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.
"A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem." J. Symbolic Logic 70 (3) 946 - 952, September 2005. https://doi.org/10.2178/jsl/1122038921