Just to add to this, Wikipedia reminds me that there are two distinct lattice packings that are special cases of the 2^aleph_0 packings that Adam refers to below: the cubic close-packing and the hexagonal close-packing: < http://en.wikipedia.org/wiki/Sphere_packing >. These are defined, respectively, by whether the alternation of the layers Adam mentions is in the pattern a b c a b c ... or a b a b a b .... Also, minor nitpicking point: the enumeration of the distinct not-necessarily-lattice ways the layers can be alternated should involve the equivalence of any shifted sequence of the 3 types of layers, and also possible equivalence by rotations between two arrangements. (Finally, the optimality of a sphere packing is determined by the limit of its density in a ball B(R) of radius R, as R -> oo, when this limit exists, which is independent of the center of the ball. This means that any modification of a packing in a bounded region will have no effect on the limiting density. --Dan On 2013-06-21, at 6:59 AM, Adam P. Goucher wrote:
Both the hexagonal and square lattices of spheres, when laminated, yield the FCC lattice, which is the unique optimal lattice packing of spheres. I think that it was recently proved that it is optimal amongst all packings, although there are 2^aleph-null optimal packings.
(Proof: In the hexagonal lattice, there are two sets of `deep holes' in which we can rest another layer. Hence, a countably infinite string over the alphabet {1,2} can be encoded in an optimal lattice packing; there are 2^aleph-null such possibilities. Consequently, it is a lower bound on the number of optimal packings. Also, it is an upper bound, since it is the total number of arrangements of countably many spheres.)