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 and a square-triangle crystal containing 24 ellipsoids 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.