Minimum perimeter rectangles that enclose congruent non-overlapping circles

We use computational experiments to find the rectangles of minimum perimeter into which a given number, n , of non-overlapping congruent circles can be packed. No assumption is made on the shape of the rectangles. In many of the packings found, the circles form the usual regular square-grid or hexagonal patterns or their hybrids. However, for most values of n in the tested range n ≤ 5000 , e.g., for n = 7 , 13 , 17 , 21 , 22 , 26 , 31 , 37 , 38 , 41 , 43 , … , 4997 , 4998 , 4999 , 5000 , we prove that the optimum cannot possibly be achieved by such regular arrangements. Usually, the irregularities in the best packings found for such n are small, localized modifications to regular patterns; those irregularities are usually easy to predict. Yet for some such irregular n , the best packings found show substantial, extended irregularities which we did not anticipate. In the range we explored carefully, the optimal packings were substantially irregular only for n of the form n = k ( k + 1 ) + 1 , k = 3 , 4 , 5 , 6 , 7 , i.e. for n = 13 , 21 , 31 , 43 , and 57. Also, we prove that the height-to-width ratio of rectangles of minimum perimeter containing packings of n congruent circles tends to 1 as n → ∞ .