Towards Uniform Online Spherical Tessellations

Abstract:

We consider an online variant of the famous Thomson problem of uniformly placing a set of n points upon the sphere as uniformly as possible. Thomson’s motivation for the problem was to minimize the electrostatic potential of a set of n electrons inside an atom, but we describe several other real life motivation of the problem and especially the online variant of it. We improve the upper bound of 3.69 shown by Chen et al. to 2.84 via a recursive tessellation algorithm applied to a regular circumscribed icosahedron. The analysis uses spherical trigonometry and elementary differential geometry. We also show a lower bound of 1.618 which holds even if we have just three points and conclude with some open problems.