A better packing of ten equal circles in a square

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A better packing of ten equal circles in a square

Let S be a square of side s in the Euclidean plane. A pucking of circles in S is nothing else but a finite family of circular disks included in S whose interiors are pairwise disjoints. A natural problem related with such packings is the description of the densest ones; in particular, what is the greatest value of the common radius r of n circles forming a packing of S? Clearly, the centres of ...

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The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 [Bol04]) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of vacuitas (in Latin, the gap in the triangle outside the circles). It is interesting that these packings are not always optimal in s...

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Packing Equal Circles in a Square: I. Solution Properties

In this paper some properties of optimal solutions for the problem of packing n equal circles into the unit square will be derived. In particular, properties, which must be satissed by at least one optimal solution of the problem and stating the intuitive fact that as many circles as possible should touch the boundary of the unit square, will be introduced.

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Packing equal circles in a square: a deterministic global optimization approach

In this paper the problem of packing n equal circles into the unit square will be considered. Starting from a general rectangular branch-and-bound algorithm, many tools, which exploit the special structure of the problem, will be introduced and discussed. Computational results will be presented and, in particular, the optimality within a given tolerance of best known solutions in the literature...

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New results in the packing of equal circles in a square

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 1989

ISSN: 0012-365X

DOI: 10.1016/0012-365x(89)90288-4