Average kissing numbers for non-congruent sphere packings

Average Kissing Numbers for Non-congruent Sphere Packings

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...

Contact Numbers for Congruent Sphere Packings in Euclidean 3-Space

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit ...

Kissing numbers, sphere packings, and some unexpected proofs

The “kissing number problem” asks for the maximal number of white spheres that can touch a black 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 ineq...

Packings with large minimum kissing numbers

For each proper power of 4, n, we describe a simple explicit construction of a finite collection of pairwise disjoint open unit balls in R in which each ball touches more than 2 √ n others. A packing of balls in the Euclidean space is a finite or infinite collection of pairwise disjoint open unit balls in Rn. It is called a lattice packing if the centers of the balls form a lattice in Rn. The m...