Packing at random in curved space and frustration: A numerical study

Random packings of discs on the sphere S-2 and spheres on the (hyper-)sphere S-3 have been built on a computer using an extension of the Jodrey-Tory algorithm Structural quantities such as the volume fraction, the pair correlation function and some mean characteristics of the Voronoi cells have been calculated for various packings containing up to N = 8192 units. While the disc packings on S-2 converge continuously, but very slowly, to the regular triangular lattice, the sphere packings on S-3 converge to the disordered frustrated Bernal's packing, of volume fraction c similar or equal to 0.645, in the (N = infinity) flat space limit. In the S-3 case, the volume fraction exhibits maxima for particular values of N, for which the corresponding packings have a narrower histogram for the number of edges of Voronoi polyhedra faces.