Cubic close packing is the packing of equal touching spheres whose centers form a face-centered cubic
lattice (FCC), also called face-centered cubic packing (Krishna and Pandey).
For comparison, the following table gives the basis vectors
and packing densities for sphere
packings on all three types of cubic lattices.
Simple cubic (SC) and body-centered cubic (BCC) packings are not close-packed. Now
that the Kepler conjecture has been established,
hexagonal close packing and face-centered
cubic packing, both of which have packing density 
, 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 of radius 
with their centers at integer Cartesian coordinates.
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.
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|
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 polyhedron
vertex) and six hemispheres. The total volume
of spheres in the unit cell
is therefore
 |  |  |
(1)
|
 |  |  |
(2)
|
The face diagonal 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 (Wells 1986, p. 237; Steinhaus 1999, pp. 203-205).
Joining neighboring sphere centers in face-centered cubic packing gives the edge skeleton of the tetrahedral-octahedral honeycomb. Its cells are regular tetrahedra and regular octahedra.
Subtracting the packed spheres from these cells partitions the interstitial space into tetrahedral and octahedral voids. For generating spheres
of radius 
, the edges
of the cells have length 
. A tetrahedral void is bounded by four concave spherical triangles and four planar curvilinear
triangles. An octahedral void is bounded by six concave spherical
polygons, each with four sides, and eight planar curvilinear triangles. The planar
boundaries lie on the faces of the tetrahedral-octahedral
honeycomb, where neighboring voids communicate. Kuchel (2026, pp. 7-13)
calls these bodies spherically truncated tetrahedra and spherically truncated octahedra.
Their volumes are
 |  |  |
(6)
|
 |  |  |
(7)
|
Counting each void once gives two tetrahedral voids and one octahedral void per packed sphere, so the total void volume per sphere is
 |
(8)
|
The total volume per sphere is 
, giving the void fraction
 |
(9)
|
In cubic body-centered packing, each sphere is surrounded by eight other spheres. The figure above shows 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
 |  |  |
(10)
|
 |  |  |
(11)
|
The space diagonal of the unit cell is 
, so each side is of length 
. The volume of
the unit cell is therefore
 |
(12)
|
giving a packing density 
of
 |  |  |
(13)
|
 |  |  |
(14)
|
(OEIS A268508).
|
|
The Voronoi diagram of the sphere centers partitions space into cubes for simple cubic packing, rhombic dodecahedra for face-centered cubic packing (left figure above), and trapezo-rhombic dodecahedra for hexagonal close packing (right figure above; Steinhaus 1999, p. 206). Each cell consists of the points at least as close to its sphere center as to any other sphere center.
The trapezo-rhombic dodecahedron consists of six rhombi and six trapezoids.
It can be obtained from the rhombic dodecahedron
by slicing in half and rotating the two halves 
with respect to each other. The lengths
of the short and long edges of the trapezoids
are 2/3 and 4/3 times the length of an edge
of a rhombus. Both the rhombic
dodecahedron and trapezo-rhombic
dodecahedron are space-filling polyhedra.
