Advertisement

Siberian Mathematical Journal

, Volume 56, Issue 4, pp 569–574 | Cite as

The set of nondegenerate flexible polyhedra of a prescribed combinatorial structure is not always algebraic

  • V. A. AlexandrovEmail author
Article

Abstract

We construct some example of a closed nondegenerate nonflexible polyhedron P in Euclidean 3-space that is the limit of a sequence of nondegenerate flexible polyhedra each of which is combinatorially equivalent to P. This implies that the set of nondegenerate flexible polyhedra combinatorially equivalent to P is not algebraic.

Keywords

flexible polyhedron dihedral angle Bricard octahedron algebraic set 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Connelly R., “Conjectures and open questions in rigidity,” in: Proc. Intern. Congr. Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 407–414.Google Scholar
  2. 2.
    Schlenker J.-M., “La conjecture des soufflets (d’après I. Sabitov),” Astérisque, 294, 77–95 (2004) (Bourbaki Seminar. Volume 2002/2003. Exposes 909–923. Paris Société Math. de France).MathSciNetGoogle Scholar
  3. 3.
    Sabitov I. Kh., “Algebraic methods for solution of polyhedra,” Russian Math. Surveys, 66, No. 3, 445–505 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gaifullin A. A., “Sabitov polynomials for volumes of polyhedra in four dimensions,” Adv. Math., 252, 586–611 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Shor L. A., “On flexibility of convex polyhedra with boundary,” Mat. Sb., 45, No. 4, 471–488 (1958) (English translation is available in [6] as Appendix 3).MathSciNetGoogle Scholar
  6. 6.
    Alexandrov A. D., Convex Polyhedra, Springer-Verlag, Berlin (2005).zbMATHGoogle Scholar
  7. 7.
    Poznyak Eh. G., “Infinitesimally non-rigid closed polyhedra,” Vestnik Mosk. Univ. Ser. I, 15, No. 3, 14–18 (1960).zbMATHGoogle Scholar
  8. 8.
    Gluck H., “Almost all simply connected closed surfaces are rigid,” Lect. Notes Math., 438, 225–239 (1975).MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bricard R., “Mémoire sur la théorie de l’octaèdre articulé,” J. Math., 3, 113–148 (1897).zbMATHGoogle Scholar
  10. 10.
    Lebesgue H., “Octaèdres articulés de Bricard,” Enseign. Math. II Sér., 13, 175–185 (1967).MathSciNetzbMATHGoogle Scholar
  11. 11.
    Stachel H., “Zur Einzigkeit der Bricardschen Oktaeder,” J. Geom., 28, No. 1, 41–56 (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sabitov I. Kh., “Local theory on bendings of surfaces,” in: Geometry III. Theory of Surfaces. Encycl. Math. Sci., 48, Springer-Verlag, Berlin, 1992, pp. 179–250.Google Scholar
  13. 13.
    Cromwell P. R., Polyhedra, Cambridge Univ. Press, Cambridge (1999).zbMATHGoogle Scholar
  14. 14.
    Alexandrov V., “The Dehn invariants of the Bricard octahedra,” J. Geom., 99, No. 1–2, 1–13 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Maksimov I. G., “Description of the combinatorial structure of algorithmically 1-parametric polyhedra of spherical type,” Siberian Math. J., 53, No. 4, 718–731 (2012).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Russia Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations