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Mysterious Movable Models

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References

  1. Chen, H.-W. 1991, Kinematics and introduction to dynamics of a movable pair of tetrahedra. M. Eng. Thesis, Dept. of Mechanical Engineering, McGill University, Montreal, Canada. This provides a thorough description of the kinematics and dynamics of the pair of tetrahedra applied in a novel robotic manipulator called double tetrabot.

  2. Fuller, R. B. 1975, Synergetics: Exploration in the geometry of thinking, p. 7. New York: Macmillan.

  3. Hilton, P., Pedersen, J. 2010, Mathematical Tapestry: Demonstrating the Beautiful unity of Mathematics, Cambridge University Press. Figures 5 and 6 are reproduced, with permission, from this book.

  4. Howard, Ian P. 1978–1979, Articulating models, J. Recreational Mathematics, 11:3, 190–195.

  5. Hyder, A., Zsombor-Murray, P. J. 1990, Design, mobility analysis and animation of a double equilateral tetrahedral mechanism. In Proceedings of International Symposium on Robotics and Manufacturing, ASME Press Series, Vol. 3, ISSN 1052-4150, pp. 49–56. This shows how to design a real double tetrahedral mechanism satisfying precisely all geometrical requirements (e.g., exact intersection of corresponding edges) and how to use it as a robotic joint.

  6. Makai, E., Tarnai, T. 2000, Overconstrained sliding mechanisms. In IUTAM-IASS Symposium on Deployable Structures: Theory and Applications. Proceedings of the IUTAM Symposium held in Cambridge, U.K., 6–9 September 1998 (S. Pellegrino, S. D. Guest, eds.), pp. 261–270. Dordrecht: Kluwer Academic Publishers. This presents a generalization of the pair of tetrahedra. It investigates motions of bar structures consisting of two congruent tetrahedra, whose edges are defined by face diagonals of a rectangular parallelepiped.

  7. Stachel, H. 1988, Ein bewegliches Tetraederpaar. Elemente der Mathematik 43, 65–75. This provides a geometrical proof pertaining to finite motions of the pair of teterahedral frameworks, and maps typical surfaces traced by an edge midpoint and by a vertex.

  8. Tarnai, T., Makai, E. 1988, Physically inadmissible motions of a movable pair of tetrahedra. In Proceedings of the Third International Conference on Engineering Graphics and Descriptive Geometry (S. M. Slaby, H. Stachel, eds.), vol 2, pp. 264–271. Vienna: Technical University. This paper deals with those motions of the pair of tetrahedra that cannot be realized by a physical model, that is, with motions where the points of intersection of straight lines of some respective edges are not internal points of the edges.

  9. Tarnai, T., Makai, E. 1989a, A movable pair of tetrahedra. Proceedings of the Royal Society of London A423, 419–442. This proves the existence of all finite motions of a pair of tetrahedra.

  10. Tarnai, T. Makai, E. 1989b, Kinematical indeterminacy of a pair of tetrahedral frames. Acta Technica Acad. Sci. Hung. 102, 123–145. This investigates infinitesimal movability of a pair of tetrahedra.

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Correspondence to Jean Pedersen.

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Pedersen, J., Tarnai, T. Mysterious Movable Models. Math Intelligencer 34, 62–66 (2012). https://doi.org/10.1007/s00283-012-9302-3

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