Semidefinite Programming Yields Exact Bounds for Optimal Circle Packing and Phyllotaxis — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: 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
Author: Caldin, Andrew Stewart
Language: English
Subfield: Industrial and Manufacturing Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203650871
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