Please note, due to essential maintenance online transactions will not be possible between 01:30 and 03:00 BST, on Tuesday 28th January 2020 (19:30-21:00 EDT). We apologise for any inconvenience.
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.