In a Class 1 tensegrity, the tendons connect to a continuous network. Ideally, a structure deploys (at least) three separate tendons from each 'knot' to a compression member. A six-strut icosahedron would need 24 tendons, yet a continuos tendon can shape one as well (but that's another story...).
A six-strut tetrahedron still has 18 tendons, which means a 3:1 ratio of tendons to strut (at least three tendons, 18 must be the smaller number of connections between 12 points). I struggled a lot building those, using 4 triangular loops and 6 tendons. Maximising tendons proved well for large structures with few compression elements, but smaller models can be build more economical.
Tensegrities derived from regular geometrical shapes 'cut' the corners off, with as many struts joining as the number of edges joining in a corner. Three edges join the corner of a tetrahedron, hence the corner is opened into a triangle. Having the four corners equally sized balances the structure well, and, to simplify matters more, only six tendons remain for final tuning of the model.
Using 10 instead of 18 hypothetical tendons make life already easier, but somehow I insisted to complicate my tensegrity build experiments more than necessary so far. Besides the x-module, the tensul and the six-strut icosahedron, all symmetrical shapes I encountered only need one tendon per strut, weaving another to either end to complete three tendons per joint.
|Toolkit variation with 6, 9 and 12 struts|
After revisiting the java app at xozzox I wanted to test the versatility of a single elastic tendon per strut approach by building a 'trigonal dipyramid'. Basically, it's a 3-stage tensul tower, yet with fewer connections than I build in larger scale so far. It has 5 corners, 2 triangular and 3 squared, and squishes down nicely when pushed along the triangular corners.
I went to build the trigonal prism using similar components, and finished much faster than the still quite laborious first go. Can I really build all platonic solids (tetrahedron, cube, octahedron, icosahedron and dodecahedron) with the simple one tendon per strut strategy?
One approach to convert a platonic solid into a tensegrity structure transforms each edge into a compression element, and the corners into tension loops. Following this definition, I already made an icosahedron and a dodecahedron, using one tendon for each of the thirty struts required. (The six strut icosahedron connects two of its twelve corners, using a 'shortcut' through the centre of the structure)
After studying some of my tetrahedral models I concluded that it can be done, and again got stunned by the simplicity of the process. Once I figured out the 'weaving' pattern, I just needed some trust in the stability of these structures, and everything came together easily.
Building a cube and an octahedron as final proof of concept happened after some side explorations. I used physical models as template, and re-used the components of prior experiments.
|Toolkit top view|
When I did the last bit of prototyping, I used the same measures as for structures with more compression elements. I doubt it would hold up spheres with more than 48 struts, yet it works great for the platonic solids, combinations thereof, and seemingly many other composite structures. Assembling any of the above models (even with some additional stability measures and tuning) took much less than 30 minutes.
Using single tendons can confuse easily, but following simple rules and pattern make it easy to build all platonic solids with just 30 identical elements. The elastic cord offers enough tension for quite a high number of elements (towers still tend to be floppy), and hooking/unhooking the tendons took little effort. The models end up in a size desktop suitable, I have to see how the market reacts tomorrow.