E8 Coxeter Eigenvalues: Golden Ratio and π/30, π/15 Trigonometry — E8 Intelligence Research

**FINDING:** The eigenvalues of the Coxeter element in the E8 root system are expressible in terms of trigonometric values of π/30 and π/15, which are directly related to the golden ratio φ and its powers. **MATH:** - Coxeter element eigenvalues for E8: \( e2π i m_i / h \), where \( h = 30 \) is the Coxeter number. - The exponents \( m_i \) for E8 are: 1, 7, 11, 13, 17, 19, 23, 29. - Thus eigenvalues are \( e2π i k / 30 \) for \( k ∈ \{1,7,11,13,17,19,23,29\} \). - Key trigonometric values: \[ cos(π/5) = φ/2 = {1+√5}{4} ≈ 0.809016 \] \[ cos(π/15) = 1/8(1+√5+√{30-6√5}) \] \[ cos(2π/15) = 1/8(-1+√5+√{30+6√5}) \] - Golden ratio φ = (1+√5)/2 ≈ 1.618034, and its reciprocal 1/φ = φ-1 ≈ 0.618034. **CONNECTION:** - The E8 root system has 240 roots, and its Coxeter number 30 = 2×3×5, linking to pentagonal symmetry (φ) and base-60 (30×2). - T Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com