# 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.