Editing Packing problems (section)

==Packing in infinite space==
Many of these problems, when the container size is increased in all directions, become equivalent to the problem of packing objects as densely as possible in infinite [[Euclidean space]]. This problem is relevant to a number of scientific disciplines, and has received significant attention. The [[Kepler conjecture]] postulated an optimal solution for [[sphere packing|packing spheres]] hundreds of years before it was [[mathematical proof|proven]] correct by [[Thomas Callister Hales]]. Many other shapes have received attention, including ellipsoids,<ref>{{Cite journal | last1 = Donev | first1 = A. | last2 = Stillinger | first2 = F. | last3 = Chaikin | first3 = P. | last4 = Torquato | first4 = S. | title = Unusually Dense Crystal Packings of Ellipsoids | doi = 10.1103/PhysRevLett.92.255506 | journal = Physical Review Letters | volume = 92 | issue = 25 | year = 2004 | pmid =  15245027|arxiv = cond-mat/0403286 |bibcode = 2004PhRvL..92y5506D | page=255506| s2cid = 7982407 }}</ref> [[Platonic solid|Platonic]] and [[Archimedean solid]]s<ref name="Torquato"/> including [[tetrahedron packing|tetrahedra]],<ref>{{Cite journal | doi = 10.1038/nature08641 | last1 = Haji-Akbari | first1 = A. | last2 = Engel | first2 = M. | last3 = Keys | first3 = A. S. | last4 = Zheng | pmid = 20010683 | first4 = X. | last5 = Petschek | first5 = R. G. | last6 = Palffy-Muhoray | first6 = P. | last7 = Glotzer | first7 = S. C. | title = Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra | year = 2009 | journal = Nature | volume = 462 | issue = 7274 | pages = 773–777 |bibcode = 2009Natur.462..773H |arxiv = 1012.5138 | s2cid = 4412674 }}</ref><ref>{{Cite journal | last1 = Chen | first1 = E. R. | last2 = Engel | first2 = M. | last3 = Glotzer | first3 = S. C. | title = Dense Crystalline Dimer Packings of Regular Tetrahedra | journal = [[Discrete & Computational Geometry]] | volume = 44 | issue = 2 | pages = 253–280 | year = 2010 | doi = 10.1007/s00454-010-9273-0| doi-access=free | arxiv = 1001.0586 | bibcode = 2010arXiv1001.0586C | s2cid = 18523116 }}</ref> [[Tripod packing|tripods]] (unions of [[cube]]s along three positive axis-parallel rays),<ref>{{citation|last=Stein|first=Sherman K.|author-link= Sherman K. Stein |date=March 1995|department=Mathematical entertainments|doi=10.1007/bf03024896|issue=2|journal=[[The Mathematical Intelligencer]]|pages=37–39|title=Packing tripods|volume=17|s2cid=124703268}}. Reprinted in {{citation|last=Gale|first=David|editor1-first=David|editor1-last=Gale|doi=10.1007/978-1-4612-2192-0|isbn=0-387-98272-8|mr=1661863|pages=131–136|publisher=Springer-Verlag|title=Tracking the Automatic ANT|year=1998}}</ref> and unequal-sphere dimers.<ref>{{Cite journal | last1 = Hudson | first1 = T. S. | last2 = Harrowell | first2 = P. | doi = 10.1088/0953-8984/23/19/194103 | pmid = 21525553 | title = Structural searches using isopointal sets as generators: Densest packings for binary hard sphere mixtures | journal = Journal of Physics: Condensed Matter | volume = 23 | issue = 19 | page = 194103 | year = 2011 | bibcode = 2011JPCM...23s4103H | s2cid = 25505460 }}</ref>

===Hexagonal packing of circles===
[[File:Circle packing (hexagonal).svg|thumb|right|The hexagonal packing of circles on a 2-dimensional Euclidean plane.]]
These problems are mathematically distinct from the ideas in the [[circle packing theorem]]. The related [[circle packing]] problem deals with packing [[circle]]s, possibly of different sizes, on a surface, for instance the [[plane (geometry)|plane]] or a [[sphere]].

The [[N-sphere|counterparts of a circle]] in other dimensions can never be packed with complete efficiency in [[dimension]]s larger than one (in a one-dimensional universe, the circle analogue is just two points). That is, there will always be unused space if people are only packing circles. The most efficient way of packing circles, [[Circle packing|hexagonal packing]], produces approximately 91% efficiency.<ref>{{Cite web | url=http://mathworld.wolfram.com/CirclePacking.html |title = Circle Packing}}</ref>

===Sphere packings in higher dimensions===
{{Main|Sphere packing}}

In three dimensions, [[Close-packing of spheres|close-packed]] structures offer the best ''lattice'' packing of spheres, and is believed to be the optimal of all packings. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible definable packings.<ref>{{cite journal | last1 = Smalley | first1 = I.J. | year = 1963 | title = Simple regular sphere packings in three dimensions | journal = Mathematics Magazine | volume = 36 | issue = 5| pages = 295–299 | doi = 10.2307/2688954 | jstor = 2688954 }}</ref> The 8-dimensional [[E8 lattice]] and 24-dimensional [[Leech lattice]] have also been proven to be optimal in their respective real dimensional space.

===Packings of Platonic solids in three dimensions===
Cubes can easily be arranged to fill three-dimensional space completely, the most natural packing being the [[cubic honeycomb]]. No other [[Platonic solid]] can tile space on its own, but some preliminary results are known. [[Tetrahedra]] can achieve a packing of at least 85%. One of the best packings of regular [[dodecahedron|dodecahedra]] is based on the aforementioned face-centered cubic (FCC) lattice.

Tetrahedra and [[octahedra]] together can fill all of space in an arrangement known as the [[tetrahedral-octahedral honeycomb]].

{| class="wikitable"
|-
! Solid
! Optimal density of a lattice packing
|-
| [[icosahedron]] 
| 0.836357...<ref name="Betke">{{cite journal|last1=Betke|first1=Ulrich|last2=Henk|first2=Martin|doi=10.1016/S0925-7721(00)00007-9|doi-access=free|issue=3|journal=[[Computational Geometry (journal)|Computational Geometry]]|mr=1765181|pages=157–186|title=Densest lattice packings of 3-polytopes|volume=16|year=2000|arxiv=math/9909172|s2cid=12118403}}</ref>
|-
| dodecahedron
| {{math|1=(5 + {{sqrt|5}})/8 = 0.904508...}}<ref name="Betke"/>
|-
| octahedron
| 18/19 = 0.947368...<ref>Minkowski, H. Dichteste gitterförmige Lagerung kongruenter Körper. ''Nachr. Akad. Wiss. Göttingen Math. Phys. KI. II'' 311–355 (1904).</ref>
|}

Simulations combining local improvement methods with random packings suggest that the lattice packings for icosahedra, dodecahedra, and octahedra are optimal in the broader class of all packings.<ref name="Torquato">{{Cite journal| first1 = S. | first2 = Y.| last2 = Jiao| title = Dense packings of the Platonic and Archimedean solids| volume = 460| last1 = Torquato| journal = Nature| issue = 7257| pages = 876–879| date=Aug 2009 | issn = 0028-0836| pmid = 19675649| doi = 10.1038/nature08239|bibcode = 2009Natur.460..876T |arxiv = 0908.4107 | s2cid = 52819935}}</ref>