Two regular tetrahedra can be joined by any of their faces. The resultant body is an equilateral triangular hexahedron called the triangular dipyramid in the Johnson classification, which assigns the number 12 to it:
As the six faces of this body are equal, adding a third tetrahedron to any of them results always in the same object:
We will build several «curves» using this object as a module or building block. First, the following segment, which can be prolonged, uses three modules to make a «curve» which progresses following a straight line, but has three contiguous edges which describe three corresponding helices (blue rods):
Next, four modules make a segment, also prolongable, having a contiguous edge which describes a circular arc (blue rods):
Promising as they seem, these curves have somewhat frustrating properties. One would expect the circular one to close, forming a full circle, after certain number of repetitions. However, after 32 segments, it just falls a bit short, and cannot be closed.
The helical one, on the other hand, would be expected to repeat the same position, or at least to have a parallel face, after not too many repetitions, but it doesn't. This property would have been very useful in a variety of building situations.
Also they are a bit too weak, and not very fault-tolerant: just a misoriented rod is usually fatal. In any case, the circular one is irreplaceable for certain constructions, as, for instance, the domes of chapels three, four, and five.
Two building examples follow, both using the circular curve. Three segments are built up from a base regular icosahedron which has three tetrahedrical legs. A slightly different position of the base attachments makes the left-hand curves cross in space without touching one another, while the right-hand ones incide in a common vertex:
Building these curves to double scale is easy: plainly, the tetrahedra which they are made of must also be built to double scale. A special reinforced icosahedron is required for a base, however. Two views of the incident version to double scale:
Common vertex detail:
We can even nest the single-scale version inside the double-scale one: