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