Lower Bound on Translative Covering Density of Octahedron

In this paper, we present the first nontrivial lower bound on the translative covering density of octahedron. To this end, we show the lower bound, in any translative covering of octahedron, on the density relative to a given parallelehedron. The resulting lower bound on the translative covering density of octahedron is $1+6.6\times {10}^{-8}$.