For inscribed convex polytopes with face circumcenters inside or on each face, cable-and-strut constraints force every same-homotopy realization to be a rotation of the original.
The Stress-Flex Conjecture
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abstract
Recently, it has been proven that a tensegrity framework that arises from coning the one-skeleton of a convex polytope is rigid. Since such frameworks are not always infinitesimally rigid, this leaves open the question as to whether they are at least prestress stable. We prove here that this holds subject to an intriguing new conjecture about coned polytope frameworks, that we call the stress-flex conjecture. Multiple numerical experiments suggest that this conjecture is true, and most surprisingly, seems to hold even beyond convexity and also for higher genus~polytopes.
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Spiderwebs on the Sphere and an Isoperimetric Theorem
For inscribed convex polytopes with face circumcenters inside or on each face, cable-and-strut constraints force every same-homotopy realization to be a rotation of the original.