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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