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Dense packings of congruent circles in a circle. (English) Zbl 0901.52017
Two methods based on the maximization of the minimum pairwise distance $$d$$ among $$n$$ points spread in a circle are proposed to generate packings of congruent circles in the circle. To overcome the difficulty caused by the nonsmoothness of that objective function, in the first method it is approximated by a smooth one, which can be regarded as a potential energy relative to repulsion forces between the points. The second method simulates the movement of the circles as billiard balls inside a circle enlarged so as to force all centers to belong to the original circle; as disks move, $$d$$ is increased and a packing is obtained in the limit.
Pictures showing the best known packings of 21-65 circles, as well as a table listing the properties of the best packings of 2-65 circles, are provided.

##### MSC:
 52C15 Packing and covering in $$2$$ dimensions (aspects of discrete geometry) 05B40 Combinatorial aspects of packing and covering 90C99 Mathematical programming
##### Keywords:
packings; congruent circles
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