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 (120 K) here:

Joint paper (pre-print) with D.A. Kottwitz.