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{\bf David W. Boll,   
Jerry Donovan , 
Ronald L. Graham and 
Boris D. Lubachevsky }
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{\bf Improving Dense Packings of Equal Disks in a Square}
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We describe a new numerical procedure for generating dense packings of
disks and spheres inside various geometric shapes. We believe that in
some of the smaller cases, these packings are in fact optimal. When
applied to the previously studied cases of packing $n$ equal disks in a
square, the procedure confirms all the previous record packings [NO1]
[NO2] [GL], except for $n =$ 32, 37, 48, and 50 disks, where better
packings than those previously recorded are found. For $n =$ 32 and 48,
the new packings are minor variations of the previous record packings.
However, for $n =$ 37 and 50, the new patterns differ substantially. For
example, they are mirror-symmetric, while the previous record packings
are not.


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