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Hyperboloids
The Geomag forum member nicknamed GrahamB invented this very curious, surprisingly strong structure:
 Tower of 8 hyperboloid modules, view 1 1518 pieces: 246 balls, 1272 rods (8.09 kg) Tower of 8 hyperboloid modules, view 2
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:
 modules rods balls 1 58 246 2 84 390 3 110 534 4 136 678 5 162 822 6 188 966 7 214 1110 8 240 1254 9 266 1398 10 292 1542 ... ... ... n 26 × n + 32 144 × n + 102
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.
 Four hyperboloid modules in a tetrahedric arrangement, view 1 1129 pieces: 214 balls, 915 rods (6.14 kg) Four hyperboloid modules in a tetrahedric arrangement, view 2 Four hyperboloid modules in a tetrahedric arrangement, view 3 Four hyperboloid modules in a tetrahedric arrangement, view 4
• The original pictures by GrahamB, and his many other constructions, can be found at his archive in flickr.