Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

Sphere packings in 3-space

In this paper we survey results on packings of congruent spheres in 3-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 and the combinatorial Kepler problem; Is...

Rigidity of Circle Polyhedra in the 2-sphere and of Hyperideal Polyhedra in Hyperbolic 3-space

We generalize Cauchy’s celebrated theorem on the global rigidity of convex polyhedra in Euclidean 3-space E to the context of circle polyhedra in the 2-sphere S. We prove that any two convex and proper non-unitary c-polyhedra with Möbiuscongruent faces that are consistently oriented are Möbius-congruent. Our result implies the global rigidity of convex inversive distance circle packings in the ...

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 ...

Average Kissing Numbers for Non-congruent Sphere Packings

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...