On flexible polyhedral surfaces

  • M. I. ShtogrinEmail author


We construct a closed orientable polyhedral surface of arbitrary genus that is embedded in three-dimensional Euclidean space and admits a one-parameter bending under which all its handles bend. This surface admits no other bendings. We also construct a flexible closed nonorientable polyhedral surface of arbitrary genus such that all its handles and Möbius strips bend during its bending.


Dihedral Angle STEKLOV Institute Projective Plane Plane Angle Polyhedral Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. D. Alexandrov, Convex Polyhedra (Gostekhizdat, Moscow, 1950; Springer, Berlin, 2005).Google Scholar
  2. 2.
    V. A. Aleksandrov, “A new example of a flexible polyhedron,” Sib. Mat. Zh. 36 (6), 1215–1224 (1995) [Sib. Math. J. 36, 1049–1057 (1995)].CrossRefGoogle Scholar
  3. 3.
    M. Berger, Géométrie, Vol. 3: Convexes et polytopes, polyedres réguliers, aires et volumes (Cedic, Paris, 1978); Engl. transl.: Geometry (Springer, Berlin, 1987), Vol. I.Google Scholar
  4. 4.
    R. Bricard, “Mémoire sur la théorie de l’octaèdre articulé,” J. Math. Pures Appl., Sér. 5, 5 (3), 113–148 (1897).Google Scholar
  5. 5.
    A. L. Cauchy, “Sur les polygones et les polyèdres. Second mémoire,” J. éc. Polytech. 9, 87–98 (1813).Google Scholar
  6. 6.
    R. Connelly, “An immersed polyhedral surface which flexes,” Indiana Univ. Math. J. 25 (10), 965–972 (1976).CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    R. Connelly, “A counterexample to the rigidity conjecture for polyhedra,” Publ. Math. Inst. Hautes étud. Sci. 47, 333–338 (1977).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    R. Connelly, “The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surfaces,” Adv. Math. 37 (3), 272–299 (1980).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    N. P. Dolbilin, M. A. Shtan’ko, and M. I. Shtogrin, “Rigidity of zonohedra,” Usp. Mat. Nauk 51 (2), 157–158 (1996) [Russ. Math. Surv. 51, 326–328 (1996)].CrossRefMathSciNetGoogle Scholar
  10. 10.
    N. P. Dolbilin, M. A. Shtan’ko, and M. I. Shtogrin, “On rigidity of polyhedral spheres with even-angle faces,” Usp. Mat. Nauk 51 (3), 197–198 (1996) [Russ. Math. Surv. 51, 543–544 (1996)].CrossRefMathSciNetGoogle Scholar
  11. 11.
    N. P. Dolbilin, M. A. Shtan’ko, and M. I. Shtogrin, “Rigidity of a quadrillage of the torus,” Usp. Mat. Nauk 54 (4), 167–168 (1999) [Russ. Math. Surv. 54, 839–840 (1999)].CrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Gluck, “Almost all simply connected closed surfaces are rigid,” in Geometric Topology: Proc. Conf., Park City, 1974 (Springer, Berlin, 1975), Lect. Notes Math. 438, pp. 225–239.CrossRefGoogle Scholar
  13. 13.
    D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie (J. Springer, Berlin, 1932); Engl. transl.: Geometry and the Imagination (Chelsea, New York, 1952).CrossRefzbMATHGoogle Scholar
  14. 14.
    R. Connelly, “An attack on rigidity. I, II,” Preprint (Cornell Univ., New York, 1974),, Scholar
  15. 15.
    N. H. Kuiper, “Sphères polyédriques flexibles dans E3, d’après Robert Connelly,” in Séminaire Bourbaki 1977/78 (Springer, Berlin, 1979), Exp. 514, Lect. Notes Math. 710, pp. 147–168.CrossRefGoogle Scholar
  16. 16.
    E. G. Poznyak, “Nonrigid closed polyhedra,” Vestn. Mosk. Univ., Ser 1: Mat. Mekh., No. 3, 14–19 (1960).Google Scholar
  17. 17.
    I. Kh. Sabitov, Volumes of Polyhedra (MTsNMO, Moscow, 2002) [in Russian].Google Scholar
  18. 18.
    I. Kh. Sabitov, “Algebraic methods for solution of polyhedra,” Usp. Mat. Nauk 66 (3), 3–66 (2011) [Russ. Math. Surv. 66, 445–505 (2011)].CrossRefMathSciNetGoogle Scholar
  19. 19.
    M. I. Shtogrin, “Rigidity of a quadrillage of the pretzel,” Usp. Mat. Nauk 54 (5), 183–184 (1999) [Russ. Math. Surv. 54, 1044–1045 (1999)].CrossRefMathSciNetGoogle Scholar
  20. 20.
    M. I. Shtogrin, “Bending of a piecewise developable surface,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 275, 144–166 (2011) [Proc. Steklov Inst. Math. 275, 133–154 (2011)].MathSciNetGoogle Scholar
  21. 21.
    M. Shtogrin, “Flexible surfaces with active handles,” in Geometry, Topology, and Applications: Abstracts Int. Conf., Yaroslavl, Sept. 23–27, 2013 (Yaroslav. Gos. Univ., Yaroslavl, 2013), pp. 105–106.Google Scholar
  22. 22.
    M. Shtogrin, “A flexible disk with a handle,” Usp. Mat. Nauk 68 (5), 177–178 (2013) [Russ. Math. Surv. 68, 951–953 (2013)].CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

Personalised recommendations