Banach-Tarski Paradox

First stated in 1924, the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. The number of pieces was subsequently reduced to five by Robinson (1947), although the pieces are extremely complicated. (Five pieces are minimal, although four pieces are sufficient as long as the single point at the center is neglected.) A generalization of this theorem is that any two bodies in  that do not extend to infinity and each containing a ball of arbitrary size can be dissected into each other (i.e., they are equidecomposable).

REFERENCES

Banach, S. and Tarski, A. "Sur la décomposition des ensembles de points en parties respectivement congruentes." Fund. Math. 6, 244-277, 1924.

Czyz, J. Paradoxes of Measures and Dimensions Originating in Felix Hausdorff's Ideas. Singapore: World Scientific, 1993.

Erickson, G. W. and Fossa, J. A. Dictionary of Paradox. Lanham, MD: University Press of America, pp. 16-17, 1998.

French, R. M. "The Banach-Tarski Theorem." Math. Intell. 10, No. 4, 21-28, 1988.

Gardner, M. The Sixth Book of Mathematical Games from Scientific American. Chicago, IL: University of Chicago Press, p. 48, 1984.

Hertel, E. "On the Set-Theoretical Circle-Squaring Problem." http://www.minet.uni-jena.de/Math-Net/reports/sources/2000/00-06report.ps.Kirsch, A. "Das Paradoxon von Hausdorff, Banach und Tarski: Kann man es 'verstehen'?" Math. Semesterber. 37, 216-239, 1990.

Robinson, R. M. "On the Decomposition of Spheres." Fund. Math. 34, 246-260, 1947.

Sierpiński, W. "On the Congruence of Sets and their Equivalence by Finite Decomposition." In Congruence of Sets and Other Monographs. New York: Chelsea.Stromberg, K. "The Banach-Tarski Paradox." Amer. Math. Monthly 86, 3, 1979.

Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 103, 2004.

 http://www.mathematicaguidebooks.org/.Wagon, S. "A Hyperbolic Interpretation of the Banach-Tarski Paradox." Mathematica J. 3, 58-60, 1993.Wagon, S. The Banach-Tarski Paradox. New York: Cambridge University Press, 1993.