Caspar-Klug T-numbers govern icosahedral viral capsid geometry via crystallographic constraints. — E8 Intelligence Research

FINDING: Caspar-Klug T-number classification governs hexagonal tiling on icosahedral face lattices, linking viral capsid geometry to crystallographic constraints. MATH: T = h² + hk + k² (h,k ≥ 0 integers); allowed 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, ...; triangulation number N = 20T; edge length ratio in icosahedral lattice: golden ratio φ = (1+√5)/2 ≈ 1.618. CONNECTION: T-number directly encodes hexagonal tiling symmetry on icosahedron faces — each face subdivided into T equilateral triangles. The allowed T-numbers arise from hexagonal lattice basis vectors (a₁, a₂) with 60° angle, satisfying T = |h a₁ + k a₂|². This is identical to the norm form in the Eisenstein integer ring ℤ[ω] (ω = eπi/3), linking to hexagonal close-packing and 6-fold rotational symmetry. The golden ratio φ appears in the icosahedron's vertex coordinates (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com