Caspar-Klug Geometry: T-Number, Pentamer Defects, and Icosahedral Capsid Curvature — E8 Intelligence Research

Abstract

FINDING: Icosahedral capsid geometry is governed by quasi-equivalence theory, requiring 12 pentamers and variable hexamers (T-number) to close a curved lattice. | MATH: T = h² + hk + k² (Caspar-Klug), where h,k are integers; curvature ratio = pentamer defect / hexamer count; Euler characteristic χ = V - E + F = 2 for sphere; pentamer introduces +12° disclination (discrete curvature). | CONNECTION: T-numbers (1,3,4,7,9,12,13,16,19,21…) map to icosahedral symmetry group (order 60); pentamer-hexamer ratio relates to golden ratio φ = 1.618 via icosahedron's 12 vertices (pentamers) and 20 faces (hexamer arrays); curvature defect per pentamer = 2π/12 = π/6 ≈ 0.5236 rad, linking to base-60 angular divisions. | DEPTH: 8 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com