A Continuous Movement Version of the Banach-Tarski Paradox: A Solution to de Groot's Problem

Journal of Symbolic Logic 70 (3):946 - 952 (2005)
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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