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: \( e^{2\pi i m_i / h} \), 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 \( e^{2\pi i k / 30} \) for \( k \in \{1,7,11,13,17,19,23,29\} \). - Key trigonometric values: \[ \cos(\pi/5) = \frac{\varphi}{2} = \frac{1+\sqrt{5}}{4} \approx 0.809016 \] \[ \cos(\pi/15) = \frac{1}{8}\left(1+\sqrt{5}+\sqrt{30-6\sqrt{5}}\right) \] \[ \cos(2\pi/15) = \frac{1}{8}\left(-1+\sqrt{5}+\sqrt{30+6\sqrt{5}}\right) \] - 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