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.
翻译:我们研究了在常规多边形内包装大量相近和非重叠圆圈的问题。我们设计了高效的算法,允许一个人在常规多边形内生成以美元为密度的密集圆圈的配置,我们进行了多个多边形的密集数字实验(这里认为最大的边数为16美元),最多达200美元圆圈(在对等三角和普通六边形的特殊情况下为400美元圆)。我们发现的一些配置可能不是包装部分的全球最高值,特别是1美元,因为问题的计算复杂程度很大,但是它们应该以给定的1美元为包装部分提供更低的宽度。这是在常规多边形中包装的第一次系统性数字研究,以前只对等边三角、方形和圆进行这种系统的数字研究。