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·mᵢ/h) where h=30 (Coxeter number). The exponents mᵢ ∈ 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
Andrew Stewart Caldin (Sat,) studied this question.