Möbius Transformations and Icosahedral Symmetry in Viral Capsid Geometry — E8 Intelligence Research

FINDING: Möbius transformations (PSL(2,C)) encode icosahedral symmetry as a finite subgroup, linking conformal geometry to viral capsid architecture via discrete spherical conformal factors. MATH: - Group: PSL(2,C) ≅ Möbius group of the Riemann sphere. - Finite subgroups: cyclic, dihedral, tetrahedral (A₄), octahedral (S₄), icosahedral (A₅). - Icosahedral group order: 60 (rotational), 120 (full). - Conformal factor on sphere: \( ds^2 = e^{2\phi} |dz|^2 / (1+|z|^2)^2 \) for Möbius-transformed metric. - Discrete conformal factor: \(\phi\) piecewise constant on spherical triangulation (e.g., capsid). - Key ratio from icosahedron: edge/radius = \( \sqrt{10-2\sqrt{5}}/2 \approx 1.0515 \); golden ratio \(\varphi = (1+\sqrt{5})/2 \approx 1.618\) appears in vertex coordinates. CONNECTION: - Icosahedral symmetry is crystallographic (H₃ Coxeter group) and yields golden ratio \(\varphi\) in its root system. - Möbius transformations preserve cross-ratio, linking to harmonic ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com