  Consider n points on the unit 2-sphere. The potential of the interaction of
two points is a function f(r) of the distance r between the points. The total
energy E of n points is the sum of the pairwise energies. The question is how
to place the points on the sphere to minimize the energy E. For the Coulomb
potential f(r)=1/r, the problem goes back to Thomson (1904). The results for n
< 5 are well known. We focus on the case n=5, which turns out to be difficult.
In this case, the following results have been obtained. For n=5, Dragnev, Legg,
and Townsend (2002) give a solution of the problem for f(r)=-log r known as
Whyte's problem. Hou and Shao (2009) give a rigorous computer-aided solution
for f(r)=-r. Schwartz (2010) gives a rigorous computer-aided solution of
Thompson's problem. We give a solution for biquadratic potentials.
