[1][2] While the circle has a relatively low maximum packing density, it does not have the lowest possible, even among centrally-symmetric convex shapes: the smoothed octagon has a packing density of about 0.902414, the smallest known for centrally-symmetric convex shapes and conjectured to be the smallest possible.
At the other extreme, Böröczky demonstrated that arbitrarily low density arrangements of rigidly packed circles exist.
A related problem is to determine the lowest-energy arrangement of identically interacting points that are constrained to lie within a given surface.
The Thomson problem deals with the lowest energy distribution of identical electric charges on the surface of a sphere.
Packing circles in simple bounded shapes is a common type of problem in recreational mathematics.
The influence of the container walls is important, and hexagonal packing is generally not optimal for small numbers of circles.
[12] Robert J. Lang has used the mathematics of circle packing to develop computer programs that aid in the design of complex origami figures.