Ellipsoid packing explained

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^[1] anda square-triangle crystal containing 24 ellipsoids^[2] 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 \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 \alpha:\sqrt{\alpha}: 1 and \alpha \in ( 1 . 365 , 1 . 5625 ) . Any ellipsoids with aspect ratios larger than one can pack denser than spheres.

See also

• Packing problems
• Sphere packing
• Tetrahedron packing

Notes and References

1. Donev . Aleksandar . Stillinger . Frank H. . Chaikin . P. M. . Torquato . Salvatore . Unusually Dense Crystal Packings of Ellipsoids . Physical Review Letters . 23 June 2004 . 92 . 25 . 255506 . 10.1103/PhysRevLett.92.255506. cond-mat/0403286 .
2. Jin . Weiwei . Jiao . Yang . Liu . Lufeng . Yuan . Ye . Li . Shuixiang . Dense crystalline packings of ellipsoids . Physical Review E . 22 March 2017 . 95 . 3 . 033003 . 10.1103/PhysRevE.95.033003. 1608.07697 .