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^[1] and a 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.

References[edit]

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

[1] Donev, Aleksandar; Stillinger, Frank H.; Chaikin, P. M.; Torquato, Salvatore (23 June 2004). "Unusually Dense Crystal Packings of Ellipsoids". Physical Review Letters. 92 (25): 255506. arXiv:cond-mat/0403286. doi:10.1103/PhysRevLett.92.255506.

[2] Jin, Weiwei; Jiao, Yang; Liu, Lufeng; Yuan, Ye; Li, Shuixiang (22 March 2017). "Dense crystalline packings of ellipsoids". Physical Review E. 95 (3): 033003. arXiv:1608.07697. doi:10.1103/PhysRevE.95.033003.

[1]

[2]

v t e Packing problems
Abstract packing	Bin Set
Circle packing	In a circle / equilateral triangle / isosceles right triangle / square Apollonian gasket Circle packing theorem Tammes problem (on sphere)
Sphere packing	Apollonian Finite In a sphere In a cube In a cylinder Close-packing Kissing number Sphere-packing (Hamming) bound
Other 2-D packing	Square packing
Other 3-D packing	Tetrahedron Ellipsoid
Puzzles	Conway Slothouber–Graatsma

See also[edit]

References[edit]