Sabitov polynomials for volumes of polyhedra in four dimensions

Abstract

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 combinatorial type and on edge lengths of the polyhedron only. Moreover, the coefficients of this polynomial are polynomials in edge lengths of the polyhedron. This result implies that the volume of a simplicial polyhedron with fixed combinatorial type and edge lengths can take only finitely many values. In particular, this yields that the volume of a flexible polyhedron in a 3-dimensional Euclidean space is constant. Until now it has been unknown whether these results can be obtained in dimensions greater than 3. In this paper we prove that all these results hold for polyhedra in a 4-dimensional Euclidean space.Comment: 23 pages; misprints corrected, Lemma 6.1 slightly rewritten, title change

Necessary conditions for the extendibility of a first-order flex of a polyhedron to its flex

“…Since P t has triangular faces only, 3F = 2E. Substituting the latter relation to (14), we obtain E = 3V − 3χ 3V − 6. This inequality implies that the set of (3V − 6) × (3V − 6) minors of the rigidity matrix of P t is non-empty.…”

“…However, the real flowering of the theory of flexible polyhedra began in the mid-1970s when Robert Connelly constructed a sphere-homeomorphic self-intersection free flexible polyhedron [7]. Now we already know that any flexible polyhedron in Euclidean 3-space preserves the total mean curvature (see [1]), enclosed volume (there are especially many articles devoted to this issue, so we point out several of them in chronological order: [23], [24], [25], [10], [26], [27], [14], and [13]), and Dehn invariants (see [16]). The reader can find more details about the theory of flexible polyhedra in the above mentioned articles, as well as in the following review articles, which we list in chronological order: [21], [8], [29], [28], and [15].…”

Flexible cross-polytopes in spaces of constant curvature

“…Another proof was given in [10]. Recently the author has generalized Sabitov's theorem to flexible polyhedra in Euclidean spaces of arbitrary dimensions n ≥ 4 [11], [12]. Nevertheless, it remained unknown whether this result is non-empty, i. e., whether there exists at least one flexible polyhedron in E n for n ≥ 5.…”

“…Second, we need to recover the set of edge lengths ℓ from the matrices G, H, and E. We start from the second problem. For a given butterfly, edge lengths ℓ bpbq are recovered uniquely from E by (11). Hence we need to recover a butterfly from G and H.…”

Volumes of Polytopes in Spaces of Constant Curvature

“…A. Gaifullin [20] was published in arXiv. He proved an analog of Theorem 1.2 for the generalized volume of a 4-dimensional polyhedron.…”