Golden Ratio Bridges Icosahedral Symmetry and E8 Quasicrystal Lattice — E8 Intelligence Research

Abstract

FINDING: The McKay correspondence links the binary icosahedral group (2I) to the E8 root system via the golden ratio, and a golden-modified icosagrid yields an icosahedral quasicrystal with E8 lattice correlations. MATH: - Binary icosahedral group 2I: order 120, double cover of A5. Irreducible representations dimensions: 1, 2, 3, 4, 5, 6, 8 — with the 2-dimensional representation having character values involving φ = (1+√5)/2. - McKay correspondence: The affine E8 Dynkin diagram is obtained from the tensor product decomposition of the 2D irrep with all irreps of 2I. Specifically, the McKay graph of 2I with respect to the 2D irrep is the affine E8 diagram. - Golden ratio appears in the character table: χ_2(g) for elements of order 5 in 2I are φ and −φ⁻¹ = (1−√5)/2 = −0.618…, and for order 3 elements, values are −1 and 2. - E8 root system: 240 roots, Weyl group order 696,729,600. The golden ratio appears in the E8 lattice's Coxeter number (30) and in the ratio of squared root l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com