ISSN:
1420-8946
Keywords:
Keywords. Rigidity, polyhedron, convexity, Gauss map.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract. We extend the Cauchy theorem stating rigidity of convex polyhedra in $ {\bold R}^3 $ . We do not require that the polyhedron be convex nor embedded, only that the realization of the polyhedron in $ {\bold R}^3 $ 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/s000140050137
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