Algebraic Methods for Solution of Polyhedra

Russian Math. Surveys 66:3 445{505 ⃝c 2011 RAS(DoM) and LMS Uspekhi Mat. Nauk 66:3 3{66 DOI 10.1070/RM2011v066n03ABEH004748 Algebraic methods for solution of polyhedra I. Kh. Sabitov Abstract. By analogy with the solution of triangles, the solution of polyhedra means a theory and methods for calculating some geometric parameters of polyhedra in terms of other parameters of them. The main content of this paper is a survey of results on calculating the volumes of polyhedra in terms of their metrics and combinatorial structures. It turns out that a far-reaching generalization of Heron's formula for the area of a trian- gle to the volumes of polyhedra is possible, and it underlies the proof of the conjecture that the volume of a deformed flexible polyhedron remains constant. Bibliography: 110 titles. Keywords: polyhedra, combinatorial structure, metric, volume, bending, bellows conjecture, volume polynomials, generalization of Heron's formula. Contents 1. Some history 446 2. Polyhedra in R3 and their generalized volume 448 3. Motivations for choosing the path to the solution of the problem 449 4. The central theorem and its consequences 452 5. The algebraic meaning of the central theorem 453 6. Main lemma 453 7. The Cayley{Menger determinant 454 8. The idea of the proof of the main lemma 455 9. Proof of the central Theorem1 458 10. Another proof of the central Theorem1 459 11. Several conclusions 463 12. Canonical volume polynomials 464 13. Calculating the volume polynomial for some polyhedra 467 14. Calculating the lengths of diagonals of polyhedra 472 15. Generalizing the equations for volumes and diagonals 478 16. Polyhedra with diagonals whose lengths are known 479 17. Isometric immersion problem for a polyhedral metric 480 This work was partially supported by the Russian Foundation for Basic Research (grants no. 09-01-00179 and 10-01-91000АНФ). AMS 2010 Mathematics Subject Classification. Primary 51M20, 52C25; Secondary 51M10, 52B11. 446 I. Kh. Sabitov 18. Some algebro-geometric properties of volume polynomials 482 18.1. Roots of a polynomial as the volumes of polyhedra 482 18.2. Roots of a volume polynomial for a flexible polyhedron 484 19. The strong bellows conjecture 486 20. Another invariant of flexible polyhedra 487 21. Polyhedra in the spherical and the hyperbolic spaces 491 22. Polytopes in multidimensional spaces 493 23. Some unsolved problems 495 Bibliography 498 1. Some history By analogy with the solution of triangles, the solution of polyhedra refers to a recently emergent area in the metric theory of polyhedra which investigates the possibilities for calculating some parameters of a polyhedron (its volume, the lengths of the edges and diagonals, dihedral angles, and so on) it terms of other metric char- acteristics of it, with account taken of the combinatorial structure of the polyhedron, of course. For the present the metric of the polyhedron and its combinatorial structure are usually assumed to be known, and such questions as the existence of a polyhedron with prescribed metric and combinatorial structure, calculation of its volume and the lengths of its diagonals, establishment of its flexibility or inflexibility, and determination of its isometric deformations are solved on this basis. Historically, the initial question here was the uniqueness of a polyhedron with given metric and combinatorial structure, first investigated by Legendre. In an appendix (Note XII) of the first edition of his well-known textbook [1] he analyzes Euclid's definition of equality of polyhedra1 and concludes that, first, this definition must in fact be regarded as a theorem to be proved, and second, we must assume in advance that the polyhedra under consideration are convex, otherwise Definition 10 contradicts an earlier example (1756) due to the English mathematician R. Simson: a cube with a quadrangular pyramid with apex outside the cube and base coinciding with a face of the cube (removed beforehand) and an equal cube with a similar pyramid but with apex inside the cube, have faces \equal in multitude and in magnitude" but obviously are not equal. Further, in the same Note XII Legendre proved for certain classes of polyhedra that two convex polyhedra with the same combinatorial structure and equal corresponding faces can be obtained one from the other by a motion, and he conjectured that this might be true for any convex polyhedra. For reasons unknown to us, Legendre left out this part of Note XII in subsequent editions (2nd to 9th) of his book, and restored it only after Cauchy in 1813 proved the equality for all convex polyhedra [3], essentially using Legendre's ideas in the proof. Hence it would be fair historically to call this result of Cauchy the Legendre{Cauchy theorem, and that is what we shall call it in what follows.2 At the same time, the Legendre{Cauchy theorem asserts that a (strictly) convex polyhedron is inflexible, because in a flex (that is, a continuous deformation of 1From Book XI of Euclid's Elements [2]: Definition 10. Equal and similar solid figures are those contained by similar planes equal in multitude and in magnitude. 2Several authors have devoted papers to a thorough analysis of Definition 10 and its relations to the Legendre{Cauchy theorem (see, for instance, [4]{[6]). Algebraic methods for solution of polyhedra 447 the polyhedron during which it has absolutely rigid faces) the polyhedron remains convex at the initial instants of the deformation, so that by the Legendre{Cauchy theorem it keeps its original shape and the flex reduces to a simple motion of itas a solid body. Once the Legendre{Cauchy theorem was established, the following question nat- urally arose: does this inflexibility hold for all polyhedra, not necessarily convex, or do there exist flexible polyhedra? (Equality in the class of all, not necessarily convex, polyhedra is refuted by Simson's example.) This question remained open for quite a long period of time until at the end of the 19th century the Belgian mathematician and engineer Bricard proved [7] the existence of flexible octahedra, providing at the same time a complete classification of them. However, all these lie in R3 but not as embedded or even immersed polyhedra, so the question of the existence of `proper' (non-self-intersecting) flexible polyhedra still remained open. Moreover, in 1975 Gluck proved [8] that almost all polyhedra homeomorphic to a sphere are inflexible, and thus the chances for the existence of embedded flexible polyhedra were considered negligible. Nevertheless, the American mathematician Connelly managed to construct examples of flexible polyhedra, first immersed ones [9] (1976) and then embedded ones [10] (1977). Some new examples of flexible polyhedra have been constructed since then (the reader can find details in [11]), and for all of them it was discovered that flexes do not change their volume. This observation provided some grounds for the conjecture that all the flexible polyhedra have the property of volume invariance (see [12] (February 1978), where the author of this conjecture was not mentioned, and [13] (August 1978), where a reference to D. Sullivan was made; however, even today we do not know3 who actually was the first to put forward the conjecture). In view of the well-known physics law pV = Const, where p is the pressure and V the gas volume, Connelly [13] called this the `bellows conjecture': a flexible polyhedron of mathematically ideal design (with a hole in one of the faces, of course) cannot be used as a bellows. This beautiful conjecture was apparently a topic of study for many geometers for almost two decades, but this remained their personal secret, since only in the two publications [16] and [17] by V. A. Alexandrov (Novosibirsk) did the author state explicitly that his ultimate aim was to prove or refute the conjecture. More- over, [18] presents examples of polyhedra which can easily be deformed but whose volume changes during the deformation | so one could assume that they also were constructed with an aim to disprove the conjecture (however, these deformations turned out not to be isometries). Finally, in 1996, a proof of the conjecture was given in [19] for flexible polyhedra of any topological genus (the first advances date back to [20] and more detailed proofs can be found in [21] and [22]). The method of the proof turned out to be very useful also in other questions of the metric theory of polyhedra, and our goal in this paper is to present a short proof of the bellows conjecture4 and a survey of other results in this area of geometry that have appeared during the last 10{15 years. We also state many unsolved problems. The surveys [24], [14], and [25] are closest to our topic. The reader can find a more popular discussion of these equations in our pamphlet [11], and [26] treats similar questions for planar polygons and also gives a large list of unsolved 3The authors of [14] and [15] do not know this either. 4Another proof appeared in [23] and we also present it in what follows. 448 I. Kh. Sabitov problems. The reader who is interested in a wider range of problems in the metric theory of polyhedra and polygons can be referred to the extensive text [15] (to our knowledge there are plans to publish it as a book, but it is presently available as a preprint). We leave unmentioned many papers relating to flexes and infinitesimal flexes of polyhedra. 2. Polyhedra in R3 and their generalized volume We start by introducing the main notions relating to polyhedra in three dimen- sions. A polyhedron in R3 is characterized by its combinatorial structure or scheme and the geometric realization of this scheme in space.

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