Flexible polyhedra and frameworks in different spaces

This project is devoted to flexible structures like polyhedra and overconstrained frameworks. Which conditions are necessary and sufficient for flexibility, which metric or combinatorial properties must change or remain constant under flexing? To recall, a polyhedron - or 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., in such a way that every face remains congruent to itself during the flex. The question whether the edge lengths of a framework determine its planar or spatial shape uniquely, is also important for many engineering applications - not only for mechanical or constructional engineers, but also for biologists in protein modeling or for the analysis of isomers in chemistry. 1897 R. Bricard proved that there are exactly three types of flexible octahedra in the Euclidean 3-space E3 . However, they all have self-intersections. 1977 R. Connelly constructed the first flexing sphere, a flexible polygonal embedding of the 2-sphere into the E3 . This polyhedron as well as another example presented by K. Steffen (1980) was a compound of flexible Bricard octahedra. 1985 R. Alexander proved that every flexible polyhedron in E3 preserves its total mean curvature during the flex.1996 I. Sabitov proved that for every flexible polyhedron in E3 the volume keeps constant during the flex. This was a consequence of his generalization of Heron`s formula: For any orientable simplicial polyhedron in E3 there exists a polynomial in which the coefficients are polynomials in the squared edge-lengths and which has the volume as a root. This polynomial depends only on the combinatorial structure of the polyhedron. 1997 V. Alexandrov disproved this for the spherical 3-space. The main aim of the new project is a systematic investigation of flexible octahedra and their higher-dimensional counterparts, the cross-polytopes, in Euclidean, elliptic and hyperbolic spaces. We have seen in the past that they are of basic importance for all known flexible polyhedra. Furthermore, new examples of flexible polytopes would shed light into the question whether the constancy of the volume of flexible polyhedra remains restricted to the Euclidean 3-space E3 or not. In particular the following main problems will be attacked: Are the three types of octahedra presented 2004 by H. Stachel the only flexible ones in the hyperbolic 3-space H 3 ? Are the flexible cross polytopes presented 2000 by H. Stachel the only flexible ones in the Euclidean 4-space? Can it be proved that no flexible cross-polytope exists in Euclidean spaces of dimension >4 ? Geometric analysis and visualization of flexible octahedra and cross-polytopes. Relations between flexible polyhedra and Kokotsakis meshes, i.e., polygonal structures consisting of a central polygon P and a surrounding belt of polygons such that 4 faces meet at each vertex of P. One of the methods applied will be based on Ivory`s Theorem, which is valid in all spaces of constant curvature, and on a certain converse, a Configuration Theorem for bipartite frameworks. This combination has already been successfully used in the past to reprove R. Bricard`s classical result.

This project was devoted to the question under which necessary and sufficient conditions geometric structures like polyhedra or more general frameworks can be flexible, though generically they are rigid. The members of the Austrian group mainly concentrated on the conditions for the flexibility of such overconstrained structures, and they analysed the kinematics of the related mechanisms. Contrary to this, the Russian members focused on the problem, which metric or combinatorial properties remain invariant under such self-motions (flexions) and which vary. The Austrian group was most successful at the analysis of those frameworks which are used at so-called Stewart-Gough-platforms, a particular family of parallel robots. In many cases, a complete classification of all types of possible self-motions could be achieved. Also the results on the analysis of flexible quad-meshes, i.e., of polyhedral structures with quadrangular faces, raised high international attention, though a complete classification of all flexible cases is still open, even in the case of 3x3-complexes, so-called Kokotsakis meshes. A particular break-through has to be mentioned at the analysis of closed serial-chains with overconstrained mobility: It could be proved that there is a close connection with the decomposition of polynomials over the ring of dual numbers into linear factors. Based on this result a new theory for the analysis of overconstrained serial-chains was introduced, the so-called bond theory, which was also adapted for the study of Stewart-Gough-platforms with self-motions. One problem originally posed in our project is still open: Which are the flexible octahedra in the hyperbolic space (= Lobachevsky 3-space)? In the Euclidean space this problem could be solved using a particular converse of Ivorys Theorem. However, it turned out that the proof of the hyperbolic counterpart of this converse needs a cumbersome case analysis. Hence, this question was postponed in favor of the above mentioned problems, since the activities there were more fruitful as well as of higher actuality. Though, it must be mentioned that flexible octahedra have not lost their actuality; surprisingly, they often act as the true reason for self-motions of Stewart-Gough-platforms. The main results obtained by the Russian project members can be divided into two groups: The algebraic component of the problem becomes apparent at Sabitovs volume-polynomial of polyhedra. In this respect, the 60 pages long survey article on flexible polynomials must be mentioned which was published in the highly esteemed Russian Mathematical Surveys. The analytic component of the addressed problem plays a role at the study of invariants under flexions of polyhedra, in particular of suspensions.