Bart, thanks for a great question. If the structure actually came out to be a regular icosahedron, it would be easy to figure out the ratio: the struts are just the diagonals of regular pentagons made up of five of the sides of the icosahedron, so the struts would be (1+√5)/2 times as long as the elastic edges of the icosahedron. (That’s the golden ratio, approximately 1.618.)

In practice, though, the parallel struts move closer together, so they will be somewhat more than 1.618 times as long as the separation between parallel struts, and somewhat less than 1.618 times as long as the other connections between struts (which should all be the same length by symmetry). Unfortunately, I don’t know how to calculate what these exact separations should be — if I did, I’d have a better chance of designing a perfectly icosahedral tensegrity.

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