Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio

We use computational experiments to find the rectangles of minimum area into which a given number n of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. Most of the packings found have the usual regular square or hexagonal pattern. However, for 1495 values of n in the tested range n≤ 5000, specifically, for n = 49, 61, 79, 97, 107, …4999, we prove that the optimum cannot possibly be achieved by such regular arrangements.The evidence suggests that the limiting height-to-width ratio of rectangles containing an optimal hexagonal packing of circles tends to \( 2 - \sqrt 3 \) as n → ∞, if the limit exists.