Editing Close-packing of equal spheres (section)

{{Short description|Dense arrangement of congruent spheres in an infinite, regular arrangement}}
[[File:Close packing box.svg|thumb|Illustration of the close-packing of equal spheres in both HCP (left) and FCC (right) lattices]]
In [[geometry]], '''close-packing of equal [[sphere]]s''' is a dense arrangement of congruent spheres in an infinite, regular arrangement (or [[Lattice (group)|lattice]]). [[Carl Friedrich Gauss]] proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a [[Lattice (group)|lattice]] packing is
:<math>\frac{\pi}{3\sqrt 2} \approx 0.74048</math>.
The same [[packing density]] can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction.  The [[Kepler conjecture]] states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by [[Thomas Callister Hales|Thomas Hales]].<ref>{{cite arXiv |last=Hales |first=T. C. |author-link=Thomas Callister Hales |eprint=math/9811071v2 |title=An overview of the Kepler conjecture |year=1998}}</ref><ref>{{cite journal |title=Mathematics: Does the proof stack up? | volume=424|issue=6944 |doi=10.1038/424012a |journal=Nature |pages=12–13|bibcode=2003Natur.424...12S|last1=Szpiro |first1=George |year=2003 | pmid=12840727|doi-access=free }}</ref> Highest density is known only for 1, 2, 3, 8, and 24 dimensions.<ref>{{cite journal |first1=H. |last1=Cohn |first2=A. |last2=Kumar |first3=S. D. |last3=Miller |first4=D. |last4=Radchenko |first5=M. |last5=Viazovska |title=The sphere packing problem in dimension 24 |journal=Annals of Mathematics |volume=185 |issue=3 |year=2017 |pages=1017–1033 |doi=10.4007/annals.2017.185.3.8 |arxiv=1603.06518|s2cid=119281758 }}</ref>

Many [[crystal]] structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one another in energy, and it may be difficult to predict which form will be preferred from first principles.
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