Icosahedral Symmetry and Quasi-Equivalence: Geometric Principles of Viral Capsid Architecture — E8 Intelligence Research

Abstract

FINDING: Icosahedral symmetry is the dominant geometric principle in viral capsid architecture, with structural constraints dictated by quasi-equivalence theory. MATH: - Icosahedral symmetry group: order 60 (rotational) or 120 (full). - Caspar-Klug triangulation numbers: T = h² + hk + k², where h,k are non-negative integers (e.g., T=1,3,4,7,9,12,13,16,19,21…). - Pentamers (12) + hexamers (10(T-1)) = 60T capsid subunits. - Golden ratio φ = (1+√5)/2 ≈ 1.618 appears in coordinates of icosahedron vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1). - Dihedral angles: 138.19° (pentagon-hexagon), 120° (hexagon-hexagon). CONNECTION: - Icosahedral symmetry is a direct geometric realization of the golden ratio φ, which governs the 3-5-2 symmetry axes. - The Caspar-Klug T-number series generates all possible icosahedral lattices on a sphere, linking to hexagonal close-packing and the 60°-120° rhombus tiling. - The ratio of pentamer to hexamer curvature (0.618 vs 1.0) mirrors the Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com