Perhaps because it is the simplest among the platonic solids, the regular tetrahedron seems to have played a significant role in the original Geomag design. Four balls exactly fit in the inner space of a scale-1 tetrahedron (left-hand picture), which implies very precise geometric relations among the shapes and dimensions of the Geomag pieces. It is hard to believe that this happened by chance.

The midpoints of the regular tetrahedron edges are the vertices of a regular octahedron, made of green rods in the left-hand picture. If we superimpose two tetrahedra in such a way that their edges are perpendicular and cut at midpoints (yellow and red in the right-hand picture), the intersection is that very same octahedron.

We could also regard the double-scale tetrahedron as built by joining four single-scale ones by their vertices. The inner octahedron turns out to be, simply, the empty space which remains in between. Similarly, to build a quadruple scale tetrahedron, we can join four double-scale ones; the empty space is now a double-scale octahedron (see left-hand picture).

The funny thing is that it seems there is no theoretical limit to this method of scaling up the tetrahedron, although the proportion of hollow space grows in each iteration. Notice that we will build the octuple-scale tetrahedron using the quadruple-scale ones just shown. Besides a hollow quadruple-scale inner octahedron, this body will contain the four hollow double-scale octahedra belonging in the component tetrahedra. And so on.

Bodies so built are called Sierpinski tetrahedra or tetrix, as they are similar to the first iterations of a so-called ideal recursive geometrical object. The next iteration would be made of 6144 rods, and would have a hollow span measuring 16 rods, just like the size of the base of the previous body.