Equilibrium Configurations of Equal Charges on a Sphere 2025, 39 - Geometric Properties
The spherical boundary conditions and the O(4) symmetry of the Coulomb interaction, the exponential growth of the multiplicity of solutions M(N) shows that the restrictions are principally compatible with many geometries. Unfortunately, most systems can't accommodate strict polyhedric solutions. Archimedean polyhedra have dual forms by joining a point above the center of each face of the polyhedron to equivalent points above all the neighboring faces. The connecting lines are constraint to intersecting the edges of the original polyhedron. The resulting duals of semi-regular polyhedra have congruent faces none of which are regular polygons.
The dipole Moment of the surface Coulomb states is elemental to determining the irregularity of configurations for large N, and whether the equivalent random networks of points on a sphere, which approach universal asymptotic statistical distribution independent of the pairwise repulsions. The average value of the dipole moment of a random configuration of N unit changes on a sphere is an increasing function of N. Systematic variations of the dipole moments exist depending on the strength of the force acting on the charges. Another measure of regularity is the angular diversity ratio, based in the specific locations of two charges on the surface of a sphere with unit radius. The degeneracy of the describing set is a measure of the symmetry of the configuration. The fraction of common angles V(N) comes out to up to 0.9.
When approximating through polyhedra, instead of mapping each point, one can instead summarized three points at a time into one plane of the figure. These planes intersect uniquely with the center of the shape, and assuming a vector k ranging over the positions of all charges not included in some triplet j characterized by another vector c extending from the center and meeting it perpendicularly, then we can generalize that kc ≤ jc, and the plane in question is the face of the polyhedron. The inequality expresses that the spherical cap over the face contain no other charges. These polyhedra are not generally regular. A dominant geometric trend sees increasing irregularity for large particle number, leaving only a kernel of ordered patterns (conjectural, up to simulation capabilities).
The Tammes problem is equivalent to finding the largest angular diameter of N congruent caps packed on the surface of a sphere without overlap. There is an analogous angle for the surface Coulomb problem through the minimum angular separation between neighboring charges in a locally stable configuration. Both are optimization problems, the one in the surface Coulomb problem derived from the total energy, the one in the Tammes problem derived from the nearest-neighbor separations. The resulting angle is a non-monotonic, decreasing function of N with an asymptotic estimate of the Tammes result. It can be viewed analogously, searching for a regular pattern for large N. Complementary to the Coulomb angle, the hole angle ΘH(N) is the angular diameter of the largest spherical cap containing no charges. The set of these regions aid in the construction of the Coulomb polyhedra. For regular configurations the ratio between hole and Coulomb angles is close to 1.