Journal of Symbolic Logic

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

Trevor M. Wilson

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Abstract

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.

Article information

Source
J. Symbolic Logic Volume 70, Issue 3 (2005), 946-952.

Dates
First available in Project Euclid: 22 July 2005

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1122038921

Digital Object Identifier
doi:10.2178/jsl/1122038921

Mathematical Reviews number (MathSciNet)
MR2155273

Zentralblatt MATH identifier
1134.03028

Citation

Wilson, Trevor M. A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem. J. Symbolic Logic 70 (2005), no. 3, 946--952. doi:10.2178/jsl/1122038921. http://projecteuclid.org/euclid.jsl/1122038921.


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References

  • Theodorus Jozef Dekker Paradoxical decompositions of sets and spaces, Ph.D. thesis, Drukkerij Wed. G. van Soest, Amsterdam,1958.
  • 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. 10726--10728.
  • Alexander S. Kechris Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York,1995.
  • Miklós Laczkovich Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem, Journal für die Reine und Angewandte Mathematik, vol. 404 (1990), pp. 77--117.
  • Stan Wagon The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge,1985.