Dehn invariant of flexible polyhedra
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866917669142265856 |
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| author | Gaifullin, Alexander A. Ignashchenko, Leonid |
| author_facet | Gaifullin, Alexander A. Ignashchenko, Leonid |
| contents | We prove that the Dehn invariant of any flexible polyhedron in Euclidean space of dimension greater than or equal to 3 is constant during the flexion. In dimensions 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 Connelly in 1979. It was believed that this conjecture was disproved by Alexandrov and Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in either sphere or Lobachevsky space of dimension greater than or equal to 3 is constant during the flexion if and only if this polyhedron satisfies the usual Bellows Conjecture, i.e., its volume is constant during every flexion of it. Using previous results due to the first listed 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 greater than or equal to 3. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1710_11247 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Dehn invariant of flexible polyhedra Gaifullin, Alexander A. Ignashchenko, Leonid Metric Geometry Algebraic Geometry 52C25, 52B45 (Primary), 51M25, 32D99 (Secondary) We prove that the Dehn invariant of any flexible polyhedron in Euclidean space of dimension greater than or equal to 3 is constant during the flexion. In dimensions 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 Connelly in 1979. It was believed that this conjecture was disproved by Alexandrov and Connelly in 2009. However, we find an error in their counterexample. Further, we show that the Dehn invariant of a flexible polyhedron in either sphere or Lobachevsky space of dimension greater than or equal to 3 is constant during the flexion if and only if this polyhedron satisfies the usual Bellows Conjecture, i.e., its volume is constant during every flexion of it. Using previous results due to the first listed 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 greater than or equal to 3. |
| title | Dehn invariant of flexible polyhedra |
| topic | Metric Geometry Algebraic Geometry 52C25, 52B45 (Primary), 51M25, 32D99 (Secondary) |
| url | https://arxiv.org/abs/1710.11247 |