Dense Periodic Packings of Tetrahedra with Small Repeating Units

We present a one-parameter family of periodic packings of regular tetrahedra, with the packing fraction 100/117 approximate to 0.8547, that are simple in the sense that they are transitive and their repeating units involve only four tetrahedra. The construction of the packings was inspired from results of a numerical search that yielded a similar packing. We present an analytic construction of the packings and a description of their properties. We also present a transitive packing with a repeating unit of two tetrahedra and a packing fraction 139+40 root 10/369 approximate to 0.7194.