New Global Minima for Thomson's Problem of Charges on a Sphere

Using numerical arguments we find that for $N$ = 306 a tetrahedral configuration ($T_h$) and for N=542 a dihedral configuration ($D_5$) are likely the global energy minimum for Thomson's problem of minimizing the energy of $N$ unit charges on the surface of a unit conducting sphere. These would be the largest $N$ by far, outside of the icosadeltahedral series, for which a global minimum for Thomson's problem is known. We also note that the current theoretical understanding of Thomson's problem does not rule out a symmetric configuration as the global minima for N=306 and 542. We explicitly find that analogues of the tetrahedral and dihedral configurations for $N$ larger than 306 and 542, respectively, are not global minima, thus helping to confirm the theory of Dodgson and Moore (Phys. Rev. B 55, 3816 (1997)) that as $N$ grows dislocation defects can lower the lattice strain of symmetric configurations and concomitantly the energy. As well, making explicit previous work by ourselves and others, for $N<1000$ we give a full accounting of icosadeltahedral configuration which are not global minima and those which appear to be, and discuss how this listing and our results for the tetahedral and dihedral configurations may be used to refine theoretical understanding of Thomson's problem.Comment: 1- Manuscript revised. 2- A new global minimum found for a dihedral (D_5) configuration found for N=54

A Novel Symmetric Four Dimensional Polytope Found Using Optimization Strategies Inspired by Thomson's Problem of Charges on a Sphere

Inspired by, and using methods of optimization derived from classical three dimensional electrostatics, we note a novel beautiful symmetric four dimensional polytope we have found with 80 vertices. We also describe how the method used to find this symmetric polytope, and related methods can potentially be used to find good examples for the kissing and packing problems in D dimensions