Suppose n equal non overlapping circles are placed on a (unit) sphere. Imagine that the circles expand at an equal rate, sliding as needed to prevent overlapping. Eventually no further expansion is possible and we obtain a packing of the n circles on the sphere. The vast majority of the circles will be locked into position by their neighbors and unable to move. In this sense a packing is rigid.
However, there may be a few circles that are not locked into position but are free to rattle in the small space between them and their nearest neighbors. These are called rattlers. For example, the best known packing of 20 circles on a sphere has 2 rattlers. It was found by van der Waerden in 1952. (Yes, the same van der Waerden whose abstract algebra book is now a classic.)
For a given value of n there may be many different packings. We consider one packing to be better than another if its circles have a larger radius. Two packings are considered the same, if one can be transformed into the other by a rigid motion of the sphere, or by an inversion in the sphere's center followed by a rigid motion. Thus packings are the same if some orthogonal matrix transforms one to the other. Otherwise the packings are different.
Packings are local maxima in the sense that all sufficently small perturbations of the circle centers break the packing; that is, cause the circles to overlap unless they are made smaller. Perturbations which only move the rattlers are excepted. For each n we wish to find the best packing, i.e. the global maximum from among all the local maxima. This problem is equivalent to maximizing the minimum distance between any pair of n points on the sphere. This is also known as Tammes' problem.
Further introductory material. (no link yet)
Picture of the best known packing for n = 150.
Data for the best known packings of circles on a sphere.
High precision data for the best known packings. (no link yet)
Minimal polynomials of best known packings.
High precision data for packings of multiplicty greater than 1.
Pictures, including dynamic ones, for packings of multiplicity greater than 1.
D. A. Kottwitz and I have written a joint paper entitled "Multiplicity and
Symmetry Breaking in (Conjectured) Densest Packings of Congruent Circles on a Sphere".
You can dowload it as a pdf file (123 K) here:
Joint paper (pre-print) with D.A. Kottwitz.
Return to packing types page.