Multiplicity Greater Than 1

Suppose n equal non-overlapping circles are placed on a (unit) sphere.  Such a packing is best (or optimal) if no other packing of n non-overlapping  equal circles on the unit sphere has circles of larger radius.

Rigorously proofed best packings only exist for n < 15 and for n = 24.  Extremely good packings, often presumed best, exist for many other values of nIn the vast majority of cases the conjecturally best packings are unique up to a rigid motion of the sphere, or an inversion in the center followed by a rigid motion.  (I.e. up to the action of orthogonal matrices).

In rare situations it may happen that two different best packings of n circles have exactly the same minimal edge length.  So far, the only know cases of the this "multiplicity greater than 1" occur for n = 15 (found by D.A. Kottwitz in 1990) and for n = 62, 76, and 117, found by D.A. Kottwitz and Jim Buddenhagen (myself).  We have submitted a paper discussing this.

All cases where this is known to happen are associated with certain hexagons that occur which we call "toggle hexagons".  Briefly, these are hexagons whose edges are minimal in the packing but which admit more than one "central point" at the same minimal distance from 3 vertices of the hexagon..  This central point we call a "toggle point".  In one position the packing is highly symmetric.  In the other, the symmetry is broken.  A new packing is formed, with less symmetry but exactly the same minimal edge.

In two cases, n = 41 and n = 54, toggle hexagons exist where an equivalent (not different) packing occurs when the toggle is switched.  We know of no other instances of this.

High precision data for the circle centers (projected to the sphere) are given here. Each is a text file with x, y, z coordinates of vertices.  There is one coordinate per record.  Thus 3 records for vertex 1, then 3 more records for vertex 2 etc.

Toggles with no increase in multiplicity:

Toggles with increase in multiplicity: