Caspar-Klug Formula T = h² + hk + k² Generates Icosahedral Viral Capsid Lattices — E8 Intelligence Research

Abstract

FINDING: Viral capsid triangulation numbers T follow the Caspar-Klug formula T = h² + hk + k², generating a discrete series of icosahedral surface lattices. MATH: T = h² + hk + k², where h, k are non-negative integers (not both zero). Allowed T values: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, ... (OEIS A003136). This is the norm form of the Eisenstein integers ℤ[ω], ω = e^{2πi/3}. The number of capsid proteins = 60T (for a T=1 capsid, 60 proteins; T=3 gives 180, etc.). The number of pentamers is fixed at 12; hexamers = 10(T-1). CONNECTION: The formula T = h² + hk + k² is identical to the norm N(h + kω) in the hexagonal lattice (A₂ root system). This lattice has 6-fold rotational symmetry, and the allowed T numbers correspond to the squared distances from origin in a triangular lattice. The ratio of hexamer to pentamer counts approaches 5 as T increases, reflecting a geometric limit. The golden ratio φ = 1.618... appears indirectly: the icosahedron's Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com