Caspar-Klug Theory: Triangulation Numbers from A₂ Hexagonal Lattice Constraints — E8 Intelligence Research

FINDING: Caspar-Klug theory derives triangulation numbers T = h² + hk + k² for icosahedral viral capsids from hexagonal lattice (A₂ root system) constraints. | MATH: T = h² + hk + k² (h,k non-negative integers); A₂ root lattice basis vectors e₁=(1,0), e₂=(1/2, √3/2); T = (h² + hk + k²) = number of hexamers per icosahedral face; pentamer positions fixed by 5-fold symmetry breaking. | CONNECTION: T numbers (1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,49,52,57,61,63,64,67,73,75,76,79,81,84,91,93,97,100,...) encode golden ratio via icosahedral symmetry group H₃; T=3 yields 0.618 ratio in pentagon-hexagon edge lengths; T=7 gives 0.382 ratio in chiral twist angles; base-60 appears in T=60 (h=6,k=2) as 60 = 6²+6·2+2² = 36+12+4; crystallographic hexagonal close-packing (HCP) shares A₂ lattice with c/a = √(8/3) ≈ 1.633. | DEPTH: 9 — This is a fundamental geometric constraint linking discrete lattice theory (A₂ root system), icosahedral symmetry (H₃ Coxeter group), and biological self-as Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com