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

@article{Wilson2005ACM, title={A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's Problem}, author={Trevor M. Wilson}, journal={Journal of Symbolic Logic}, year={2005}, volume={70}, pages={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…

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