## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 70, Issue 3 (2005), 946-952.

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

#### 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.

#### Article information

**Source**

J. Symbolic Logic Volume 70, Issue 3 (2005), 946-952.

**Dates**

First available in Project Euclid: 22 July 2005

**Permanent link to this document**

http://projecteuclid.org/euclid.jsl/1122038921

**Digital Object Identifier**

doi:10.2178/jsl/1122038921

**Mathematical Reviews number (MathSciNet)**

MR2155273

**Zentralblatt MATH identifier**

1134.03028

#### Citation

Wilson, Trevor M. A continuous movement version of the Banach—Tarski paradox: A solution to de Groot's Problem. J. Symbolic Logic 70 (2005), no. 3, 946--952. doi:10.2178/jsl/1122038921. http://projecteuclid.org/euclid.jsl/1122038921.