The Banach-Tarski paradox is a seemingly bizarre and outrageous claim that it is possible to take a ball, break it into a number of pieces and then reassemble those pieces to make two identical copies of the ball. The claim can be made even stronger: it is possible to decompose a ball the size of a marble and then reassemble the pieces to make another ball the size of the Earth, or, indeed, the size of the known universe!
Before writing off Messieurs Banach and Tarski as being either very bad mathematicians or very good practical jokers, it's important to understand that this is not a claim about what can actually be done with a real ball, a sharp knife, and some dabs of glue. Nor is there any chance of some entrepreneur being able to slice up a gold ingot and assemble in its place two new ones like the original. The Banach-Tarski paradox tells us nothing new about the physics of the world around us but a great deal about how "volume," "space," and other familiar-sounding things can assume unfamiliar guises in the strange abstract world of mathematics.
Stefan Banach and Alfred Tarski announced their startling conclusion in 1924, having built on earlier work by Felix Hausdorff who proved that it's possible to chop up the unit interval (the line segment from 0 to 1) into countably many pieces, slide these bits around, and fit them together to make an interval of length 2. The Banach-Tarski paradox, which mathematicians often refer to as the Banach-Tarski decomposition because it's really a proof not a paradox, highlights the fact that among the infinite set of points that make up a mathematical ball, the concept of volume and of measure can't be defined for all possible subsets. What this boils down to is that quantities that can be measured in any familiar sense are not necessarily preserved when a ball is broken down into subsets and then those subsets reassembled in a different way using just translations (slides) and rotations (turns). These unmeasurable subsets are extremely complex, lacking reasonable boundaries and volume in the ordinary sense, and thus are not attainable in the real world of matter and energy. In any case, the Banach-Tarski paradox doesn't give a prescription for how to produce the subsets: it only proves their existence and that there must be at least five of them to produce a second copy of the original ball. The fact that the Banach-Tarski paradox depends on the axiom of choice (AC), yet is so strongly counterintuitive, has been used by some mathematicians to suggest that AC must be wrong; however, the benefits of adopting AC are so great that such black sheep of the mathematical family as the paradox are generally tolerated.
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86: 3, 1979.
2. Wagon, S. The Banach-Tarski Paradox. New York: Cambridge University Press, 1993.