In the past, ellipsoids with equal axes have been closely packed, under the force of gravity, into various structures. Critchlow in his book Order in Space (1970) illustrates ellipsoids with equal axes arranged in (a) a simple 4-ellipsoid with equal axes tetrahedral configuration, (b) a simple 6-ellipsoid with equal axes octahedral configuration, and (c) a simple 13-ellipsoid with equal axes cuboctahedral configuration, referring to each simple configuration as a distinct regular pattern. In discussing these arrangements, Critchlow indicates that the tetrahedral configuration is the most economic grouping of--ellipsoids of equal axes--while "the next most economic regular grouping of--ellipsoids of equal axes--is six in the octahedral configuration."
A preliminary examination of the ellipsoids of equal axes arranged in a tetrahedral configuration with a triangular base and in a 4-sided pyramid configuration with a square base would appear to support Critchlow's characterizations and distinctions. The lines connecting the centerpoints of the three ellipsoids of equal axes which form the "base" of the simple 4-ellipsoid tetrahedron form an equilateral triangle. On the other hand, the lines connecting the centerpoints of the four ellipsoids of equal axes which form the "base" of a simple 5-ellipsoid pyramid (i.e. a one-half octahedron) form a square.
In the past, a lattice structure based on a tetrahedral configuration and a lattice structure based on a pyramidal, or one-half octahedral configuration were viewed as different.
The fact that a tetrahedral configuration, an octahedral (or pyramidal) configuration and a cuboctahedral configuration yield precisely the same lattice structure when extended into space or merged together, however, has remained unknown and conspicuously unsuggested, especially as applied to ellipsoids of influence under the influence of gravity.