
The regular tetrahedron
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 scale1 tetrahedron (lefthand 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 inner four balls could be regarded as another regular tetrahedron, made with zerolength rods, that is, with no rods at all. So, this zerorod tetrahedron fits inside a single rod one. The very same geometric relations ensure that the single rod tetrahedron will fit into a double rod one, which in turn will fit into a triple rod one, an so on (righthand picture).
The midpoints of the regular tetrahedron edges are the vertices of a regular octahedron, made of green rods in the lefthand picture. If we superimpose two tetrahedra in such a way that their edges are perpendicular and cut at midpoints (yellow and red in the righthand picture), the intersection is that very same octahedron.
In other words, if we add four singlescale regular tetrahedra on alternate faces of a regular octahedron, we obtain a doublescale regular tetrahedron (again, lefthand picture). If we add a tetrahedron on every face of the octahedron, we obtain the body shown in the righthand picture, equivalent to two intersecting doublescale tetrahedra, and called stella octangula.
We could also regard the doublescale tetrahedron as built by joining four singlescale 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 doublescale ones; the empty space is now a doublescale octahedron (see lefthand picture).
Adding another four doublescale tetrahedra on the hollow faces of this octahedron, we get the doublescale stella octangula shown in the righthand picture, in which the inner octahedron is also hollow.
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 octuplescale tetrahedron using the quadruplescale ones just shown. Besides a hollow quadruplescale inner octahedron, this body will contain the four hollow doublescale octahedra belonging in the component tetrahedra. And so on.
It is possible, however, for the practical limit to be at something like the fourth iteration, that is, at the regular tetrahedron to scale 16, as the one shown in the following pictures:
Bodies so built are called Sierpinski tetrahedra or tetrix, as they are similar to the first iterations of a socalled 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.
The nth iteration would have 6 × 4^{n} rods and 2 × 4^{n} + 2 balls.
Building objects with big planar bases, like this one, requires a truly planar table. When I built it, I was searching for a defective or misplaced piece until I realized that the construction didn't perfectly fit on my table at a certain position, while it did at others.
