Improving Dense Packings of Equal Disks in a Square
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.