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.

 is the maximum distance from the origin to a point on the circumference of circle i

 is the distance between the centers of any two circles minus their radii

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,  must be equal to or greater than zero.

Hence the entire set of constraints

Hence the optimized radius of the circumscribing circle is