Circle packing in regular polygons.

This article focuses on the numerical study of packing congruent, non-overlapping circles inside regular polygons with 3 to 16 sides, extending previous work limited mainly to triangles, squares, and circles. It introduces and applies three algorithms—an energy-minimization method with border repulsion (Algorithm 1), a shaking-inspired refinement (Algorithm 2), and a variance-minimizing refinement (Algorithm 3)—to generate dense packing configurations for up to 200 circles (400 for triangles and hexagons). The study analyzes packing fractions, border effects, and topological properties of the resulting configurations, employing Euler's theorem to relate Voronoi cell structures and topological charges within the polygons. Upper bounds on packing densities are derived using known geometric inequalities, and numerical results reveal that polygons like the equilateral triangle, regular hexagon, dodecagon, and pentadecagon exhibit distinctive packing behaviors, including the presence of "necklace" configurations and variations in border crowding. The data and configurations produced provide near-optimal lower bounds for packing fractions and offer insights into the interplay between geometry, topology, and packing efficiency in finite polygonal domains.