TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 265, Number 2, June 1981 TENSEGRITY FRAMEWORKS BY B. ROTH1 AND W. WHITELEY Abstract. A tensegrity framework consists of bars which preserve the distance between certain pairs of vertices, cables which provide an upper bound for the distance between some other pairs of vertices and struts which give a lower bound for the distance between still other pairs of vertices. The present paper establishes some basic results concerning the rigidity, flexibility, infinitesimal rigidity and infinitesimal flexibility of tensegrity frameworks. These results are then applied to a number of questions, problems and conjectures regarding tensegrity frameworks in the plane and in space. 1. Introduction. The rigidity and flexibility of frameworks have been extensively studied in recent years and, although many questions remain unanswered, there now exists a substantial body of useful and interesting results. However, our knowledge of tensegrity frameworks (consisting of bars which preserve the distance between certain pairs of vertices, cables which provide an upper bound for the distance between some pair of vertices and struts which give a lower bound for the distance between other pairs of vertices) is in an embryonic state. Tensegrity frameworks are obviously of interest to architects and engineers (for example, see Calladine [4] and Fuller [6]); perhaps more surprising is their appearance in the work of the sculptor Kenneth Snelson. The symbiosis of frameworks and tensegrity frameworks has only lately become evident. For example, recent work of Connelly [5] on the rigidity of frameworks given by triangulated surfaces relies heavily on tensegrity frameworks. On the other hand, one theme of the present paper is that knowledge of frameworks frequently enhances our understanding of tensegrity frameworks. This mutually advantageous interplay between the study of frame- works and that of tensegrity frameworks seems likely to remain a prominent feature of the subject. The present paper establishes some basic results concerning tensegrity frame- works and then applies these to several open problems and conjectures in the plane and in space. More specifically, in §3 we define rigidity and flexibility for tensegrity frameworks and show that these notions are invariant under various changes in the definitions. In §4, after defining infinitesimal rigidity and infinitesimal flexibility for tensegrity frameworks, we establish the equivalence of infinitesimal rigidity and static rigidity for tensegrity frameworks using standard results from finite-dimen- sional convexity theory. §5 deals with the relationships between infinitesimal Received by the editors October 30, 1979. 1980 Mathematics Subject Classification. Primary 51F99; Secondary 52A25, 70B15. 'The first author is grateful to Branko Grünbaum and the University of Washington for their hospitality during his sabbatical year. © 1981 American Mathematical Society 0002-9947/81/0000-02 56/S08.00 419 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use