Abstract
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).
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References
V. Chvátal,Linear Programming. Freeman, San Francisco, CA, 1983.
S. T. Coffin.The Puzzling World of Polyhedral Dissections, Oxford University Press, Oxford, 1991.
R. H. Crowell and R. H. Fox.Introduction to Knot Theory, Blaisdell, New York, 1965.
R. Dawson. On removing a ball without disturbing the others.Mathematics Magazine, 57 (1): 27–30, 1984.
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.
L. Fejes-Toth and A. Heppes. Uber stabile Körpersysteme,Compositio Mathematica, 15 (2): 119–126, 1963.
J. B. Fraleigh.A First Course in Abstract Algebra. Addison-Wesley, Reading, MA, 1982.
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.
P. McMullen and G. C. Shephard,Convex Polytopes and the Upper Bound Conjecture. Cambridge University Press, Cambridge, 1971.
L. S. H. d. Mello.Computer-Aided Mechanical Assembly Planning. Kluwer, Boston, 1991.
B. Mishra, J. T. Schwart, and M. Sharir. On the existence and synthesis of multifinger positive grips.Algorithmica, 2: 541–558, 1987.
B. K. Natarajan. On planning assembles.Proceedings of the Fourth Annual ACM Symposium on Computational Geometry, pages 299–308, 1988.
C. H. Papadimitriou and K. Steiglitz,Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, 1982.
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.
F. P. Preparata. Planar point location revisited.International Journal of Foundations of Computer Science, 1 (1): 71–86, 1990.
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.
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.
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The research of J. Snoeyink was supported in part by an NSERC Research Grant. J. Stolfi was previously at DEC Systems Research Center, Palo Alto, CA, USA.
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Snoeyink, J., Stolfi, J. Objects that cannot be taken apart with two hands. Discrete Comput Geom 12, 367–384 (1994). https://doi.org/10.1007/BF02574386
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DOI: https://doi.org/10.1007/BF02574386