2012/08/24 by Alexander Tumanov, Tumanov, Alexander · 1 citation
Materials Science · Mathematics · Physics and Astronomy · #52A40 #52C35 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Graphite, nuclear technology, radiation studies #Mathematical Approximation and Integration #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.1208.5044
openalex publication_date 2012/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.