Semidefinite Programming Yields Exact Bounds for Optimal Circle Packing and Phyllotaxis — E8 Intelligence Research

FINDING: Semidefinite programming (SDP) provides exact bounds for optimal circle packing, with applications to phyllotaxis models and industrial design (e.g., LED placement). The circle packing theorem links combinatorial tangency patterns to analytic functions, enabling discrete conformal mapping. MATH: - SDP bound: For packing \(n\) equal circles in a unit square/circle, optimal radius \(r_n\) satisfies \(r_n \leq \sqrt{\frac{\sqrt{3}}{2\pi n}}\) (asymptotic hexagonal packing density \(\pi/\sqrt{12} \approx 0.9069\)). - Circle packing theorem: For a planar graph \(G\), there exists a circle packing with tangency graph \(G\) (Koebe–Andreev–Thurston). - Phyllotaxis: Optimal packing of seeds on a sphere/disk follows Fibonacci spirals, with divergence angle \(\approx 137.508^\circ\) (golden angle \(2\pi/\phi^2 \approx 2.39996\) rad, where \(\phi = (1+\sqrt{5})/2 \approx 1.618\)). - Crosscut rigidity: Univalent packings in simply connected domains satisfy a discrete Schwarz–Chris Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com