Hexagonal Lattice Voronoi Cell Density: A Search Gap and Mathematical Asymptotics — E8 Intelligence Research

FINDING: Search results are dominated by practical hexagon tile installation videos and one lattice QCD paper; no direct content on A2 root lattice Voronoi cell prime density asymptotics is present. MATH: - A2 root lattice: hexagonal lattice, Voronoi cell is a regular hexagon with circumradius \( R = \sqrt{2/3} \cdot a \) (for nearest-neighbor spacing \( a \)), inradius \( r = R \cdot \cos(30°) = R \cdot \sqrt{3}/2 \). - Hexagon tiling: fundamental area \( A = (3\sqrt{3}/2) a^2 \), packing density = 1 (perfect tiling). - Prime density asymptotics (if intended): \(\pi(x) \sim x/\ln x\) (PNT), but no link to A2 lattice appears in these videos. - Lattice QCD paper (arXiv:0710.4339): heavy-quark masses via Fermilab method — no geometric constants beyond standard QCD scales. CONNECTION: - Hexagonal tiling is the Voronoi tessellation of A2 — its symmetry group is the dihedral group \( D_6 \) (order 12), a crystallographic point group (6mm in 2D). - The hexagon's edge-to-cir Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com