Objects that cannot be taken apart with two hands

ABSTRACT

It has been conjectured that every configuration C of convex objects in 3-space with disjoint interiors can be taken apart by translation with two hands: that is, some proper subset of C 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).

REFERENCES

• 1: S. T. Coffin. The Puzzling World of Polyhedral Dissections. Oxford University Press, 1991.
• 2: R. Dawson. On removing a ball without disturbing the others. Math. Mag., 57(1):27-30, Jan. 1984.
• 3: N. G. de Bruijn. Nieuw Archie{ voor Wiskunde, 2:67, 1954. Problems 17 and 18. Answers in Wiskundige Opgaven met de oplossingen, 20:19-20, 1955.
• 4: L. Fejes Toth and A. Heppes. Uber stabile KSrpersysteme. Compositio Mathematica, 15(2):119-126, 1963.
• 5: Leo J. Guibas , F. Frances Yao, On translating a set of rectangles, Proceedings of the twelfth annual ACM symposium on Theory of computing, p.154-160, April 28-30, 1980, Los Angeles, California, United States [doi>10.1145/800141.804663]
• 6: L. S. H. d. Mello. Computer-Aided Mechanical Assembly Planning. Kluwer Academic Publishers, Boston, 1991.
• 7: B. Mishra, J. T. Schwartz, and M. Sharir. On the existence and synthesis of multifinger positive grips. Aigorithmica, 2(541-558), 1987.
• 8: B. K. Natarajan, On planning assemblies, Proceedings of the fourth annual symposium on Computational geometry, p.299-308, June 06-08, 1988, Urbana-Champaign, Illinois, United States [doi>10.1145/73393.73424]
• 9: Jocelyne Pertin-Trocaz, Grasping: a state of the art, The robotics review 1, MIT Press, Cambridge, MA, 1989
• 10: F. P. Preparata. Planar point location revisited. Int. J. Found. Comp. $ci., 1(1):71-86, 1990.
• 11: R. H. Wilson and T. Matsui. Partitioning an assembly for infinitesimal motions in translation and rotation. In IEEE International Conference on Intellegent Robots and Systems, pages 1311-1318, 1992.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Geometrical problems and computations

Additional Classification:
  F. Theory of Computation
  F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY
      F.2.2 Nonnumerical Algorithms and Problems
          Subjects: Computations on discrete structures

General Terms:
Algorithms, Theory, Verification