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 Trudy Mat. Inst. Steklova, 2018, Volume 302, Pages 143–160 (Mi tm3933)

Dehn invariant and scissors congruence of flexible polyhedra

Alexander A. Gaifullinabcd, Leonid S. Ignashchenkod

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Institute for Information Transmission Problems (Kharkevich Institute), Russian Academy of Sciences, Bol'shoi Karetnyi per. 19, str. 1, Moscow, 127051 Russia
c Skolkovo Institute of Science and Technology, ul. Nobelya 3, Moscow, 121205 Russia
d Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 119991 Russia

Abstract: We prove that the Dehn invariant of any flexible polyhedron in $n$-dimensional Euclidean space, where $n\ge 3$, is constant during the flexion. For $n=3$ and $4$ this implies that any flexible polyhedron remains scissors congruent to itself during the flexion. This proves the Strong Bellows Conjecture posed by R. Connelly in 1979. It was believed that this conjecture was disproved by V. Alexandrov and R. Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in the $n$‑dimensional sphere or $n$-dimensional Lobachevsky space, where $n\ge 3$, is constant during the flexion whenever this polyhedron satisfies the usual Bellows Conjecture, i.e., whenever its volume is constant during every flexion of it. Using previous results of the first named author, we deduce that the Dehn invariant is constant during the flexion for every bounded flexible polyhedron in odd-dimensional Lobachevsky space and for every flexible polyhedron with sufficiently small edge lengths in any space of constant curvature of dimension at least $3$.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-51-55017 Ministry of Education and Science of the Russian Federation ÌÄ-2907.2017.1 The work of the first author was supported by the Russian Foundation for Basic Research (project no. 16-51-55017) and by a grant of the President of the Russian Federation (project no. MD-2907.2017.1).

DOI: https://doi.org/10.1134/S0371968518030068

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 130–145

Bibliographic databases:

UDC: 514.174
MSC: Primary 52C25, 52B45; Secondary 51M25, 32D99

Citation: Alexander A. Gaifullin, Leonid S. Ignashchenko, “Dehn invariant and scissors congruence of flexible polyhedra”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Trudy Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 143–160; Proc. Steklov Inst. Math., 302 (2018), 130–145

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3933
• https://doi.org/10.1134/S0371968518030068
• http://mi.mathnet.ru/eng/tm/v302/p143

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This publication is cited in the following articles:
1. Alexandrov V., “Necessary Conditions For the Extendibility of a First-Order Flex of a Polyhedron to Its Flex”, Beitr. Algebr. Geom.
2. V. Alexandrov, “A sufficient condition for a polyhedron to be rigid”, J. Geom., 110:2 (2019), UNSP 38
3. V. Alexandrov, “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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