Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions

Gaifullin, Alexander A.

doi:10.48550/arxiv.1210.5408

Cited by 8 publications

(26 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(The number N and the polynomials a i depend on the combinatorial structure of the polyhedron.) The same result for polyhedra of dimensions n ≥ 4 has recently been obtained by one of the authors [2], [3].…”

Section: Introductionsupporting

confidence: 80%

Deformations of Period Lattices of Flexible Polyhedral Surfaces

Gaifullin

Sergey

2014

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

Abstract. In the end of the 19th century Bricard discovered a phenomenon of flexible polyhedra, that is, polyhedra with rigid faces and hinges at edges that admit non-trivial flexes. One of the most important results in this field is a theorem of Sabitov asserting that the volume of a flexible polyhedron is constant during the flexion. In this paper we study flexible polyhedral surfaces in R 3 two-periodic with respect to translations by two non-colinear vectors that can vary continuously during the flexion. The main result is that the period lattice of a flexible two-periodic surface homeomorphic to a plane cannot have two degrees of freedom.

show abstract

Section: Introductionsupporting

confidence: 80%

Deformations of Period Lattices of Flexible Polyhedral Surfaces

Gaifullin

Sergey

2014

Discrete Comput Geom

Self Cite

View full text Add to dashboard Cite

show abstract

“…It is not hard to check that signs in formulae ( 12) and ( 13) are consistent so that the volumes of simplices [A F 1 G] and [A F 2 G] enter the left-hand side of ( 12) with the opposite signs for any two facets F 1 and F 2 with a common (m − 2)-dimensional face G. But the simplices [A F 1 G] and [A F 2 G] are congruent to each other. Hence their volumes are equal to each other, which implies equality (12).…”

Section: 2mentioning

confidence: 96%

“…A multi-dimensional generalization of the Bellows Conjecture, that is, the assertion of the constancy of the volume of an arbitrary flexible polyhedron in E n , n ≥ 4, was proved by the author [11], [12]. The question naturally arises if the analogue of the Bellows Conjecture holds in the spheres S n and in the Lobachevsky spaces Λ n , n ≥ 3.…”

Section: Introductionmentioning

confidence: 99%

Embedded flexible spherical cross-polytopes with nonconstant volumes

Gaifullin¹

2015

Proc. Steklov Inst. Math.

Self Cite

View full text Add to dashboard Cite

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, unlike in the spheres, in the Euclidean spaces and the 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 built only in dimension 3 (Alexandrov, 1997), and 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 among which there are our counterexamples to the usual Bellows Conjecture. By the way, we obtain several geometric results on flexible cross-polytopes of the simplest type. In particular, we write relations on the volumes of their faces of codimensions 1 and 2. 4

show abstract

“…Remark 7.2. Definition 7.1 is a natural analogue of the standard definition of a flexible polyhedron in the Euclidean space used in [27]- [29], [14], [17], [18]. The only difference is that in the Euclidean case we use affine mappings instead of pseudo-linear.…”

Section: Flexible Polyhedra and Their Volumesmentioning

confidence: 99%

“…If X n = E n or Λ n , then the Bellows conjecture in X n says that the volume of any flexible polyhedron in X n is constant during the flexion. The author [17], [18] proved the Bellows conjecture in all Euclidean spaces E n , n ≥ 4.…”

Section: Introductionmentioning

confidence: 99%

The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

Gaifullin¹

2015

Sb. Math.

View full text Add to dashboard Cite

A flexible polyhedron in an n-dimensional space X n of constant curvature is a polyhedron with rigid (n− 1)-dimensional faces and hinges at (n− 2)-dimensional faces. The Bellows conjecture claims that, for n ≥ 3, the volume of any flexible polyhedron is constant during the flexion. The Bellows conjecture in Euclidean spaces E n was proved by Sabitov for n = 3 (1996) and by the author for n ≥ 4 (2012). Counterexamples to the Bellows conjecture in open hemispheres S n + were constructed by Alexandrov for n = 3 (1997) and by the author for n ≥ 4 (2015). In this paper we prove the Bellows conjecture for bounded flexible polyhedra in odd-dimensional Lobachevsky spaces. The proof is based on the study of the analytic continuation of the volume of a simplex in the Lobachevsky space considered as a function of the hyperbolic cosines of its edge lengths.

show abstract

Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions

Cited by 8 publications

References 0 publications

Deformations of Period Lattices of Flexible Polyhedral Surfaces

Deformations of Period Lattices of Flexible Polyhedral Surfaces

Embedded flexible spherical cross-polytopes with nonconstant volumes

The analytic continuation of volume and the Bellows conjecture in Lobachevsky spaces

Contact Info

Product

Resources

About