  Dense polyhedron packings are useful models of a variety of condensed matter
and biological systems and have intrigued scientists mathematicians for
centuries. Recently, organizing principles for the types of structures
associated with the densest polyhedron packings have been put forth. However,
finding the maximally dense packings for specific shapes remains nontrivial and
is a subject of intensive research. Here, we analytically construct the densest
known packing of truncated tetrahedra with $\phi=207/208=0.995~192...$, which
is amazingly close to unity and strongly implies that this packing is maximally
dense. This construction is based on a generalized organizing principle for
polyhedra that lack central symmetry. Moreover, we find that the holes in this
putative optimal packing are small regular tetrahedra, leading to a new tiling
of space by regular tetrahedra and truncated tetrahedra. We also numerically
study the equilibrium melting properties of what apparently is the densest
packing of truncated tetrahedra as the system undergoes decompression. Our
simulations reveal two different stable crystal phases, one at high densities
and the other at intermediate densities, as well as a first-order
liquid-crystal phase transition.
