On the translative packing densities of tetrahedra and cuboctahedra

In 1900, as a part of his 18th problem, Hilbert asked the question to determine the density of the densest tetrahedron packings. However, up to now no mathematician knows the density δ^t(T)${\delta }^t(T)$ of the densest translative tetrahedron packings and the density δ^c(T)${\delta }^c(T)$ of the densest congruent tetrahedron packings. This paper presents a local method to estimate the density of the densest translative packings of a general convex solid. As an application, we obtain the upper bound in

0.3673469?≤δ^t(T)≤0.3840610?,$0.3673469?\le {\delta }^t(T)\le 0.3840610?,$

where the lower bound was established by Groemer in 1962, which corrected a mistake of Minkowski. For the density δ^t(C)${\delta }^t(C)$ of the densest translative cuboctahedron packings, we obtain the upper bound in

0.9183673?≤δ^t(C)≤0.9601527?.$0.9183673?\le {\delta }^t(C)\le 0.9601527?.$

In both cases we conjecture the lower bounds to be the correct answer.