An important problem in various Finite Element Methods is refining a polyhedral mesh to get a better approximation to the solution. For that purpose it is ideal to look at polyhedrons which can be subdivided into copies of themselves. The next best compromise is to have a subdivision process that doesn't create too many "classes" of polyhedrons.
In 2D, this is pretty easy because any triangle and any parallelogram can be subdivided into scaled copies of itself. In 3D, this stops being true with the tetrahedron. Of course, the hypercubes work in any dimension for this problem. But is there a polyhedron with this property that has fewer vertices than the cube? And in general can we say anything about such polyhedrons?