  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.
