Analytical Construction of A Dense Packing of Truncated Tetrahedra 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 207/208=0.995 192, which is amazingly close to unity and strongly implies its optimality. This construction is based on a generalized organizing principle for polyhedra that lack central symmetry that we introduce here. The packing characteristics and equilibrium melting properties of the putative optimal packing as the system undergoes decompression are discussed.