# Embedded flexible spherical cross-polytopes with nonconstant volumes

- 27 Downloads
- 2 Citations

## Abstract

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.

## Preview

Unable to display preview. Download preview PDF.

### References

- 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.R. Alexander, “Lipschitzian mappings and total mean curvature of polyhedral surfaces. I,” Trans. Am. Math. Soc.
**288**, 661–678 (1985).CrossRefMATHGoogle Scholar - 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.K. Aomoto, “Analytic structure of Schläfli function,” Nagoya Math. J.
**68**, 1–16 (1977).MATHMathSciNetGoogle Scholar - 5.G. T. Bennett, “Deformable octahedra,” Proc. London Math. Soc., Ser. 2,
**10**, 309–343 (1912).CrossRefGoogle Scholar - 6.R. Bricard, “Mémoire sur la théorie de l’octaèdre articulé,” J. Math. Pures Appl., Sér. 5,
**3**, 113–148 (1897).MATHGoogle Scholar - 7.R. Connelly, “A counterexample to the rigidity conjecture for polyhedra,” Publ. Math., Inst. Hautes étud. Sci.
**47**, 333–338 (1977).CrossRefMATHMathSciNetGoogle Scholar - 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.R. Connelly, I. Sabitov, and A. Walz, “The bellows conjecture,” Beitr. Algebra Geom.
**38**(1), 1–10 (1997).MATHMathSciNetGoogle Scholar - 10.H. S. M. Coxeter, “The functions of Schläfli and Lobatschefsky,” Q. J. Math.
**6**, 13–29 (1935).CrossRefMathSciNetGoogle Scholar - 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.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.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.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.W. S. Massey,
*Algebraic Topology: An Introduction*(Harcout, Brace & World, New York, 1967).MATHGoogle Scholar - 16.V. V. Prasolov,
*Elements of Combinatorial and Differential Topology*(Am. Math. Soc., Providence, RI, 2006), Grad. Stud. Math.**74**.CrossRefMATHGoogle Scholar - 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.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.I. Kh. Sabitov, “The volume as a metric invariant of polyhedra,” Discrete Comput. Geom.
**20**(4), 405–425 (1998).CrossRefMATHMathSciNetGoogle Scholar - 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.H. Stachel, “Flexible cross-polytopes in the Euclidean 4-space,” J. Geom. Graph.
**4**(2), 159–167 (2000).MATHMathSciNetGoogle Scholar - 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