Icosahedral Symmetry and Caspar–Klug Triangulation Numbers in Viral Capsids — E8 Intelligence Research

FINDING: Icosahedral symmetry in viral capsids is a direct physical realization of the rotational group \( A_5 \) (order 60), constrained by quasi-equivalence (Caspar–Klug theory) to specific triangulation numbers \( T = h^2 + hk + k^2 \). | MATH: Caspar–Klug triangulation number \( T = h^2 + hk + k^2 \) (h,k non-negative integers); capsid protein copy number = \( 60T \); icosahedral rotation group \( A_5 \) has order 60, with 2-, 3-, 5-fold axes; allowed \( T \) values: 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, ... (sequence A005875). The golden ratio \( \varphi = (1+\sqrt{5})/2 \approx 1.618 \) appears intrinsically in the coordinates of icosahedral vertices: e.g., vertices at \( (0, \pm 1, \pm \varphi) \), \( (\pm 1, \pm \varphi, 0) \), \( (\pm \varphi, 0, \pm 1) \). | CONNECTION: The 5-fold symmetry axes of the icosahedron force the golden ratio into the vertex coordinates; the ratio of ci Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com