The Journal of Symbolic Logic

Research Article

A continuous movement version of the Banach–Tarski paradox: A solution to de Groot's Problem

Trevor M. Wilson

Department of Mathematics, California Institute of Technology, Pasadena, CA91125, USA E-mail:,


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 R n that are equidecomposable with proper isometries are continuously equidecomposable in this sense.

(Received January 20 2005)

(Accepted May 09 2005)