E8 Coxeter Eigenvalues Yield Golden Ratio at π/30 and π/15 — E8 Intelligence Research

FINDING: E8 Coxeter element eigenvalues yield exact trigonometric values at π/30 and π/15, linking golden ratio to exceptional root system symmetries. MATH: Eigenvalues of Coxeter element in E8: exp(2πi·m_i/h) where h=30 (Coxeter number). The exponents m_i ∈ {1,11,19,29} for E8 produce cos(π/30), cos(11π/30)=sin(π/15), etc. Exact values: cos(π/30) = (√(30+6√5)+√5-1)/8, cos(π/15) = (√(30+6√5)+√5+1)/8. Golden ratio φ = (1+√5)/2 appears: cos(π/5)=φ/2, cos(π/30) involves √5 and φ. CONNECTION: E8 root system (240 roots, 8D) has Coxeter number 30, linking to π/30. Golden ratio φ=1.618, its reciprocal 0.618, and φ²=2.618 appear in these cosines. Base-60: 30°=π/6, 15°=π/12, but π/30=6° — a base-60 subdivision. Crystallographic: E8 is the largest exceptional Lie group, its root lattice E8 is the densest sphere packing in 8D. DEPTH: 9 — Direct marriage of number theory (trigonometric exact values), algebra (Coxeter elements), and geometry (E8 root system). The golden ratio emerges naturall Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com