Local Growth in an Array of Disks
Start with a closely packed array of identical disks. Then make each disk grow at a certain rate, attempting to maintain the packing. If the growth rate varies with the position of the disk, a packing can sometimes be maintained, and sometimes not. This setup provides a good model for processes in biology and elsewhere in which there is local isotropic growth at different rates in different positions. Failure to maintain the packing shown here would indicate that the real system—say a growing plant leaf—would somehow buckle into 3D.
The condition for the packing to be maintained is that the growth rate function satisfies Laplace's equation.