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