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 Trudy Mat. Inst. Steklova, 2014, Volume 286, Pages 88–128 (Mi tm3566)

Flexible cross-polytopes in spaces of constant curvature

A. A. Gaifullinabc

a Lomonosov Moscow State University, Moscow, Russia
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
c Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia

Abstract: We construct self-intersecting flexible cross-polytopes in the spaces of constant curvature, that is, in Euclidean spaces $\mathbb E^n$, spheres $\mathbb S^n$, and Lobachevsky spaces $\Lambda ^n$ of all dimensions $n$. In dimensions $n\ge5$, these are the first examples of flexible polyhedra. Moreover, we classify all flexible cross-polytopes in each of the spaces $\mathbb E^n$, $\mathbb S^n$, and $\Lambda ^n$. For each type of flexible cross-polytopes, we provide an explicit parametrization of the flexion by either rational or elliptic functions.

 Funding Agency Grant Number Russian Foundation for Basic Research 13-01-1246913-01-91151 Ministry of Education and Science of the Russian Federation MD-2969.2014.1 Dynasty Foundation The work was partially supported by the Russian Foundation for Basic Research (project nos. 13-01-12469 and 13-01-91151), by a grant of the President of the Russian Federation (project no. MD-2969.2014.1), and by the Dynasty Foundation.

DOI: https://doi.org/10.1134/S0371968514030066

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English version:
Proceedings of the Steklov Institute of Mathematics, 2014, 286, 77–113

Bibliographic databases:

UDC: 514.114+517.583

Citation: A. A. Gaifullin, “Flexible cross-polytopes in spaces of constant curvature”, Algebraic topology, convex polytopes, and related topics, Collected papers. Dedicated to Victor Matveevich Buchstaber, Corresponding Member of the Russian Academy of Sciences, on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 286, MAIK Nauka/Interperiodica, Moscow, 2014, 88–128; Proc. Steklov Inst. Math., 286 (2014), 77–113

Citation in format AMSBIB
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• https://doi.org/10.1134/S0371968514030066
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. A. Gaifullin, “Embedded flexible spherical cross-polytopes with nonconstant volumes”, Proc. Steklov Inst. Math., 288 (2015), 56–80
2. A. A. Gaifullin, “The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces”, Sb. Math., 206:11 (2015), 1564–1609
3. H. Stachel, “Flexible polyhedral surfaces with two flat poses”, Symmetry-Basel, 7:2 (2015), 774–787
4. I. Izmestiev, “Classification of flexible Kokotsakis polyhedra with quadrangular base”, Int. Math. Res. Notices, 2017, no. 3, 715–808
5. Alexander A. Gaifullin, “The bellows conjecture for small flexible polyhedra in non-Euclidean spaces”, Mosc. Math. J., 17:2 (2017), 269–290
6. Alexander A. Gaifullin, Leonid S. Ignashchenko, “Dehn invariant and scissors congruence of flexible polyhedra”, Proc. Steklov Inst. Math., 302 (2018), 130–145
7. V. M. Buchstaber, A. P. Veselov, “Conway topograph, $\mathrm{PGL}_2(\pmb{\mathbb Z})$-dynamics and two-valued groups”, Russian Math. Surveys, 74:3 (2019), 387–430
8. V. Alexandrov, “A sufficient condition for a polyhedron to be rigid”, J. Geom., 110:2 (2019), UNSP 38
9. I. Izmestiev, “Statics and kinematics of frameworks in euclidean and non-euclidean geometry”, Eighteen Essays in Non-Euclidean Geometry, Irma Lectures in Mathematics and Theoretical Physics, 29, eds. V. Alberge, A. Papadopoulos, European Mathematical Soc, 2019, 191–233
10. Izmestiev I., “Four-Bar Linkages, Elliptic Functions, and Flexible Polyhedra”, Comput. Aided Geom. Des., 79 (2020), UNSP 101870
11. Alexandrov V., “The Spectrum of the Laplacian in a Domain Bounded By a Flexible Polyhedron in R-D Does Not Always Remain Unaltered During the Flex”, J. Geom., 111:2 (2020), 32
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