Communication: a packing of truncated tetrahedra that nearly fills all of space and its melting properties.

Dense polyhedron packings are useful models of a variety of condensed matter and biological systems and have intrigued scientists and mathematicians for centuries. Here, we analytically construct the densest known packing of truncated tetrahedra with a remarkably high packing fraction φ = 207/208 = 0.995192..., which is amazingly close to unity and strongly implies its optimality. This construction is based on a generalized organizing principle for polyhedra lacking central symmetry that we introduce here. The "holes" in the putative optimal packing are perfect tetrahedra, which leads to a new tessellation of space by truncated tetrahedra and tetrahedra. Its packing characteristics and equilibrium melting properties as the system undergoes decompression are discussed.