Algebra versus analysis in the theory of flexible polyhedra

A polyhedron (more precisely, a polyhedral surface) is said to be flexible if its spatial shape can be changed continuously due to changes of its dihedral angles only, i. e., if every face remains congruent to itself during the flex. In other words, a polyhedron P 0 is flexible if it is included in a continuous family {P t }, 0 t 1, of polyhedra P t such that, for every t, the corresponding faces of P 0 and P t are congruent while the polyhedra P 0 and P t are not congruent. The family {P t }, 0 t 1, is called the flex of P 0 . Self-intersections are possible both for P 0 and P t provided the converse is not formulated explicitly. Without loss of generality we assume that the faces of the polyhedra are triangular.

Flexible self-intersection free sphere-homeomorphic polyhedra in Euclidean 3space were constructed by R. Connelly thirty years ago [4], [6]. Since that time, various non-trivial properties of flexible polyhedra were discovered in the Euclidean 3-space [10] and 4-space [12] (for results in the hyperbolic 3-space, see [13]). Let us formulate two of them in a form suitable for our purposes.

Let P be a closed oriented polyhedron in R 3 , let E be the set of its edges, let |ℓ| be the length of the edge ℓ, and let α(ℓ) be the dihedral angle of P at the edge ℓ measured from inside of P . The sum

is called the total mean curvature of P . Theorem 1. Let P 0 be a flexible polyhedron in R 3 and let {P t }, 0 t 1, be its flex. The total mean curvature M (P t ) is independent of t.

Theorem 1 was proved by R. Alexander [2] for all Euclidean n-spaces, n 3, though no example of a flexible polyhedron is known for n 5.

Let P 0 be a flexible polyhedron and let {P t }, 0 t 1, be its flex. Let r t be the point of the polyhedron P t which corresponds to the point r 0 ∈ P 0 . It follows from the definition of a flexible polyhedron that, for any curve γ 0 ⊂ P 0 , the length of the curve γ t = {r t |r 0 ∈ γ 0 } ⊂ P t is independent of t. The reader can easily verify the well-known fact that, for any curve γ 0 ⊂ P 0 , the length of the curve γ t = {r 0 + tv|r 0 ∈ γ 0 } is stationary at t = 0, where v = d dt | t=0 r t is the velocity vector of the point r t at t = 0. Obviously, the vector field v is linear on every face of P 0 . This leads to the following well-known definition: a vector field w on a polyhedron P , which is linear on every face of P , is said to be its infinitesimal flex if, for any curve γ ⊂ P , the length of the curve γ(t) = {r + tw|r ∈ γ} is stationary at t = 0, see [7] for more detail. Of course, the velocity vector field of a flexible polyhedron is its infinitesimal flex (but the converse is not necessarily true).

In [2] Theorem 1 was obtained as an obvious corollary of the following Theorem 2 proved for R n , n 3.

Theorem 2. Let P be a closed oriented polyhedron in R 3 , let w be its infinitesimal flex, and let P (t) = {r + tw|r ∈ P }. Then d dt | t=0 M P (t) = 0. In [2] Theorem 2 was proved with the help of the Stokes theorem. Later several authors observed (see, e. g., [3], [11]) that Theorem 2 follows immediately from the Schläfli differential formula, which, in turn, is based on the Stokes theorem. In any case, all known proofs of Theorems 1 and 2 belong to Analysis.

In [8] I.Kh. Sabitov proved another highly non-trivial property of the flexible polyhedra that may be formulated as follows and whose many-dimensional analog is not known yet.

Theorem 3. If {P t } is a flex of an orientable polyhedron in R 3 , then the oriented volume of P t is constant in t.

In [8], [9], and later in [5] Theorem 3 was obtained as an obvious corollary of the following Theorem 4 valid since every continuous mapping, whose image is a finite set, is constant.

Theorem 4. For the set P K of all (not necessarily flexible) closed polyhedra in R 3 with triangular faces and with a prescribed combinatorial structure K there exists a universal polynomial p K of a single variable whose coefficients are universal polynomials in the edge lengths of a polyhedron P ∈ P K and such that the oriented the volume of any P ∈ P K is a root of p K .

In [8] and [9] Theorem 4 was proved with the help of the theory of resultants, while in [5] it was proved with the help of the theory of places. In any case, all known proofs of Theorems 3 and 4 belong to Algebra. Now recall that the derivative of the volume of a deformable domain in R 3 equals one third of the flux across the boundary of the velocity vector of a point of the boundary of the domain, i. e., equals one third of the integral of the normal component of the velocity over the boundary. Hence, an ‘infinitesimal version’ of Theorem 3 should read that, for every orientable polyhedron P , the flux P (w, n) dP of any infinitesimal flex w equals zero. Theorem 5 shows that this is not the case and, thus, that Theorem 3 can not be proved by means of Analysis like Theorem 1.

Theorem 5. There is a closed oriented polyhedron P in R 3 with the following properties:

(i) the flux acros

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