Packing tripods: Narrowing the density gap

In 1984, S. K. Stein and his co-authors posed a problem concerning simple three-dimensional shapes, known as semicrosses, or tripods. By definition, a tripod is formed by a corner and the three adjacent edges of an integer cube. How densely can one fill the space with nonoverlapping tripods of a given size? In particular, is it possible to fill a constant fraction of the space as the tripod size tends to infinity? In this paper, we settle the second question in the negative: the fraction of the space that can be filled with tripods must be infinitely small as the size grows. We also make a step towards the solution of the first question, by improving the currently known asymptotic lower bound on tripod packing density, and by presenting some computational results on small-size packings.