Editing Close-packing of equal spheres (section)

==FCC and HCP lattices==
[[File:Square circle grid spheres.png|120px|thumb|left|FCC arrangement seen on 4-fold axis direction]]
{| class=wikitable align=right width=360
!colspan=2|FCC
!HCP
|-
|[[File:Cuboctahedron B2 planes.png|120px]]||[[File:Cuboctahedron 3 planes.png|120px]]
|[[File:Triangular orthobicupola wireframe.png|120px]]
|-
|colspan=3|The ''FCC'' arrangement can be oriented in two different planes, square or triangular. These can be seen in the [[cuboctahedron]] with 12 vertices representing the positions of 12 neighboring spheres around one central sphere. The ''HCP'' arrangement can be seen in the triangular orientation, but alternates two positions of spheres, in a [[triangular orthobicupola]] arrangement.
|}
There are two simple regular lattices that achieve this highest average density. They are called '''face-centered cubic''' ('''FCC''') (also called '''[[cubic crystal system|cubic]] close packed''') and '''[[Hexagonal crystal system|hexagonal]] close-packed''' ('''HCP'''), based on their [[symmetry]]. Both are based upon sheets of spheres arranged at the vertices of a triangular tiling; they differ in how the sheets are stacked upon one another.  The FCC lattice is also known to mathematicians as that generated by the A<sub>3</sub> [[root system]].<ref>{{cite book |author-link1=John Horton Conway |last1=Conway |first1=John Horton |author-link2=Neil Sloane |last2=Sloane |first2=Neil James Alexander |last3=Bannai |first3=Eiichi |title=Sphere packings, lattices, and groups |url=https://archive.org/details/spherepackingsla0000conw_b8u0 |url-access=registration |publisher=Springer |year=1999 |isbn=9780387985855 |at=Section 6.3}}</ref>

=== Cannonball problem ===
{{Main|Cannonball problem}}
[[Image:Fortres Monroe 1861 - Cannon-balls.jpg|thumb|Cannonballs piled on a triangular ''(front)'' and rectangular ''(back)'' base, both [[Face-centered cubic|FCC]] lattices.]]
The problem of close-packing of spheres was first mathematically analyzed by [[Thomas Harriot]] around 1587, after a question on piling cannonballs on ships was posed to him by Sir [[Walter Raleigh]] on their expedition to America.<ref>{{cite encyclopedia |title=Cannonball Problem |encyclopedia=The Internet Encyclopedia of Science |first=David |last=Darling |url=http://www.daviddarling.info/encyclopedia/C/Cannonball_Problem.html }}</ref> 
Cannonballs were usually piled in a rectangular or triangular wooden frame, forming a three-sided or four-sided pyramid. Both arrangements produce a face-centered cubic lattice – with different orientation to the ground.  Hexagonal close-packing would result in a six-sided pyramid with a hexagonal base. 
[[File:Pyramid of Snowballs Large.png|thumb|Collections of [[snowball]]s arranged in pyramid shape. The front pyramid is hexagonal close-packed and rear is face-centered cubic. ]]

The [[cannonball problem]] asks which flat square arrangements of cannonballs can be stacked into a square pyramid. [[Édouard Lucas]] formulated the problem as the [[Diophantine equation]] <math>\sum_{n=1}^{N} n^2 = M^2</math> or <math>\frac{1}{6} N(N+1)(2N+1) = M^2</math> and conjectured that the only solutions are <math>N = 1, M = 1,</math> and <math>N = 24, M = 70</math>. Here <math>N</math> is the number of layers in the pyramidal stacking arrangement and <math>M</math> is the number of cannonballs along an edge in the flat square arrangement.
<!-- TO DO: Who first proposed the hcp arrangement? -->

=== 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
|- style="text-align:center;"
| [[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.]]