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Embedded flexible spherical cross-polytopes with nonconstant volumes

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Abstract

We construct examples of embedded flexible cross-polytopes in the spheres of all dimensions. These examples are interesting from two points of view. First, in dimensions 4 and higher, they are the first examples of embedded flexible polyhedra. Notice that, in contrast to the spheres, in the Euclidean and Lobachevsky spaces of dimensions 4 and higher still no example of an embedded flexible polyhedron is known. Second, we show that the volumes of the constructed flexible cross-polytopes are nonconstant during the flexion. Hence these cross-polytopes give counterexamples to the Bellows Conjecture for spherical polyhedra. Earlier a counterexample to this conjecture was constructed only in dimension 3 (V.A. Alexandrov, 1997), and it was not embedded. For flexible polyhedra in spheres we suggest a weakening of the Bellows Conjecture, which we call the Modified Bellows Conjecture. We show that this conjecture holds for all flexible cross-polytopes of the simplest type, which includes our counterexamples to the ordinary Bellows Conjecture. Simultaneously, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write down relations for the volumes of their faces of codimensions 1 and 2.

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Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    Alexander A. Gaifullin

  2. Moscow State University, Moscow, 119991, Russia

    Alexander A. Gaifullin

  3. Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol’shoi Karetnyi per. 19, Moscow, 127994, Russia

    Alexander A. Gaifullin

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Gaifullin, A.A. Embedded flexible spherical cross-polytopes with nonconstant volumes. Proc. Steklov Inst. Math. 288, 56–80 (2015). https://doi.org/10.1134/S0081543815010058

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