The Geomag forum member nicknamed GrahamB invented this very curious, surprisingly strong structure:
The tower is made of eight modules. The platforms are hexagonal antiprisms with tetrahedra attached on the lateral faces. Each platform is joined with the adjacent ones by means of twelve six-rod segments. The six inner segments join the vertices of the hexagons, while the six outer ones join the free vertices of the tetrahedra. The six inner segments are twisted clockwise (as seen from above), and the outer ones are very slightly twisted counterclockwise.
When a straight line rotates around a non-incident, non-parallel, non-perpendicular axis, it describes a hyperboloid of revolution (see the second figure at this MathWorld article). Each set of six segments, inner and outer, outlines a hyperboloid of revolution. Hence the title of this page, for lack of a better one.
The upper platform of the tower in the previous pictures has six superfluous tetrahedra, which would be required to extend the tower another level. In the following table we give the piece counts required to build towers of different sizes, without excess tetrahedra in the last level, and using the same foot shown in the pictures:
In the following structure (also by GrahamB) four modules are placed around a truncated regular tetrahedron, each one attached by a hexagonal antiprism. The structure is very stable not only when completed but also, unexpectedly, at every stage of construction, as it doesn't require any scaffolding or additional support.