Finite and Uniform Stability of Sphere Packings

Sphere packings revisited

In this paper we survey most of the recent and often surprising results on packings of congruent spheres in d-dimensional spaces of constant curvature. The topics discussed are as follows: Hadwiger numbers of convex bodies and kissing numbers of spheres; Touching numbers of convex bodies; Newton numbers of convex bodies; One-sided Hadwiger and kissing numbers; Contact graphs of finite packings ...

Maximizing the Packing Density on a Class of Almost Periodic Sphere Packings

We consider the variational problem of maximizing the packing density on some finite dimensional set of almost periodic sphere packings. We show that the maximal density on this manifold is obtained by periodic packings. Since the density is a continuous, but a nondifferentiable function on this manifold, the variational problem is related to number theoretical questions. Every sphere packing i...

Finite Sphere Packings and Sphere Coverings

Deriving Finite Sphere Packings

Sphere packing problems have a rich history in both mathematics and physics; yet, relatively few analytical analyses of sphere packings exist, and answers to seemingly simple questions are unknown. Here, we present an analytical method for deriving all packings of n spheres in R3 satisfying minimal rigidity constraints (≥ 3 contacts per sphere and ≥ 3n− 6 total contacts). We derive such packing...

Disks vs. Spheres: Contrasting Properties of Random Packings

Collections of random packings of rigid disks and spheres have been generated by computer using a previously described concurrent algorithm. Particles begin as infinitesimal moving points, grow in size at a uniform rate, undergo energynonconserving collisions, and eventually jam up. Periodic boundary conditions apply, and various numbers of particles have been considered (N~<2000 for disks, N~<...