Circle packing in regular polygons

TL;DR

This paper presents efficient algorithms and extensive numerical experiments for packing a large number of congruent circles inside regular polygons, providing new insights and lower bounds for packing densities in these shapes.

Contribution

It introduces the first systematic numerical study of circle packing in regular polygons beyond basic shapes, with algorithms capable of handling up to 16-sided polygons and hundreds of circles.

Findings

Developed efficient algorithms for dense circle packing in regular polygons.

Conducted extensive numerical experiments across various polygons and circle counts.

Provided lower bounds for packing fractions in regular polygons.

Abstract

We study the packing of a large number of congruent and non--overlapping circles inside a regular polygon. We have devised efficient algorithms that allow one to generate configurations of $N$ densely packed circles inside a regular polygon and we have carried out intensive numerical experiments spanning several polygons (the largest number of sides considered here being $16$) and up to $200$ circles ($400$ circles in the special cases of the equilateral triangle and the regular hexagon) . Some of the configurations that we have found possibly are not global maxima of the packing fraction, particularly for $N \gg 1$, due to the great computational complexity of the problem, but nonetheless they should provide good lower bounds for the packing fraction at a given $N$. This is the first systematic numerical study of packing in regular polygons, which previously had only been carried out for the equilateral triangle, the square and the circle.