This applet shows polyhedra based on n points on a sphere that correspond to a spherical code. The following optimization criteria were used:

- Packings: maximize minimum distance between any pair of points

- Coverings: minimize the maximum distance of any point on the sphere from the closest one of the n points

- Minimal energy (minimal 1/r potential): minimize the potential energy Sum 1 / dist(P_i, P_j )

- Maximum volume: maximize the volume of the convex hull

- Packings with antipodal constellations (PackAnti)

To run enter a numeric value between 4 and 64 in the "Points?" field and hit enter. The middle tile contains the colors of the first 12 edge lengths, this is to help spotting symmetries. The last tile contains a data report.

To see the difference between the various optimization criteria, set e.g. n=8.

Background on packings: This way of packing circles on a sphere results in a spherical code. The problem of spherical packing is sometimes known as the Fejes Tóth's or Tammes’ problem. There are exact solutions for 4, 6, and 12, but the general problem has not been solved. Best known solutions are available here and they are used in this applet. Exception: PackAnti with 16 points designed by “Magic/Shapeways”, PackAnti with 18 points designed by Alea Kybos.

Sources:

Spherical codes based on N. J. A. Sloane, with the collaboration of R. H. Hardin, W. D. Smith and others, Tables of Spherical Codes, published electronically at www.research.att.com/~njas/

The applet is based on code by Bob Allanson, modified with permission. http://members.ozemail.com.au/~llan/mpol.html