but crystallographic symmetry of E8 (8D lattice) projects to 3D with golde Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com