G2 Root System: Hexagonal Symmetry and Dual Lattice via 30° Rotation — E8 Intelligence Research

FINDING: The G2 root system is the unique rank-2 exceptional root system, encoding hexagonal lattice symmetry with long:short root ratio √3, and its 30° rotation maps the hexagonal lattice onto its dual triangular lattice. | MATH: G2 has 12 roots: 6 short (length 1) at angles 0°,60°,...,300° and 6 long (length √3) at 30°,90°,...,330°. Long:short ratio = √3 ≈ 1.732. The Weyl group is D6 (dihedral of order 12), generated by reflections across lines at 30° increments. The root lattice is the hexagonal lattice A2; rotating by 30° maps it to its reciprocal lattice (dual), with reciprocal spacing ratio 1/√3. The Cartan matrix is [[2,-3],[-1,2]] with determinant 1 (unimodular — self-dual lattice). | CONNECTION: The 30° rotation is precisely the angle that interchanges the two sublattices of the hexagonal lattice (triangular vs. honeycomb). The ratio √3 = 2cos(30°) appears; its reciprocal 1/√3 ≈ 0.577. The golden ratio does NOT appear in G2 (it appears in H2, H3, H4 — icosahedral systems). How Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com