Tetrahedron packing

[9][10] A further improvement was made in 2009 by Torquato and Jiao, who compressed Chen's structure using a computer algorithm to a packing fraction of 78.2021%.

[11] In mid-2009 Haji-Akbari et al. showed, using MC simulations of initially random systems that at packing densities >50% an equilibrium fluid of hard tetrahedra spontaneously transforms to a dodecagonal quasicrystal, which can be compressed to 83.24%.

For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%.

[1] The Chen, Engel and Glotzer result currently stands as the densest known packing of hard, regular tetrahedra.

Surprisingly, the square-triangle tiling[12] packs denser than this double lattice of triangular bipyramids when tetrahedra are slightly rounded (the Minkowski sum of a tetrahedron and a sphere), making the 82-tetrahedron crystal the largest unit cell for a densest packing of identical particles to date.