On the problem of the densest packing of spherical segments into a sphere

Abstract

The paper considers a particular variant of the classical optimal packing problem when the container is a sphere, the packed elements are equal spherical caps, and the optimality criterion is to maximize their geodesic radius. At the same time, we deal with a special integral metric to determine the distance between points, which becomes Euclidean in the simplest case. We propose a heuristic numerical algorithm based on the construction of spherical Voronoi diagrams, which makes it possible to obtain a locally optimal solution to the problem under consideration. Numerical calculations show the operability and effectiveness of the proposed method and allow us to draw some conclusions about the properties of packings.