Icosahedral Symmetry and Caspar-Klug Theory in Viral Capsid Assembly — E8 Intelligence Research

FINDING: Icosahedral symmetry group Ih (order 120) governs viral capsid self-assembly, with mathematical constraints from group theory and Caspar-Klug theory dictating capsid geometry. MATH: - Icosahedral symmetry group: Ih order 120 (rotational subgroup I order 60). - Caspar-Klug triangulation numbers: T = h² + hk + k², where h,k ≥ 0 integers (e.g., T=1,3,4,7,9,12,13,16...). - Capsid radius scaling: R ∝ √T. - Pentamer/hexamer ratio: 12 pentamers + 10(T-1) hexamers. - Key constants: golden ratio φ = (1+√5)/2 ≈ 1.618, its reciprocal φ⁻¹ ≈ 0.618, and φ² ≈ 2.618 appear in icosahedral vertex coordinates. CONNECTION: - Golden ratio φ (1.618) and its powers (0.618, 2.618) are intrinsic to icosahedron geometry (edge/radius ratios, face angles). - T-numbers encode hexagonal lattice distortions; T=3 capsids (e.g., MS2 phage) have 180 subunits, T=7 (e.g., adenovirus) 420 subunits. - Base-60 not directly present, but icosahedral symmetry relates to 5-fold axes (pentagonal symmet Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com