Periodica Mathematica Hungarica

, Volume 39, Issue 1–3, pp 225–240 | Cite as


  • Hellmuth Stachel


More than hundred years ago R. Bricard determined all continuously flexible octahedra. On the other hand, also the geometric characterization of first-order flexible octahedra has been well known for a long time. The objective of this paper is to analyze the cases between, i.e., octahedra which are infinitesimally flexible of order n > 1 but not continuously flexible. We prove explicit necessary and sufficient conditions for the orders two, three and even for all n < 8, provided the octahedron under consideration is not totally flat. Any order ≥ 8 implies already continuous flexibility, as the configuration problem for octahedra is of degree eight.


Geometric Characterization Configuration Problem Order Flexibility Flexible Octahedra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. T. Bennet, Deformable Octahedra, Proc. London math. soc., Sec. Series 10 (1912), 309–343.Google Scholar
  2. [2]
    W. Blaschke, Über affine Geometrie XXVI: Wackelige Achtflache, Math. Z. 6 (1920), 85–93.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    O. Bottema and B. Roth, Theoretical Kinematics, North-Holland Publishing Company, Amsterdam, 1979.zbMATHGoogle Scholar
  4. [4]
    R. Bricard, Mémoire sur la théorie de l'octaèdre articulé, J. math. pur. appl., Liouville 3 (1897), 113–148.zbMATHGoogle Scholar
  5. [5]
    R. Bricard, Leçon de cinématique II, Paris, 1927.Google Scholar
  6. [6]
    R. Connelly and H. Servatius, Higher-order rigidity — What is the proper definition? Discrete Comput. Geom. 11 (1994), no. 2, 193–200.zbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Karger and M. Husty, On Self-Motions of a Class of Parallel Manipulators, in: J. Lenarčič, V. Parenti-Castelli (eds.): Recent Advances in Robot Kinematics, Kluwer Acad. Publ., 1996, pp. 339–348.Google Scholar
  8. [8]
    A. Karger and M. Husty, Singularities and Self-motions of Stewart-Gough Platforms, in: J. Angeles, E. Zakhariev (eds.): Computational methods in Kinematics, NATO Advanced Study Institute, Varna, Bulgaria, June 1997, vol. II, 279–288.Google Scholar
  9. [9]
    I. Kh. Sabitov, Local Theory of Bendings of Surfaces, in: Yu.D. Burago, V. A. Zalgaller (eds.): Geometry III, Theory of Surfaces, Encycl. of Math. Sciences, vol. 48, Springer-Verlag, 1992, pp. 179–250.Google Scholar
  10. [10]
    J. Graver, B. Servatius and H. Servatius, Combinatorial rigidity, Graduate Studies in Mathematics, vol. 2, American Mathematical Society, Providence, 1993.zbMATHGoogle Scholar
  11. [11]
    H. Stachel, Bemerkungen über zwei räumliche Trilaterationsprobleme, Z. Angew. Math. Mech. 62 (1982), 329–341.zbMATHMathSciNetGoogle Scholar
  12. [12]
    H. Stachel, Zur Einzigkeit der Bricardschen Oktaeder, J. Geom. 28 (1987), 41–56.zbMATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    H. Stachel, Infinitesimal Flexibility of Higher Order for a Planar Parallel Manipulator, Institut für Geometrie, TU Wien, Technical Report 64 (1999).Google Scholar
  14. [14]
    H. Stachel, Higher-Order Flexibility of a Bipartite Planar Framework, in: A. Kecskeméthy, M. Schneider and C. Woernle (eds.): Advances in Multibody Systems and Mechatronics, Inst. f. Mechanik und Getriebelehre, TU Graz, Duisburg, 1999 (ISBN 3–9501108–0–1), pp. 345–357.Google Scholar
  15. [15]
    W. Wunderlich, Starre, kippende, wackelige und bewegliche Achtflache, Elem. Math. 20 (1965), 25–32.zbMATHMathSciNetGoogle Scholar
  16. [16]
    W. Wunderlich, Fast bewegliche Oktaeder mit zwei Symmetrieebenen, Rad Jugosl. Akad. Zagreb 428, Mat. Znan. 6 (1987), 129–135.zbMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Hellmuth Stachel

There are no affiliations available

Personalised recommendations