Advertisement

Geometriae Dedicata

, Volume 170, Issue 1, pp 335–345 | Cite as

Continuous deformations of polyhedra that do not alter the dihedral angles

  • Victor Alexandrov
Original Paper
  • 124 Downloads

Abstract

We prove that, both in the hyperbolic and spherical 3-spaces, there exist nonconvex compact boundary-free polyhedral surfaces without selfintersections which admit nontrivial continuous deformations preserving all dihedral angles and study properties of such polyhedral surfaces. In particular, we prove that the volume of the domain, bounded by such a polyhedral surface, is necessarily constant during such a deformation while, for some families of polyhedral surfaces, the surface area, the total mean curvature, and the Gauss curvature of some vertices are nonconstant during deformations that preserve the dihedral angles. Moreover, we prove that, in the both spaces, there exist tilings that possess nontrivial deformations preserving the dihedral angles of every tile in the course of deformation.

Keywords

Dihedral angle Flexible polyhedron Hyperbolic space Spherical space Tessellation 

Mathematics Subject Classification (2010)

52C25 52B70 52C22 51M20 51K05 

References

  1. 1.
    Andreev, E.M.: On convex polyhedra in Lobachevskij spaces (in Russian). Mat. Sb., N. Ser. 81, 445–478 (1970). An English translation in Math. USSR, Sb. 10, 413–440 (1970)Google Scholar
  2. 2.
    Berger, M.: Geometry. I, II. Corrected 4th Printing. Universitext. Springer, Berlin (2009)Google Scholar
  3. 3.
    Connelly, R.: A counterexample to the rigidity conjecture for polyhedra. Publ. Math. IHES. 47, 333–338 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Dolbilin, N., Frettlöh, D.: Properties of Böröczky tilings in high-dimensional hyperbolic spaces. Eur. J. Comb. 31(4), 1181–1195 (2010)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dupont, J.L., Sah, C.-H.: Three questions about simplices in spherical and hyperbolic 3-space. In: The Gelfand Mathematical Seminars, 1996–1999, pp. 49–76. Birkhäuser, Boston (2000)Google Scholar
  6. 6.
    Dupont, J.L.: What is ...a Scissors Congruence? Notices Am. Math. Soc. 59(9), 1242–1244 (2012)Google Scholar
  7. 7.
    Mazzeo, R., Montcouquiol, G.: Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhedra. J. Differ. Geom. 87(3), 525–576 (2011)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. 2d ed. Graduate Texts in Mathematics, 149. Springer, New York (2006)Google Scholar
  9. 9.
    Sabitov, I.Kh.: Algebraic methods for the solution of polyhedra (in Russian). Uspekhi Mat. Nauk 66(3), 3–66 (2011). An English translation in. Russian Math. Surveys 66(3), 445–505 (2011)Google Scholar
  10. 10.
    Souam, R.: The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. Differ. Geom. Appl. 20(1), 31–45 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Stoker, J.J.: Geometrical problems concerning polyhedra in the large. Commun. Pure Appl. Math. 21, 119–168 (1968)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

Personalised recommendations