Kissing numbers, sphere packings, and some unexpected proofs

Kissing Numbers, Sphere Packings, and Some Unexpected Proofs, vol. 51, number 8

T he “kissing number problem” asks for the maximal number of blue spheres that can touch a red sphere of the same size in n-dimensional space. The answers in dimensions one, two, and three are classical, but the answers in dimensions eight and twenty-four were a big surprise in 1979, based on an extremely elegant method initiated by Philippe Delsarte in the early seventies which concerns inequa...

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

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...

1 3 M ay 1 99 4 Average kissing numbers for non - congruent sphere packings

(The appearance of the number of the beast in the lower bound is purely coincidental.) The supremal average kissing number k is defined in any dimension, as are kc, the supremal average kissing number for congruent ball packing, and ks, the maximal kissing number for a single ball surrounded by congruent balls with disjoint interiors. (Clearly, kc ≤ k and kc ≤ ks.) It is interesting that k is a...