Geometriae Dedicata

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

Continuous deformations of polyhedra that do not alter the dihedral angles

  • Victor AlexandrovEmail author
Original Paper


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.


Dihedral angle Flexible polyhedron Hyperbolic space Spherical space Tessellation 

Mathematics Subject Classification (2010)

52C25 52B70 52C22 51M20 51K05 


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

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

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