Hexagonal Lattice Geometry Unites Viral Capsids and Aperiodic Tiling — E8 Intelligence Research

FINDING: Viral capsid geometry is governed by quasi-equivalence and hexagonal tiling constraints, while the "Einstein tile" (hat) is a chiral aperiodic monotile whose existence was proven via hexagonal-lattice arithmetic constraints. | MATH: Caspar-Klug theory: capsid triangulation number T = h² + hk + k² (h,k ≥ 0 integers), where the hexagonal lattice basis vectors e₁, e₂ give T = |h·e₁ + k·e₂|². For the hat tile: its vertices lie on a hexagonal lattice; the tile's edges are unit steps in the Eisenstein integers ℤ[ω], ω = e^{2πi/3} = −1/2 + i√3/2. The hat's aperiodicity proof uses the "matching rules" derived from the norm N(a + bω) = a² − ab + b², and the tile's area is 1 (in units of the lattice fundamental parallelogram). The hat is a polykite (union of 8 half-hexagons), with 13 sides, and its chiral pair is its mirror image. | CONNECTION: Direct: hexagonal lattice (root lattice A₂), Eisenstein integers (norm form a² − ab + b²), and the golden ratio appears in the hat's geometry — Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com