Editing Sphere packing (section)

===Dense packing===
{{main|Close-packing of equal spheres}}
In three-dimensional Euclidean space, the densest packing of equal spheres is achieved by a family of structures called [[Close-packing of spheres|close-packed]] structures. One method for generating such a structure is as follows. Consider a plane with a compact arrangement of spheres on it. Call it A. For any three neighbouring spheres, a fourth sphere can be placed on top in the hollow between the three bottom spheres. If we do this for half of the holes in a second plane above the first, we create a new compact layer. There are two possible choices for doing this, call them B and C. Suppose that we chose B. Then one half of the hollows of B lies above the centers of the balls in A and one half lies above the hollows of A which were not used for B. Thus the balls of a third layer can be placed either directly above the balls of the first one, yielding a layer of type A, or above the holes of the first layer which were not occupied by the second layer, yielding a layer of type C. Combining layers of types A, B, and C produces various close-packed structures.

Two simple arrangements within the close-packed family correspond to regular lattices. One is called cubic close packing (or [[Cubic crystal system|face-centred cubic]], "FCC")—where the layers are alternated in the ABCABC... sequence. The other is called [[Hexagonal crystal family |hexagonal close packing]] ("HCP"), where the layers are alternated in the ABAB... sequence.{{Dubious|FCC and HCP are not both lattices?|date=April 2022}} But many layer stacking sequences are possible (ABAC, ABCBA, ABCBAC, etc.), and still generate a close-packed structure. In all of these arrangements each sphere touches 12 neighboring spheres,<ref name="grancrys">{{Cite journal |last1=Dai |first1=Weijing |last2=Reimann |first2=Joerg |last3=Hanaor |first3=Dorian |last4=Ferrero |first4=Claudio |last5=Gan |first5=Yixiang |date=2019-03-13 |title=Modes of wall induced granular crystallisation in vibrational packing |url=https://hal.science/hal-02355176/document/#page=6 |journal=Granular Matter |language=en |volume=21 |issue=2 |pages=26 |doi=10.1007/s10035-019-0876-8 |arxiv=1805.07865 |s2cid=254106945 |issn=1434-7636}}</ref> and the average density is
:<math>\frac{\pi}{3\sqrt{2}} \approx 0.74048.</math>

In 1611, [[Johannes Kepler]] conjectured that this is the maximum possible density amongst both regular and irregular arrangements—this became known as the [[Kepler conjecture]]. [[Carl Friedrich Gauss]] proved in 1831 that these packings have the highest density amongst all possible lattice packings.<ref>{{cite journal|first=C. F.|last=Gauß|author-link=Carl Friedrich Gauss|title=Besprechung des Buchs von L. A. Seeber: ''Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen'' usw|trans-title=Discussion of L. A. Seeber's book: ''Studies on the characteristics of positive ternary quadratic forms'' etc|journal=Göttingsche Gelehrte Anzeigen|year=1831}}</ref> In 1998, [[Thomas Callister Hales]], following the approach suggested by [[László Fejes Tóth]] in 1953, announced a proof of the Kepler conjecture. Hales' proof is a [[proof by exhaustion]] involving checking of many individual cases using complex computer calculations. Referees said that they were "99% certain" of the correctness of Hales' proof. On 10 August 2014, Hales announced the completion of a formal proof using [[automated proof checking]], removing any doubt.<ref>{{cite web|url=https://code.google.com/p/flyspeck/wiki/AnnouncingCompletion|website=Google Code Archive|title=Long-term storage for Google Code Project Hosting}}</ref>