FINDING: Coxeter element eigenvalues in E8 yield the golden ratio as a fundamental geometric constant, linking root system symmetries to pentagonal tiling and quasicrystalline order. MATH: For Coxeter group E8, the Coxeter element has eigenvalues \ (e^2 i mᵢ / h \) where \ (h = 30 \) is the Coxeter number. The set includes \ (e^2 i 6/30 = e^2 i/5 \) and \ (e^2 i 12/30 = e^2 i 2/5 \), whose real parts are \ ( (72^) = (5-1) /4 0. 309 \) and \ ( (144^) = - (5+1) /4 -0. 809 \). The golden ratio \ (= (1+5) /2 1. 618 \) appears via \ (2 (36^) = \) and \ (2 (72^) = 1/ 0. 618 \). The eigenvalue spectrum includes \ (\) and \ (1/ \) as algebraic integers in the cyclotomic field \ (Q (₅) \). CONNECTION: The golden ratio emerges from the E8 root system's 30-fold rotational symmetry (Coxeter element order 30), which projects to 5-fold symm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence. com
Andrew Stewart Caldin (Fri,) studied this question.