Combinatorial packings ofR^3by certain error spheres

This paper concerns one of the "error spheres" discussed by Golomb in 1969, his "Stein corner" in three-dimensional Euclidean space
R^{3}
. This figure, which we shall call a semicross, is defined as follows. Let
k
be a positive integer. The
(k, 3)
-semicross consists of
3k + 1
unit cubes: a corner cube together with three nonopposite arms of length
k
. (It may be thought of as a tripod.) For
k \geq 2
translates of the
(k, 3)
-semicross do not tile
R^{3}
. The question of how densely the translates pack
R^{3}
will be examined by combinatorial techniques. While the maximum density is not determined, sufficiently dense packings are produced to show that they are much denser than the densest lattice packing.