Abstract
The paradox of Banach, Tarski, and Hausdorff shows that any two bounded sets M,N \subseteq E 3 with non-empty interior are equidecomposable. The result remains true if M and N are replaced by collections of sets. We present quantified versions of the paradox by giving estimates for the minimal number of pieces in such decompositions. The emphasis is on replications of sets M , i.e., on the equidecomposability of M with k copies of M , k ≥ 2 . In particular, we discuss the problem of replicating the cube.
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Received January 21, 1999, and in revised form March 23, 2000. Online publication October 10, 2000.
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Richter, C. Simple Paradoxical Replications of Sets . Discrete Comput Geom 25, 65–83 (2001). https://doi.org/10.1007/s004540010077
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DOI: https://doi.org/10.1007/s004540010077