Analytical Constructions of a Family of Dense Tetrahedron Packings and the Role of Symmetry

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. We have discovered the densest known packings of regular tetrahedra with a density $\phi= {12250/14319} = 0.855506...$. These packings are special cases of an analytical two-parameter family of dense periodic packings with four particles per fundamental cell that we have constructed here. From this family, we can recover a set of recent packing arrangements due to Kallus {\it et al.} [arXiv:0910.5226] with density $\phi={100/117}=0.854700...$, which has higher symmetry than our densest packings, We also describe a procedure that could lead to rigorous upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.