Circle packing

[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.

The most efficient way to pack different-sized circles together is not obvious.
Identical circles in a hexagonal packing arrangement, the densest packing possible
Hexagonal packing through natural arrangement of equal circles with transitions to an irregular arrangement of unequal circles
Fifteen equal circles packed within the smallest possible square . Only four equilateral triangles are formed by adjacent circles.
A compact binary circle packing with the most similarly sized circles possible. [ 7 ] It is also the densest possible packing of discs with this size ratio (ratio of 0.6375559772 with packing fraction (area density) of 0.910683). [ 8 ]