Ellipsoid packing

In geometry, ellipsoid packing is the problem of arranging identical ellipsoid throughout three-dimensional space to fill the maximum possible fraction of space.

The currently densest known packing structure for ellipsoid has two candidates,

a simple monoclinic crystal with two ellipsoids of different orientations{{cite journal |last1=Donev |first1=Aleksandar |last2=Stillinger |first2=Frank H. |last3=Chaikin |first3=P. M. |last4=Torquato |first4=Salvatore |title=Unusually Dense Crystal Packings of Ellipsoids |journal=Physical Review Letters |date=23 June 2004 |volume=92 |issue=25 |pages=255506 |doi=10.1103/PhysRevLett.92.255506|arxiv=cond-mat/0403286 }} and

a square-triangle crystal containing 24 ellipsoids{{cite journal |last1=Jin |first1=Weiwei |last2=Jiao |first2=Yang |last3=Liu |first3=Lufeng |last4=Yuan |first4=Ye |last5=Li |first5=Shuixiang |title=Dense crystalline packings of ellipsoids |journal=Physical Review E |date=22 March 2017 |volume=95 |issue=3 |pages=033003 |doi=10.1103/PhysRevE.95.033003|arxiv=1608.07697 }} in the fundamental cell. The former monoclinic structure can reach a maximum packing fraction around $0.77073$ for ellipsoids with maximal aspect ratios larger than $\backslash sqrt\{3\}$. The packing fraction of the square-triangle crystal exceeds that of the monoclinic crystal for specific biaxial ellipsoids, like ellipsoids with ratios of the axes $\backslash alpha:\backslash sqrt\{\backslash alpha\}:1$ and $\backslash alpha\; \backslash in\; (1.365,1.5625)$. Any ellipsoids with aspect ratios larger than one can pack denser than spheres.