ON CONTACT NUMBERS OF LOCALLY SEPARABLE UNIT SPHERE PACKINGS

Contact graphs of unit sphere packings revisited

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 ...

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 ...

On Mixed Sphere Packings

Balls of two diierent radii r 1 < r 2 can trivially be packed in E 3 with density better than the classical packing density 3 L = 0:74048. .. , if r 1 =r 2 0:41::. We show that this even holds for r 1 =r 2 = 0:623. .. Further we show, under strong restrictions, that r 1 =r 2 close to 1 does not yield a density > 3 L. These results correspond to similar results on circle packings in the plane

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...

Average Kissing Numbers for Non-congruent Sphere Packings