Potential minimizing configurations of points on the sphere

Authors:

Ann Davis
Scott Malloy
Michael Neubauer
Mark Schilling
William Watkins
Joel Zeitlin
Department of Mathematics
California State University Northridge
Northridge, CA 91330-8313

"The purpose of computation is insight, not numbers" Richard Hamming

Consider n equally charged particles confined to the surface of the unit sphere in 3-space. They repel each other according to the Coulomb force (the size of which is proportional to the reciprocal of the distance between them squared). The particles move to a configuration that minimizes the total Coulomb potential. Beyond n = 6 no theoretical results are available to determine the potential minimizing configurations. Following in the footsteps of many before us, we have developed an algorithm to find potential minimizing configurations computationally. Often, there are several local minima of the potential energy function. In those cases we list them in increasing order of the potential and label them e.g. 16a, 16b, etc. in order of increasing potential. In fact, n = 16 is the smallest n for which this occurs. It is conjectured that the number of local minima increases exponentially. Since we are only dealing with relatively small values of n our indexing scheme is sufficient.

Scott's page has a real time java applet that let's users play around with electron configurations and produces the results we have listed before. (Caution: Keep the number of electrons small at first to get a feel for the applet.)

Below we provide pictures of the convex hull and cylindrical projections of these minimizing configurations. In the cylindrical projections we show points of degree 5 or less as triangles while the higher degree points are shown as dots. Below for each n and for each potential minimizing configuration we exhibit

For a quick tour of some cases that may spark further interest you may wish to view n=2, 4, 6, 8, 12, 24, 44, 48, 62 and 67.


Click here for a list of the points in MATHEMATICA format.


Copyright by Neubauer, Schilling, Watkins and Zeitlin (1998).