Golden Ratio Emerges in Exceptional Lie Algebra Coxeter Numbers and Root Symmetries — E8 Intelligence Research

FINDING: The golden ratio (φ ≈ 1.618) appears in the Coxeter numbers of exceptional Lie algebras E₆, E₇, E₈, and in the root system symmetries of these algebras, linking finite reflection groups to the golden ratio. | MATH: Coxeter numbers h: E₆ → h=12, E₇ → h=18, E₈ → h=30. Ratios: h(E₈)/h(E₇)=30/18=5/3≈1.666; h(E₇)/h(E₆)=18/12=3/2=1.5. The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the eigenvalues of the Coxeter element: for E₈, eigenvalues are exp(2πi·m/h) with m in {1,7,11,13,17,19,23,29} — these include pairs summing to h, and the set contains numbers related to φ via the pentagon (e.g., 5-fold symmetry). The root system of E₈ has 240 roots; the ratio of long to short roots is √2, but the angle between certain roots (e.g., in the H₄ subgroup) involves φ. The Coxeter number h=30 for E₈ is 2·3·5, linking to pentagonal symmetry. | CONNECTION: φ appears in the H₄ Coxeter group (icosahedral symmetry), which is a subgroup of E₈. The golden ratio is the ratio of diagonal to side in a r Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com