Caspar-Klug T-Numbers: Hexagonal Lattice Geometry and Base-60 Symmetry in Viral Capsids — E8 Intelligence Research

FINDING: Caspar-Klug triangulation numbers (T=h²+hk+k²) define icosahedral viral capsid geometry via hexagonal lattice indexing, with base-60 resonance in crystallographic symmetry. MATH: - Triangulation number: \( T = h^2 + hk + k^2 \), where (h,k) are integer coordinates on a hexagonal lattice. - Number of capsid proteins: \( 60T \). - Hexagonal lattice basis vectors: \( a_1 = (1,0), a_2 = (1/2, √3/2) \). - T values: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, ... (sequence A003136 in OEIS). - Base-60 connection: T=1,3,4,7,9,12,16,19,21,25,27,28,31,36,37,39,43,48,49 — note 12, 36, 48 are multiples of 12 (base-60 submultiples). 60 itself is not a T-number, but 60T yields protein counts that are multiples of 60. CONNECTION: - Geometric harmony ratios: The hexagonal lattice naturally produces angles of 60° and 120°, and distances scale by \(√3\). The ratio of circumradius to edge length in a regular hexagon is 1. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com