mathematics the first equations more than tree struts tetrahedron dissimilar



A few words just before we start talking equations: In his dissertation about tensegritiesValentín Gómez Jáureguihad organized three questionnaires all investigating the tensegrity. Eventually he used only two because the third, prepared for the general public, was cancelled soon after it's preparation. JŠuregui gives the following reason for it's cancellation: "..Therefore, the author had the idea of confirming the impression of excitement that people have when seeing a tensegrity structure. An informal survey was carried out in order to discover the most predominant opinions, but it was abandont not much later because the unique opinion was generalized. Every single person that saw any of the models thought that it was "really amazing", "gorgeous", "stunning", using these or similar expressions. Therefore it was not worthwhile to gather all these opinions when the point of view was basically the same."

It is completely understandable that Jáuregui stopped his survey on the general opinion, but there seems to be a paradox somewhere, because where else do the words "really amazing", "gorgeous" and "stunning" sound so boring? Nevertheless, the general conclusion is that a tensegrity is "really amazing" and "gorgeous". The word "gorgeous" emphasizing the beauty and "amazing" underlining the magic part: "how is it possible?", "is it a trick?"

In a way one can say it is a trick. And I think the key that unravels this trick is called mathematics. This section of the website contains an introduction to the mathematics behind a tensegrity, but a serious investigation about this topic is made by Robert Burkhardt. In his book A Practical Guide to Tensegrity Designhe demonstrates how to calculate the proportions between strings and struts within even the more difficult tensegrities. It is an excellent work that helped me and probabaly a lot of other people in unravelling tensegrity-problems.
Burkhardt starts the mathematical part of his book by explaining the simple T-prism as shown here on the picture. At the end of the puzzle he gives the following comment which is a good explanation of "the trick" behind a tensegrity: "..The length of the final three tendons (one for each side of the prism - the side tendons) has to be chosen carefully; otherwise, the structure will turn out to be a loose jumble of sticks and fishing line. As the two ends of the prism are twisted relative to each other, the vertices corresponding to the opposite obtuse angles initially grow closer to each other. As the twisting continues, there comes a point where they start to move apart again. If the side tendons are tied with length of fishing line which corresponds tot the minimum length reached at this point, the structure is stable since it can't move away from that configuration except by lengthening the distance between those two point, and that is prevented by the minimum-length tendon.
This is the "trick" which underlies all tensegrity design methods explored here."

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Simple T-prism. The struts are a little thicker than I usually use and the tendons are made of fishing line. The thin lines in contrast with the thick bars accentuates the "floating" character of the tensegrity.

And there are a lot more scientists and PhD's who have studied the mathematics behind a tensegrity. It appears that it is not that easy to find the shortest string around a bunch of sticks or the longest stick in a net of strings.

Personal note

One blade of grass seems to be just a simple blade of grass that is not capable of doing anything else than grow. But, in this case growing means the use of sunshine to transform water and carbon oxide into organic material. And that is something we humans are not capable of despite all our inventions and "high tech" development.
I realize it may sound a bit strange, but I compare this "blade of grass" theory with the strings in a tensegrity. They are pretty stupid, but they never make a mistake in finding the shortest way from one strut end to the other. And, for us poor humans, finding that shortest route can be an endless brain damaging investigation. At least that is my own personal experience.


This beautiful tensegrity is a music instrument called "Rope and Sound".
The following text on createdigitalmusic explains how it works: "Rope and Sound is an installation that uses rope tension to control sound. Pull on a cord, and the change in tension triggers electronic thuds and mellow chimes. The trick is conductive fibers braided into the rope; as the tension changes, the conduction of the rope changes, as well."

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On the scan below one can see just one page of Bob Burkhardt's "A practical Guide to Tensegrity Design". The page is shown here just to give you an impression of the mathematics involved.

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One page of "A practical Guide to Tensegrity Design" by Robert Burkhardt.

Marcelo Pars