Discrete Conformal Mapping of Hexagonal Tilings to Icosahedral Viral Capsids — E8 Intelligence Research

Abstract

FINDING: Discrete conformal equivalence provides a rigorous framework for mapping hexagonal tilings onto icosahedral surfaces, directly modeling viral capsid geometry via quasi-equivalence theory. | MATH: The key is the discrete conformal factor \( u_i \) on vertices, satisfying \( \ell_{ij} = e^{(u_i+u_j)/2} \ell_{ij}^0 \) for edge lengths, with the condition that the angle sum around each vertex is \( 2\pi \) (or \( 2\pi - \theta_i \) for cone angles). For icosahedral capsids, the target is a sphere with 12 cone singularities of angle \( 2\pi/5 \) at the 5-fold axes. The hexagonal tiling's curvature is concentrated at these points. The Möbius transformation group (conformal automorphisms of the sphere) is \( \text{PSL}(2,\mathbb{C}) \). | CONNECTION: The icosahedral symmetry group (order 60) is a finite subgroup of \( \text{SO}(3) \). The 5-fold axes correspond to cone angles \( 2\pi/5 = 0.4\pi \), which is a rational multiple of \( \pi \). The ratio of edge lengths in the hexagonal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com