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Dense packings of congruent circles in rectangles with a variable aspect ratio. (English) Zbl 1077.52511

Aronov, Boris (ed.) et al., Discrete and computational geometry. The Goodman-Pollack Festschrift. Berlin: Springer (ISBN 3-540-00371-1/hbk). Algorithms Comb. 25, 633-650 (2003).
Summary: 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 \(n\leq 5000\), specifically, for \(n=49,61,79,97,107,\dots4999\), 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 \(n\to\infty\), if the limit exists.
For the entire collection see [Zbl 1014.00040].

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
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