SPHERE PACKING STUDIES
The fcc packing is easily described in terms of adding successive layers of spheres to a tetrahedron. Canon balls and fruits in the grocery store are typically stacked in this fashion. A single orange nestles in the "valley" formed by three below it. A triangular layer of six oranges underlies that one and so on. Each layer adds a certain number of oranges, which may be expressed as a function of the growing tetrahedral shape's 'frequency'. The frequency is equivalent to the number of intervals between oranges along the tetrahedron's edge. * * * * * * * * * * * * * * * * * * * * Fig. 1. Triville Packing (or Pool Ball) Kepler studied sphere packing pretty intensely and knew that you get the same fcc packing if you start with a layer of spheres packed in a square arrangement and nest the next layer in the valleys so formed. If you taper off as you go upwards, this looks kind of like a Mayan Temple, so I call it "Mayan temple packing". * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Fig. 2. Squaresville Packing (or Mayan Temple) As Jim Morrissett pointed out to me during an IRC chat one morning, the Mayan temple packing forces the fcc, whereas the pool ball packing does not. This is because a "squaresville" layer presents only one set of valleys for the next layer of spheres, whereas a "triville" layer presents twice as many valleys as we will find usable  presents an alternative, allowing us to go for hcp instead of fcc or some other Barlow packing. The table below has a slightly different focus from Fuller's in Synergetics in Synergetics Principles ( section 220.00). Fuller's 2nd power derivations were of the form 2 p ff + 2 where 2 p is the number of nonpolar vertices (V2). This expresses the number of spheres in the outer layer of a shape as a function of frequency. Below, the focus is on the number of spheres added as a shape grows in size, versus the number of spheres exposed on a surface. For example, a tetrahedron starts with a nuclear sphere, and then an expanding base of 3, 6, 10... new spheres per layer. 

The halfoctahedron is like an Egyptian pyramid in shape. It's base consists of spheres arranged in a square nbyn pattern. Each layer above the base is one sphere less along each edge. In other words, a fivelayer halfocta consists of 25+9+4+1 spheres, or 39 total. Since the number of layers is one more than the frequency, our equations start with one for F=0. When F increases to one, four spheres are added, giving a total of five spheres etc. 

All of the packings described above are equivalent to the facecentered cubic. Connecting the centers of adjacent spheres with rods, and allowing the spheres to fade from view, is what gives us the isotropic vector matrix in synergetics. 
See A Study in Sphere Packing for a more computational approach to sphere packing in a cuboctahedral conformation (IVMstyle). Relevant readings:

Synergetics on the
Web
maintained by Kirby Urner
12around1 graphic by Richard Hawkins [rh] using
Alias Animator V5.1 on an SGI Indigo2 Extreme
chardhawk@nets.com
Thanks to Kevin Brown for email re Bernoulli,
and to Dr. John Conway
for a lot of historical information and uptodate nomenclature.