There are three types of cubic lattices corresponding to three types of cubic close packing, as summarized in the following table. Now
that the Kepler conjecture has been established,
hexagonal close packing and face-centered
cubic close packing, both of which have packing density
of 
,
are known to be the densest possible packings of equal spheres.
| lattice type | basis vectors | packing density |
| simple cubic (SC) |  ,  ,  |  |
| face-centered cubic (FCC) |  ,  ,  |  |
| body-centered cubic (BCC) |  ,  ,  |  |
Simple cubic packing consists of placing spheres centered on integer coordinates in Cartesian space.
Arranging layers of close-packed spheres such that the spheres of every third layer overlay one another gives face-centered cubic packing. To see where the name comes from, consider packing six spheres together in the shape of an equilateral triangle and place another sphere on top to create a triangular pyramid. Now create another such grouping of seven spheres and place the two pyramids together facing in opposite directions.
Connecting the centers of eight of the spheres, a cube emerges (Steinhaus 1999, pp. 203-204) in which the centers of the other six spheres lie at the centers of the faces of the cube. Connecting the centers of these 14 spheres gives a stella octangula, illustrated above.
|
|
Consider the cube defined by 14 spheres in face-centered cubic packing. This "unit cell," one face of which is illustrated above
in schematic form, contains eight 
-spheres (one at each polygon
vertex) and six hemispheres. The total volume
of spheres in the unit cell is therefore
 |  |  |
(1)
|
 |  |  |
(2)
|
The diagonal of a face of the unit cell is 
, so each side is of length 
. The volume of the unit
cell is therefore
 |
(3)
|
giving a packing density 
of
 |  |  |
(4)
|
 |  |  |
(5)
|
(OEIS A093825; Conway and Sloane 1993, p. 2).
|
|
In face-centered cubic packing, each sphere is surrounded by 12 other spheres. Taking a collection of 13 such spheres gives the cluster illustrated above. Connecting the centers of the external 12 spheres gives a cuboctahedron (Steinhaus 1999, pp. 203-205; Wells 1986, p. 237).
In cubic body-centered packing, each sphere is surrounded by eight other spheres. The figure above shoes the unit cell in body-centered cubic packing. In this configuration,
a single full sphere occupies the center and is surrounded by eight 
-spheres. The total volume of
spheres in the unit cell is therefore
 |  |  |
(6)
|
 |  |  |
(7)
|
The space diagonal of the unit cell is 
, so each side is of length 
. The volume of the unit
cell is therefore
 |
(8)
|
giving a packing density 
of
 |  |  |
(9)
|
 |  |  |
(10)
|
(OEIS A268508).
|
|
If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand uniformly until running into each other, they form
cubes, hexagonal prisms, and rhombic dodecahedra, respectively. In particular, if
the spheres of face-centered cubic packing are expanded until they fill up the gaps,
they form a solid rhombic dodecahedron (left
figure above), and if the spheres of hexagonal
close packing are expanded, they form a second irregular dodecahedron consisting
of six rhombi and six trapezoids (right figure above; Steinhaus 1999, p. 206)
known as the trapezo-rhombic dodecahedron.
The latter can be obtained from the former by slicing in half and rotating the two
halves 
with respect to each other. The lengths of the short and long edges of the rotated
dodecahedron have lengths 2/3 and 4/3 times the length of the rhombic faces. Both
the rhombic dodecahedron and trapezo-rhombic
dodecahedron are space-filling polyhedra.
