Packing at Random in Curved Space and Frustration: a Numerical Study

Random packings of discs on the sphere S2 and spheres on the (hyper-)sphere S3 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 S2 converge continuously, but very slowly, to the regular triangular lattice, the sphere packings on S3 converge to the disordered frustrated Bernal's packing, of volume fraction c≃0.645, in the (N=∞) flat space limit. In the S3 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.