Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles

Details Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles

2004-12-22

arxiv.org/abs/math/0412443v4

Discrete Mathematics, vol.309 (28 April 2009), 1947-1962

Mathematics

Metric Geometry

existence of irregular minimum perimeter packings for n not of the form (10) is conjectured; smallest such n is n=66; existenc

Scientific paper

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 tends to infinity.

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