What Can You Learn from a Hole in the Ground, a Tent Pole, Two Pieces of Rope, and Two Tent Pegs
Roger S. Tobie
A large class of tensile-integrity structures, popularly called tensegrities, can be derived from (not necessarily regular) n sided right prisms by rotating the tops and bottoms clockwise or counter clockwise in relationship to each other. Thus these structures are chiral and exist in both right and left handed mirror image versions of each other. The resulting structures exhibit invariant plan views when parallel projected from the "top" down or "bottom" up along their main axis of rotational symmetry. These plan views are invariant in the same sense that similar triangles are invariant in shape in that they have the same shape with constant angular values no matter what the overall size of the projected structure is and no matter what its overall height to width ratio happens to be. Also, the plan views are identical for both the right and left handed versions. This leads to something called the constant twist angle theorem. This theorem has been demonstrated using calculus, see for instance the Appendix to Chapter 1 of Kenner, Geodesic Math and How to Use It. However, using descriptive geometry and intersecting planes, there is a much simpler and more intuitively obvious way of demonstrating this constant twist angle theorem which bypasses calculus entirely. This leads to a particularly elegant and simple way of drawing a series of plan views of these structures that depends solely on the number of sticks within a structure. This drawing algorithm requires only the tools of Euclidean geometry, namely, unmarked ruler and compass, although in a some cases a graduated protractor is required, say when the circumference of a circle is to be divided into seven equal parts. The first drawing in this series of plan views bears a striking resemblance to the first figure in the first book of Euclid's Elements.
