The equilibrium configurations problem essentially asks, how can n points be uniformly distributed on a sphere? Theory has not been able to predict where the n points should go simply as a function of n. Instead we have had to imagine that the points are equally-charged electrons, repelling each other yet constrained to remain on the surface of the sphere. Watching where the electrons push each other until they reach an equilibrium state is one way to study the problem.
The applet above initially shows four randomly-placed electrons--the user can spin the sphere by dragging the mouse. Each point bears an electrostatic potential, called the Coulomb potential, which is higher if it has points close to it. Hitting the "Step" button shows one iteration of mutual repulsion among the electrons. Hitting the "Run" button shows repeated iterations of the electron perturbation until the difference in the system's potential from one iteration to the next (epsilon) falls below the threshold (the default threshold is 1/10^4, but this can be changed by the user).
The user can change the number of electrons displayed (hit "return" on the keyboard after editing the "Number of Electrons" text field), run the simulation, and see the interesting configurations which result. The user can also choose whether the electrons are initially randomly placed or are instead placed in banded configurations, choosing which intial configuration is desired. "2 mod 3", for example, will put two electrons at the poles and the other electrons in bands of three. Starting with banded configurations can help the user discern order in the final configuration.
This applet was written by Scott Malloy; it was modified from the XYZApp applet by James Gosling of Sun Microsystems which displays models of molecules. To learn more about the equilibrium configurations problem see http://www.csun.edu/~hcmth007/points.html.