Differential Geometry and Geometric Structures

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Flexible Structures

We investigate polyhedra and frameworks which are flexible or infinitesimally flexible, and problems related to the bellows conjecture.

Steffen's polyhedron

R. Connelly constructed a flexible polygonal embedding of the 2-sphere into the E³ in 1977. A simplified flexing sphere was presented by K. Steffen in 1978. The unfolding of Steffen's polyhedra is given above. Note that both flexing spheres are compound of Bricard octahedra which all have self-intersections.

Bricard

R. Bricard proved in 1897 that there are three types of flexible octahedra in E³.
Here both flat poses of a Bricard octahedron of type 3 are illustrated. Note that Bricard octahedra keep their volume constant during the flex. This is due to the Bellows Conjecture which was proven by I. Sabitov in the year 1996.

Flexible Polyhedra and Frameworks in Different Spaces

This project is part of an international cooperation funded by the Austrian Science Fund (FWF) and the Russian Foundation for Basic Research (RFBR).

FWF grant no. I 408-N13

RFBR grant no. 10-01-91000-ANF

Duration: 2010-2013

FWF-Funding: € 298 966.50

Austrian Members

Hellmuth Stachel, TU Vienna, Institute of Discrete Mathematics and Geometry
Email: stachel@geometrie.tuwien.ac.at
Hans-Peter Schröcker, University of Innsbruck, Unit Geometry and CAD
Email: hans-peter.schroecker@uibk.ac.at
Georg Nawratil, TU Vienna, Institute of Discrete Mathematics and Geometry
Email: nawratil@geometrie.tuwien.ac.at
Ekaterina Zaputryaeva , TU Vienna, Institute of Discrete Mathematics and Geometry
Email: katya@geometrie.tuwien.ac.at

Russian Members

Idzhad Sabitov, Moscow State University, Faculty of Mechanics and Mathematics
Email: isabitov@mail.ru
Victor Alexandrov, Novosibirsk State University, Sobolev Institute of Mathematics
Email: alex@math.nsc.ru
Sergey Mikhalev, Moscow State University, Faculty of Mechanics and Mathematics
Email: mikhalev@bk.ru
Dmitriy Slutskiy, Novosibirsk State University, Sobolev Institute of Mathematics
Email: dmitry.slutsky@gmail.com

Aims and Scope

The joint project is devoted to the study of 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 modelling or for the analysis of isomers in chemistry.

Project Publications

G. Nawratil: Stewart Gough platforms with linear singularity surface. In Proc. of 19th IEEE International Workshop on Robotics in Alpe-Adria-Danube Region (RAAD'2010), pages 231-235, ISBN 978-1-4244-6884-3.
G. Nawratil and H. Stachel: Composition of spherical four-bar-mechanisms. New Trends in Mechanisms Science (D. Pisla et al. eds.), pages 99-106, Springer, 2010, ISBN 978-90-481-9688-3.
V. Alexandrov. Algebra versus analysis in the theory of flexible polyhedra. Aequationes Mathematicae 79 (3) 229-235 (2010) arXiv:0902.0186v2 [math.MG], Erratum: Aequationes Mathematicae 81 (1) 199 (2011).
D. Slutskiy. An infinitesimally nonrigid polyhedron with nonstationary volume in the Lobachevsky 3-space. Sibirsk. Mat. Zh. 52 (1) 167-176 (2011) arXiv:1002.3884v3 [math.MG].
G. Nawratil: Reducible compositions of spherical four-bar linkages with a spherical coupler component. Mechanism and Machine Theory 46 (5) 725-742 (2011).
H. Stachel. The Influence of Geometry on the Rigidity or Flexibility of Structures. Proc. IWSSIP 2010 - 17th Internat. Conf. on Systems, Signals and Image Processing, June 2010, Rio de Janeiro/Brazil, (ISBN 978-85-228-0565-5) pages 24-29.
G. Nawratil: Self-motions of TSSM manipulators with two parallel rotary axes. ASME Journal of Mechanisms and Robotics 3 (3) 031007 (2011).
G. Nawratil: Flexible octahedra in the projective extension of the Euclidean 3-space. Journal for Geometry and Graphics 14 (2) 147-169 (2010).
G. Nawratil: Types of self-motions of planar Stewart Gough platforms. Meccanica 48 (5) 1177-1190 (2013).
G. Nawratil: Basic result on type II DM self-motions of planar Stewart Gough platforms. Mechanisms, Transmissions and Applications (E.Ch. Lovasz, B. Corves), pages 235-244, Springer, 2011, ISBN 978-94-007-2726-7.
G. Nawratil: Reducible compositions of spherical four-bar linkages without a spherical coupler component. Mechanism and Machine Theory 49 87-103 (2012).
H. Stachel. Remarks on flexible quad meshes. Proceedings 11th Internat. Conf. on Engineering Graphics - BALTGRAF-11, June 2011, Tallinn/Estonia, 84-92 (ISBN 978-9949-23-112-6).
H. Stachel. What lies between the flexibility and rigidity of structures. Serbian Architectural Journal 3/2, 102-115 (2011).
H. Stachel. On the Rigidity of Polygonal Meshes. South Bohemia Mathematical Letters 19/1 (1911).
G. Nawratil: Necessary conditions for type II DM self-motions of planar Stewart Gough platforms. Journal for Geometry and Graphics 16 (2) 139-151 (2012).
G. Nawratil: Planar Stewart Gough platforms with a type II DM self-motion. Journal of Geometry 102 (1) 149-169 (2011)
G. Nawratil: Self-motions of planar projective Stewart Gough platforms. Latest Advances in Robot Kinematics (J. Lenarcic, M. Husty eds.), pages 27-34, Springer, 2012, ISBN 978-94-007-4619-0.
V. Alexandrov: The Dehn invariants of the Bricard octahedra. Journal of Geometry 99 (1-2) 1-13 (2010).
V. Alexandrov: New manifestations of the Darboux's rotation and translation fields of a surface. New Zealand Journal of Mathematics 40 59-65 (2010).
I.Kh. Sabitov: Algebraic methods for solution of polyhedra. Russian Math. Surveys 66 (3) 445-505 (2011).
G. Nawratil: Review and recent results on Stewart Gough platforms with self-motions. Applied Mechanics and Materials 162 151-160 (2012).
G. Hegedüs, J. Schicho and H.-P. Schröcker: Construction of Overconstrained Linkages by Factorization of Rational Motions. Latest Advances in Robot Kinematics (J. Lenarcic, M. Husty eds.), pages 213-220, Springer, 2012, ISBN 978-94-007-4619-0.
G. Hegedüs, J. Schicho and H.-P. Schröcker: Bond Theory and Closed 5R Linkages. Latest Advances in Robot Kinematics (J. Lenarcic, M. Husty eds.), pages 221-228, Springer, 2012, ISBN 978-94-007-4619-0.
V. Alexandrov and R. Connelly: Flexible suspensions with a hexagonal equator. Illinois Journal of Mathematics 55 (1) 127-155 (2011).
V. Alexandrov: Continuous deformations of polyhedra that do not alter the dihedral angles. arXiv:1212.4676v1 [math.MG] (2012).
D. Slutskiy: A necessary flexibility condition of a nondegenerate suspension in Lobachevsky 3-space. arXiv:1208.2793v1 [math.MG] (2012).
G. Nawratil: On elliptic self-motions of planar projective Stewart Gough platforms. Transactions of the Canadian Society for Mechanical Engineering, in press.
G. Nawratil: Non-existence of planar projective Stewart Gough platforms with elliptic self-motions. Computational Kinematics (F. Thomas, A. Perez Garcia eds.), Springer, 2013, in press.
G. Nawratil: Flexible Oktaeder mit Fernelementen. Informationsblätter der Geometrie IBDG 31 (2) 28-30 (2012).

Related Publications

H. Stachel. Zur Einzigkeit der Bricardschen Oktaeder. J. Geom. 28 (1987), 41-56. [Zbl], [MR].
I. Sabitov. Local theory on bendings of surfaces. In Geometry III. Theory of surface, volume 48 of Encycl. Math. Sci., pages 179-250. 1992. translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 48, 196-270 (1989). [MR].
V. Alexandrov. An example of a flexible polyhedron with nonconstant volume in the spherical space. Beitr. Algebra Geom. 38/1 (1997), 11-18. [MR].
R. Connelly, I. Sabitov, and A. Walz. The Bellows conjecture. Beitr. Algebra Geom. 38/1 (1997), 1-10. [MR].
V. Alexandrov. Sufficient conditions for the extendibility of an n-th order flex of polyhedra. Beitr. Algebra Geom. 39/2 (1998), 367-378. [MR].
I. Sabitov. On some recent results in the metric theory of polyhedra. Rend. Circ. Mat. Palermo, II. Ser., Supplemento 65/II (2000), 167-177, Proceedings of the Third international conference in stochastic geometry, convex bodies and empirical measures, held in Mazara del Vallo, Italy, May 24-29, 1999. P. M. Gruber (Ed.). [MR].
H. Stachel. Higher order flexibility of octahedra. Period. Math. Hungar. 39 (1999), 225-240, Discrete geometry and rigidity (Budapest, 1999). [Zbl], [MR].
S. Mikhalev. Some necessary metric conditions for flexibility of suspensions. Mosc. Univ. Math. Bull. 56/3 (2001), 14-20. [MR].
H. Stachel. Flexible cross-polytopes in the Euclidean 4-space. J. Geom. Graphics 4 (2000), 159-167. [Zbl], [MR].
H. Stachel. Remarks on Bricard's flexible octahedra of type 3. In Proc. 10th ICGG, volume 1, pages 8-12, Kyiv, 2002. ISBN 966-96185-2-5. held in Kiev (Ukraine), July 28 -- Aug. 3, 2002.
H. Stachel. Ivory's theorem in the Minkowski plane. Math. Pannon. 13 (2002), 11-22. [Zbl], [MR].
H. Stachel and J. Wallner. Ivory's theorem in hyperbolic spaces. Sib. Math. J. 45 (2004), 785-794. [Zbl], [MR].
H. Stachel. Flexible Octahedra in the Hyperbolic Space. Proc. Janos Bolyai Conference in Hyperbolic Geometry, July 8-12, 2002, Budapest.
In A. Prekopa, E. Molnár (eds.), Non-euclidean geometries: János Bolyai memorial volume, Mathematics and its applications, vol. 581, Springer, New York 2006 (ISBN 0-387-29554-2) [MR]: pages 209-225, 2006. [MR].
V. Alexandrov, I.Kh. Sabitov and H. Stachel (eds.): Rigidity and Related Topics in Geometry. Special issue of European J. Combinatorics 31 (4) 1035-1204 (2010).
H. Stachel. A kinematic approach to Kokotsakis meshes. Comput. Aided Geom. Des., 27 428-437 (2010).