Rigidity of certain polyhedra in
$ {\bold R}^3 $ |
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Authors: | L Rodríguez H Rosenberg |
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Institution: | (1) Instituto de Matemática Pura e Aplicada, (IMPA), Estrada Dona Castorina 110, Rio de Janeiro, Brazil 22460, e-mail: luci@impa.br, BR;(2) Departement de Mathématiques, Université de Paris VII, 2 Place Jussieu, F-75251 Paris Cedex 05, France, e-mail: rosen@math.jussieu.fr, FR |
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Abstract: | We extend the Cauchy theorem stating rigidity of convex polyhedra in . We do not require that the polyhedron be convex nor embedded, only that the realization of the polyhedron in be linear and isometric on each face. We also extend the topology of the surfaces to include the projective plane in addition
to the sphere. Our approach is to choose a convenient normal to each face in such a way that as we go around the star of a
vertex the chosen normals are the vertices of a convex polygon on the unit sphere. When we can make such a choice at each
vertex we obtain rigidity. For example, we can prove that the heptahedron is rigid.
Received: March 3, 1999; revised: December 7, 1999. |
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Keywords: | , Rigidity, polyhedron, convexity, Gauss map, |
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