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In spite of their great importance and numerous applications in many civil, aerospace and biological systems, our understanding of tensegrity structures is still quite preliminary, fragmented and incomplete. Here we establish a unified closed-form analytical solution for the necessary and sufficient condition that ensures the existence of self-equilibrated and super-stable states for truncated regular polyhedral tensegrity structures, including truncated tetrahedral, cubic, octahedral, dodecahedral and icosahedral tensegrities.

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