25 citations · 171 across the 21 of their papers we have counts for
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Dehn invariant of flexible polyhedra
Alexander A. Gaifullin, Leonid Ignashchenko
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 im…
Flexible Polyhedra and Their Volumes
Alexander A. Gaifullin
We discuss some recent results on flexible polyhedra and the bellows conjecture, which claims that the volume of any flexible polyhedron is constant during the flexion. Also, we su…
The bellows conjecture for small flexible polyhedra in non-Euclidean spaces
Alexander A. Gaifullin
The bellows conjecture claims that the volume of any flexible polyhedron of dimension 3 or higher is constant during the flexion. The bellows conjecture was proved for flexible pol…
Embedded flexible spherical cross-polytopes with non-constant volumes
Alexander A. Gaifullin
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 high…
Flexible cross-polytopes in spaces of constant curvature
Alexander A. Gaifullin
We construct self-intersected flexible cross-polytopes in the spaces of constant curvature, that is, the Euclidean spaces, the spheres, and the Lobachevsky spaces of all dimensions…
Volume of a simplex as a multivalued algebraic function of the areas of its two-faces
Alexander A. Gaifullin
For n greater than or equal to 4, the square of the volume of an n-simplex satisfies a polynomial relation with coefficients depending on the squares of the areas of 2-faces of thi…
Deformations of period lattices of flexible polyhedral surfaces
Alexander A. Gaifullin, Sergey A. Gaifullin
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…
Generalization of Sabitov's Theorem to Polyhedra of Arbitrary Dimensions
Alexander A. Gaifullin
In 1996 Sabitov proved that the volume of an arbitrary simplicial polyhedron P in the 3-dimensional Euclidean space satisfies a monic (with respect to V) polynomial relation…
Sabitov polynomials for volumes of polyhedra in four dimensions
Alexander A. Gaifullin
In 1996 I.Kh. Sabitov proved that the volume of a simplicial polyhedron in a 3-dimensional Euclidean space is a root of certain polynomial with coefficients depending on the combin…