This application finds the best packing of unequal non-overlapping disks in a larger circle, such that the radius of the container is minimized. This is a difficult global optimization problem that demands strong solvers; this application uses Maple's Global Optimization Toolbox. You must have the Global Optimization Toolbox installed to use this application

One solution for the packing of 50 disks with the radii 1 to 50 (as found by this application) is visualized below. Other solutions are documented at http://www.packomania.com.

Packing optimization is industrially important, with applications in pallet loading, the arrangement of fiber optic cables in a tube, or the placing of blocks on a circuit board.

Number of circles

Radius of circle n is equal to n

The decision variables are the coordinates () of the centers of the circles, and the radius rc of the circumscribing circle.

The maximum distance between the furthest point on a circle's circumference and the origin must be smaller than the radius of the circumscribing circle.

For circles i and j not to overlap, distance between the centers of any two circles minus their radii must be greater than zero

Hence the entire set of constraints

Hence the optimized radius of the circumscribing circle is