Objects that cannot be taken apart with two hands
It has been conjectured that every configurationC of convex objects in 3-space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset ofC can be translated to infinity without disturbing its complement. We show that the conjecture holds for five or fewer objects and give a counterexample with six objects. We extend the counterexample to a configuration that cannot be taken apart with two hands using arbitrary isometries (rigid motions).
Unable to display preview. Download preview PDF.
- 2.S. T. Coffin.The Puzzling World of Polyhedral Dissections, Oxford University Press, Oxford, 1991.Google Scholar
- 3.R. H. Crowell and R. H. Fox.Introduction to Knot Theory, Blaisdell, New York, 1965.Google Scholar
- 5.N. G. de Bruijn. Problems 17 and 18.Nieuw Archief voor Wikskunde, 2: 67, 1954, Answers inWiskundige Opgaven met de oplossingen, 20: 19–20, 1955.Google Scholar
- 7.J. B. Fraleigh.A First Course in Abstract Algebra. Addison-Wesley, Reading, MA, 1982.Google Scholar
- 8.L. J. Guibas and F. F. Yao. On translating a set of rectangles.Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pages 154–160, 1980.Google Scholar
- 12.B. K. Natarajan. On planning assembles.Proceedings of the Fourth Annual ACM Symposium on Computational Geometry, pages 299–308, 1988.Google Scholar
- 14.J. Pertin-Troccaz. Grasping: A state of the art. In O. Khatib, J. J. Craig, and T. Lozano-Perez, editors,The Robotics Review 1, pages 71–98. MIT Press, Cambridge, MA, 1989.Google Scholar
- 16.J. Snoeyink. Video: Objects that cannot be taken apart with two hands.Proceedings of the Ninth Annual ACM Symposium on Computational Geometry, page 405, 1993. Video Review of Computational Geometry also available as DEC SRC Report 101. 4:39 animation.Google Scholar
- 17.R. H. Wilson and T. Matsui. Partitioning an assembly for infinitestimal motions in translation and rotation.IEEE International Conference on Intellegent Robots and Systems, pages 1311–1318, 1992.Google Scholar