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 i mᵢ / h \), where \ (h = 30 \) is the Coxeter number. - The exponents \ (mᵢ \) for E8 are: 1, 7, 11, 13, 17, 19, 23, 29. - Thus eigenvalues are \ (e^2 i k / 30 \) for \ (k \1, 7, 11, 13, 17, 19, 23, 29\ \). - Key trigonometric values: \ (/5) = 2 = 1+54 0. 809016 \ \ (/15) = 18 (1+5+30-65) \ \ (2/15) = 18 (-1+5+30+65) \ - 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