Some progress in the packing of equal circles in a square
نویسندگان
چکیده
منابع مشابه
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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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.
متن کاملNew results in the packing of equal circles in a square
The problem of nding the maximum diameter of n equal mutually disjoint circles inside a unit square is addressed in this paper. Exact solutions exist for only n = 1; : : : ; 9; 10; 16; 25; 36 while for other n only conjectural solutions have been reported. In this work a max{min optimization approach is introduced which matches the best reported solutions in the literature for all n 30, yields ...
متن کاملSymmetry Breaking Constraints for the Problem of Packing Equal Circles in a Square
The Packing Equal Circles in a Square (PECS) problem is a nonconvex nonlinear optimization problem which involves a high degree of symmetry. The Branch-and-Bound algorithms work bad due to the presence of symmetric optima, because the Branch-and-Bound tree becomes large, and the time to reach the leaves (i.e., the optimal solutions) increases. In this paper, we introduce some inequalities which...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1990
ISSN: 0012-365X
DOI: 10.1016/0012-365x(90)90135-5