The cubic sphere packing has centers of spheres at all points (i,j,k) where i, j, and k are integers. The centers are the integer lattice. The radius of each sphere is 1/2.
The face centered cubic packing has centers at each point on the lattice generated by the three vectors (0,1,1), (1,0,1), and (1,1,0). The fcc lattice is the set of points in the integer lattice whose coordinates sum to an even integer. Points in the fcc lattice are a distance of square root of 2 from the neighboring point, so we take spheres whose radius is half this distance.
Packing densities
The cubic lattice sphere packing can be tiled by cubes containing one sphere. The fcc lattice sphere packing can be tiled by cubes containing a total of four spheres, some cut into quarter spheres, as in the following figure.
Now we can compute the densities of these two packings.