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

Published online by Cambridge University Press:  12 March 2014

Trevor M. Wilson
Trevor M. Wilson*
Affiliation:
Department of Mathematics, California Institute of Technology, Pasadena, CA91125, USAE-mail:, trev@its.caltech.edu
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 946 - 952
Copyright
Copyright © Association for Symbolic Logic 2005

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References

REFERENCES

[1]Dekker, Theodorus Jozef, Paradoxical decompositions of sets and spaces, Ph.D. thesis, Drukkerij Wed. G. van Soest, Amsterdam, 1958.Google Scholar
[2]Dougherty, Randall and Foreman, Matthew, 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.CrossRefGoogle ScholarPubMed
[3]Kechris, Alexander S., Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.CrossRefGoogle Scholar
[4]Laczkovich, MiklÓs, Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem, Journal fÜr die Reine und Angewandte Mathematik, vol. 404 (1990), pp. 77–117.Google Scholar
[5]Laczkovich, MiklÓs, Equidecomposability of sets, invariant measures, and paradoxes, Rendiconti dell'lstituto di Matematica dell'Università di Trieste, vol. 23 (1991), no. 1, pp. 145–176, (1993), School on Measure Theory and Real Analysis (Grado, 1991).Google Scholar
[6]Wagon, Stan, The Banach-Tarski paradox, Encyclopedia of Mathematics and its Applications, vol. 24, Cambridge University Press, Cambridge, 1985.CrossRefGoogle Scholar