Discrete & Computational Geometry

, Volume 43, Issue 2, pp 436–466 | Cite as

Geometry of Configuration Spaces of Tensegrities

  • Franck DorayEmail author
  • Oleg Karpenkov
  • Jan Schepers


Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in ℝ d to admit a nonzero self-stress with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in ℝ d . In particular we find surgeries on graphs that give relations between different strata. Further we discuss questions related to geometric conditions defining the strata for plane tensegrities. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.

Tensegrity Self-stress Frameworks Stratification 


  1. 1.
    Barnabei, M., Brini, A., Rota, G.-C.: On the exterior calculus of invariant theory. J. Algebra 96(1), 120–160 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Berg, A.R., Jordán, T.: A proof of Connelly’s conjecture on 3-connected circuits of the rigidity matroid. J. Combin. Theory Ser. B 88(1), 77–97 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bochnak, J., Coste, M., Roy, M.-F.: Géométrie Algébrique Réelle. Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, vol. 12. Springer, Berlin (1987) zbMATHGoogle Scholar
  4. 4.
    Bolker, E.D., Crapo, H.: Bracing rectangular frameworks, I. SIAM. J. Appl. Math. 36(3), 473–490 (1979) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Caspar, D.L.D., Klug, A.: Physical principles in the construction of regular viruses. In: Proceedings of Cold Spring Harbor Symposium on Quantitative Biology, vol. 27, pp. 1–24 (1962) Google Scholar
  6. 6.
    Connelly, R.: Rigidity. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. 44, Chap. 1.7, pp. 223–271. North-Holland, Amsterdam (1993) Google Scholar
  7. 7.
    Connelly, R., Whiteley, W.: Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math. 9(3), 453–491 (1996) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    de Guzmán, M.: Finding tensegrity forms. Preprint (2004) Google Scholar
  9. 9.
    de Guzmán, M., Orden, D.: Finding tensegrity structures: geometric and symbolic approaches. In: Proceedings of EACA-2004, pp. 167–172 (2004) Google Scholar
  10. 10.
    de Guzmán, M., Orden, D.: From graphs to tensegrity structures: geometric and symbolic approaches. Publ. Mat. 50, 279–299 (2006) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Doubilet, P., Rota, G.-C., Stein, J.: On the foundations of combinatorial theory. IX., Combinatorial methods in invariant theory. Stud. Appl. Math. 53, 185–216 (1974) zbMATHMathSciNetGoogle Scholar
  12. 12.
    Ingber, D.E.: Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. J. Cell Sci. 104, 613–627 (1993) Google Scholar
  13. 13.
    Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Combin. Theory Ser. B 94(1), 1–29 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Maxwell, J.C.: On reciprocal figures and diagrams of forces. Philos. Mag. 4(27), 250–261 (1864) Google Scholar
  16. 16.
    Motro, R.: Tensegrity: Structural Systems for the Future. Kogan Page Science, London (2003) Google Scholar
  17. 17.
    Orden, D., Rote, G., Santos, F., Servatius, B., Servatius, H., Whiteley, W.: Non-crossing frameworks with non-crossing reciprocals. Discrete Comput. Geom. 32(4), 567–600 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Roth, B., Whiteley, W.: Tensegrity frameworks. Trans. Amer. Math. Soc. 265(2), 419–446 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Servatius, B.: Tensegrities. PAMM 7(1), 1070101–1070102 (2007) CrossRefGoogle Scholar
  20. 20.
    Skelton, R.E.: Deployable tendon-controlled structure. United States Patent 5642590 (1 July, 1997) Google Scholar
  21. 21.
  22. 22.
  23. 23.
    Sturmfels, B., Whiteley, W.: On the synthetic factorization of projectively invariant polynomials. J. Symb. Comput. 11, 439–453 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tibert, A.G.: Deployable tensegrity structures for space applications. Ph.D. thesis, Royal Institute of Technology, Stockholm (2002) Google Scholar
  25. 25.
    White, N.L., Whiteley, W.: The algebraic geometry of stresses in frameworks. SIAM J. Algebr. Discrete Methods 4(4), 481–511 (1983) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Whiteley, W.: Rigidity and scene analysis. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, Chap. 49, pp. 893–916. CRC, New York (1997) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Mathematisch InstituutUniversiteit LeidenLeidenThe Netherlands

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