The closest packing of equal circles on a sphere

Abstract

The closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom. The technique is based on the concept that the centres of the circles repel one another, and minimizing the total 'repulsion energy' leads to an approximate structure from which exact solutions can be determined. A number of improved packings have been discovered for values of x in the range 20-40. Many different types of structure are found that are of lower symmetries than those previously described. The packing density p, defined as the fraction of the spherical surface that is enclosed by the circles, is found to increase as the number of circles increases, in contrast to the conclusion from previous studies. However, this increase in p is very slight and the values remain substantially below that for an infinite number of circles, or a close-packed plane.