Embedded flexible spherical cross-polytopes with nonconstant volumes



We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, in contrast to the spheres, in the Euclidean and Lobachevsky spaces of dimensions 4 and higher still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was constructed only in dimension 3 (V.A. Alexandrov, 1997), and it was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type, which includes our counterexamples to the ordinary Bellows Conjecture. Simultaneously, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write down relations for the volumes of their faces of codimensions 1 and 2.


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  1. 1.
    D. V. Alekseevskij, E. B. Vinberg, and A. S. Solodovnikov, “Geometry of spaces of constant curvature,” in Geometry. II (Springer, Berlin, 1993), Encycl. Math. Sci. 29, pp. 1–138.CrossRefGoogle Scholar
  2. 2.
    R. Alexander, “Lipschitzian mappings and total mean curvature of polyhedral surfaces. I,” Trans. Am. Math. Soc. 288, 661–678 (1985).CrossRefMATHGoogle Scholar
  3. 3.
    V. Alexandrov, “An example of a flexible polyhedron with nonconstant volume in the spherical space,” Beitr. Algebra Geom. 38 (1), 11–18 (1997).MATHMathSciNetGoogle Scholar
  4. 4.
    K. Aomoto, “Analytic structure of Schläfli function,” Nagoya Math. J. 68, 1–16 (1977).MATHMathSciNetGoogle Scholar
  5. 5.
    G. T. Bennett, “Deformable octahedra,” Proc. London Math. Soc., Ser. 2, 10, 309–343 (1912).CrossRefGoogle Scholar
  6. 6.
    R. Bricard, “Mémoire sur la théorie de l’octaèdre articulé,” J. Math. Pures Appl., Sér. 5, 3, 113–148 (1897).MATHGoogle Scholar
  7. 7.
    R. Connelly, “A counterexample to the rigidity conjecture for polyhedra,” Publ. Math., Inst. Hautes étud. Sci. 47, 333–338 (1977).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    R. Connelly, “Conjectures and open questions in rigidity,” in Proc. Int. Congr. Math., Helsinki, 1978 (Acad. Sci. Fennica, Helsinki, 1980), Vol. 1, pp. 407–414.Google Scholar
  9. 9.
    R. Connelly, I. Sabitov, and A. Walz, “The bellows conjecture,” Beitr. Algebra Geom. 38 (1), 1–10 (1997).MATHMathSciNetGoogle Scholar
  10. 10.
    H. S. M. Coxeter, “The functions of Schläfli and Lobatschefsky,” Q. J. Math. 6, 13–29 (1935).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. A. Gaifullin, “Sabitov polynomials for volumes of polyhedra in four dimensions,” Adv. Math. 252, 586–611 (2014); arXiv: 1108.6014 [math.MG].CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    A. A. Gaifullin, “Generalization of Sabitov’s theorem to polyhedra of arbitrary dimensions,” Discrete Comput. Geom. 52 (2), 195–220 (2014); arXiv: 1210.5408 [math.MG].CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    A. A. Gaifullin, “Flexible cross-polytopes in spaces of constant curvature,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 286, 88–128 (2014) [Proc. Steklov Inst. Math. 286, 77–113 (2014)]; arXiv: 1312.7608 [math.MG].Google Scholar
  14. 14.
    A. A. Gaifullin, “Volumes of flexible polyhedra,” in Geometry Days in Novosibirsk–2014: Abstracts Int. Conf. Dedicated to the 85th Birthday of Yu.G. Reshetnyak, Novosibirsk, Sept. 24–27, 2014 (Sobolev Inst. Math., Novosibirsk, 2014), pp. 98–99.Google Scholar
  15. 15.
    W. S. Massey, Algebraic Topology: An Introduction (Harcout, Brace & World, New York, 1967).MATHGoogle Scholar
  16. 16.
    V. V. Prasolov, Elements of Combinatorial and Differential Topology (Am. Math. Soc., Providence, RI, 2006), Grad. Stud. Math. 74.CrossRefMATHGoogle Scholar
  17. 17.
    I. Kh. Sabitov, “Volume of a polyhedron as a function of its metric,” Fundam. Prikl. Mat. 2 (4), 1235–1246 (1996).MATHMathSciNetGoogle Scholar
  18. 18.
    I. Kh. Sabitov, “A generalized Heron–Tartaglia formula and some of its consequences,” Mat. Sb. 189 (10), 105–134 (1998) [Sb. Math. 189, 1533–1561 (1998)].CrossRefMathSciNetGoogle Scholar
  19. 19.
    I. Kh. Sabitov, “The volume as a metric invariant of polyhedra,” Discrete Comput. Geom. 20 (4), 405–425 (1998).CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    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
  21. 21.
    H. Stachel, “Flexible cross-polytopes in the Euclidean 4-space,” J. Geom. Graph. 4 (2), 159–167 (2000).MATHMathSciNetGoogle Scholar
  22. 22.
    H. Stachel, “Flexible octahedra in the hyperbolic space,” in Non-Euclidean Geometries: János Bolyai Memorial Volume, Ed. by A. Prékopa et al. (Springer, New York, 2006), Math. Appl. 581, pp. 209–225.CrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.Institute for Information Transmission Problems (Kharkevich Institute)Russian Academy of SciencesMoscowRussia

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