Ellipsoid packing

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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 [math]\displaystyle{ 0.77073 }[/math] for ellipsoids with maximal aspect ratios larger than [math]\displaystyle{ \sqrt{3} }[/math]. 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 [math]\displaystyle{ \alpha:\sqrt{\alpha}:1 }[/math] and [math]\displaystyle{ \alpha \in (1.365,1.5625) }[/math]. Any ellipsoids with aspect ratios larger than one can pack denser than spheres.

See also


  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. 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. doi:10.1103/PhysRevE.95.033003.