Editing Packing problems (section)

===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>