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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:, [email protected]
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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

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