A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem
Trevor M. Wilson
Source: J. Symbolic Logic
Volume 70, Issue 3
(2005), 946-952.
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.
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Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1122038921
Digital Object Identifier: doi:10.2178/jsl/1122038921
Mathematical Reviews number (MathSciNet):
MR2155273
References
Theodorus Jozef Dekker Paradoxical decompositions of sets and spaces, Ph.D. thesis, Drukkerij Wed. G. van Soest, Amsterdam ,1958.
Mathematical Reviews (MathSciNet):
MR121696
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.
Mikl\' os 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.
Mathematical Reviews (MathSciNet):
MR803509