Editing Close-packing of equal spheres (section)

=== Positioning and spacing ===
In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres ([[octahedron|octahedral]]) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is {{sqrt|{{frac|3|2}}}} for the tetrahedral, and {{sqrt|2}} for the octahedral, when the sphere radius is 1.

Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius.

The most regular ones are
*FCC = ABC ABC ABC... (every third layer is the same)
*HCP = AB AB AB AB... (every other layer is the same).

There is an uncountably infinite number of disordered arrangements of planes (e.g. ABCACBABABAC...) that are sometimes collectively referred to as "Barlow packings", after crystallographer [[William Barlow (geologist)|William Barlow]].<ref>{{cite journal|author=Barlow, William|title=Probable Nature of the Internal Symmetry of Crystals|journal=Nature|year=1883|volume=29|issue=738|pages=186–188|doi=10.1038/029186a0|bibcode=1883Natur..29..186B|url=https://zenodo.org/record/1429283|doi-access=free}}</ref>

In close-packing, the center-to-center spacing of spheres in the ''xy'' plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the ''z'' (vertical) axis, is:

:<math>\text{pitch}_Z = \sqrt{6} \cdot {d\over 3}\approx0.816\,496\,58 d,</math>

where ''d'' is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres.

The [[coordination number]] of HCP and FCC is 12 and their [[atomic packing factor]]s (APFs) are equal to the number mentioned above, 0.74.[[File:Cubic_Closest_Packing_(CCP)_and_Hexagonal_Closet_Packing_(HCP).png|763x763px|Cubic Closest Packing (CCP) and Hexagonal Closet Packing (HCP)]]

{| border="0" cellpadding="10px" style="border:1px solid gray;"
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!Comparison between HCP and FCC
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| [[Image:close packing.svg|100000x250px]]
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| '''Figure 1''' – The HCP lattice (left) and the FCC lattice (right). The outline of each respective [[Bravais lattice]] is shown in red. The letters indicate which layers are the same. There are two "A" layers in the HCP matrix, where all the spheres are in the same position. All three layers in the FCC stack are different. Note the FCC stacking may be converted to the HCP stacking by translation of the upper-most sphere, as shown by the dashed outline.
|}

[[Image:Close-packed spheres.jpg|thumb|right|341px|'''Figure&nbsp;2'''&nbsp; [[Thomas Harriot]] in ca. 1585 first pondered the mathematics of the ''cannonball arrangement'' or ''cannonball stack,'' which has an FCC lattice. Note how the two balls facing the viewer in the second tier from the top contact the same ball in the tier below. This does not occur in an HCP lattice (the left organization in ''Figure&nbsp;1'' above, and ''Figure&nbsp;4'' below).]]

[[Image:Cannonball stack with FCC unit cell.jpg|thumb|right|341px|'''Figure 3'''&nbsp; Shown here is a modified form of the cannonball stack wherein three extra spheres have been added to show all eight spheres in the top three tiers of the FCC lattice diagramed in ''Figure&nbsp;1''.]]

[[Image:Hexagonal close-packed unit cell.jpg|thumb|right|341px|'''Figure 4'''&nbsp; Shown here are all eleven spheres of the HCP lattice illustrated in ''Figure&nbsp;1''. The difference between this stack and the top three tiers of the cannonball stack all occurs in the bottom tier, which is rotated half the pitch diameter of a sphere (60°). Note how the two balls facing the viewer in the second tier from the top do not contact the same ball in the tier below.]]

[[Image:Pyramid_of_35_spheres_animation_original.gif|thumb|right|341px|'''Figure 5'''&nbsp; This animated view helps illustrate the three-sided pyramidal ([[Tetrahedron|tetrahedral]]) shape of the cannonball arrangement.]]