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Continuous deformations of polyhedra that do not alter the dihedral angles

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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.

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Notes

  1. If the reader prefers to deal with polyhedra with simply connected faces only, he can triangulate the face \(P\cap \tau \) without adding new vertices. In this case, the movement of \(T\) should be small enough so that no triangle of the triangulation becomes degenerate during the deformation.

  2. In Section 18.3.8.6 of the well known book [2] the reader may find a hypothesis that this problem should be solved in a negative form. We just want to warn the reader about a typo which may obscure: the last letter \(\pi \) in that Section 18.3.8.6 should be replaced by \(\pi ^2\).

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Correspondence to Victor Alexandrov.

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The author is supported in part by the Russian Foundation for Basic Research (project 10–01–91000–anf) and the State Maintenance Program for Young Russian Scientists and the Leading Scientific Schools of the Russian Federation (grant NSh–921.2012.1).

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Alexandrov, V. Continuous deformations of polyhedra that do not alter the dihedral angles. Geom Dedicata 170, 335–345 (2014). https://doi.org/10.1007/s10711-013-9884-8

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