Icosahedral Symmetry and Caspar-Klug Theory Govern Viral Capsid Architecture — E8 Intelligence Research

Abstract

FINDING: Icosahedral symmetry is the dominant geometric constraint for viral capsid architecture, dictated by Caspar-Klug theory. | MATH: Icosahedral symmetry group (order 60); Caspar-Klug triangulation number T = h² + hk + k², where (h,k) are non-negative integers (e.g., T=1,3,4,7,9,12...). The capsid is composed of 60T protein subunits arranged in pentamers (12 vertices) and hexamers (10(T-1) faces). The dihedral angles of the icosahedron are ~138.19° (edge) and ~116.57° (face). The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in the coordinates of the icosahedron's vertices: (0, ±1, ±φ), (±1, ±φ, 0), (±φ, 0, ±1). | CONNECTION: The icosahedron is dual to the dodecahedron; both embed φ. The ratio of edge length to circumradius is 1/(√(φ√5)) ≈ 0.618. The face angle of the icosahedron's equilateral triangles is 60°, but the dihedral angle relates to φ via arccos(-√5/3) ≈ 138.19°, whose supplementary angle (41.81°) has sine = 2/φ² ≈ 0.618. The T-number series generates all possible icosahed Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com