We obtain an upper bound to the packing density of regular tetrahedra. The
bound is obtained by showing the existence, in any packing of regular
tetrahedra, of a set of disjoint spheres centered on tetrahedron edges, so that
each sphere is not fully covered by the packing. The bound on the amount of
space that is not covered in each sphere is obtained in a recursive way by
building on the observation that non-overlapping regular tetrahedra cannot
subtend a solid angle of $4\pi$ around a point if this point lies on a
tetrahedron edge. The proof can be readily modified to apply to other polyhedra
with the same property. The resulting lower bound on the fraction of empty
space in a packing of regular tetrahedra is $2.6\ldots\times 10^{-25}$ and
reaches $1.4\ldots\times 10^{-12}$ for regular octahedra.