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A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's Problem

  • Trevor M. Wilson (a1)

In 1924 Banach and Tarski demonstrated the existence of a paradoxical decomposition of the 3-ball B, i.e., a piecewise isometry from B onto two copies of B. This article answers a question of de Groot from 1958 by showing that there is a paradoxical decomposition of B in which the pieces move continuously while remaining disjoint to yield two copies of B. More generally, we show that if n > 2, any two bounded sets in Rn that are equidecomposable with proper isometries are continuously equidecomposable in this sense.

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[2] Randall Dougherty and Matthew Foreman , Banach-Tarski paradox using pieces with the property of Baire, Proceedings of the National Academy of Sciences of the United States of America, vol. 89 (1992), no. 22, pp. 1072610728.

[3] Alexander S. Kechris , Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.

[6] Stan Wagon , The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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