Circle packing in regular polygons 

We discuss algorithms that allow one to generate configurations of congruent circles inside a regular polygon and present numerical results for polygons with up to 16 sides. Some general properties of these configurations will be highlighted and the topological charge associated with their Voronoi diagrams will be introduced. Specific attention will be given to  the equilateral triangle and the regular hexagon, that for configurations with special numbers of disks allow a perfect triangular packing.

We will show that the numerical results suggest that even regular polygons with 6j sides (j=2,3,…) support highly symmetrical configurations for the same “magic” numbers of the hexagon, up to some critical N where they fail to be optimal. These configurations are invariant under rotations of pi/3 and are closely related to the ones with

perfect hexagonal packing in the regular hexagon and to the configurations with curved hexagonal packing (CHP) in the circle found a long time ago by Graham and Lubachevsky [“Curved hexagonal packings of equal disks in a circle,” Discrete Comput. Geometry 18(2), 179–194 (1997)]. We will first discuss a purely numerical approach to this problem, based on a modification of the previous algorithms, that produces CHPs with good probability and later describe a rigorous procedure that  allows one to construct all the CHP configurations for a given polygon at once. Finally we will mention the curious cases of CHP configurations in the nonagon and in the icosiheptagon for N=37.



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