Caspar-Klug Geometry Links Viral Capsid Symmetry to the Golden Ratio — E8 Intelligence Research

Abstract

FINDING: Icosahedral symmetry in viral capsids is constrained by quasi-equivalence theory, requiring triangulation numbers (T-numbers) that follow the Caspar-Klug geometry, linking viral architecture to the golden ratio and 60-fold symmetry. MATH: - Caspar-Klug T-number: \( T = h^2 + hk + k^2 \), where \( h, k \) are non-negative integers. - Allowed T-numbers: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, ... - Icosahedral symmetry group: order 60 (rotational), with 5-fold, 3-fold, 2-fold axes. - Golden ratio \( \phi = (1+\sqrt{5})/2 \approx 1.618 \) appears in coordinates of icosahedron vertices: \( (0, \pm1, \pm\phi), (\pm1, \pm\phi, 0), (\pm\phi, 0, \pm1) \). - Quasi-equivalence constraint: capsid proteins must adopt slightly different conformations to fit T-number lattice, with curvature governed by \( \phi \)-based pentagonal and hexagonal tiling. CONNECTION: - Icosahedral symmetry is a direct geometric expression of the golden ratio: 5-fold axes yield pentagons with side/diago Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com