The Banach-Tarski Paradox

Yes, it is true. It is possible to take a cannonball apart into finitely many pieces, and put those pieces together to form a solid sphere the size of the sun. This is the Banach-Tarski Paradox, invented by the mathematician Stefan Banach and the logician Alfred Tarski in 1924. In fact it is true that any solid 3-dimensional object can be taken apart in this way and the pieces put together to form any other solid 3-D object of ANY size and shape. So for instance a pencil could be taken apart and put back together to form wooden Eiffel Tower.

As you may have guessed, the pieces are not particularly nice: they are, in some sense, inconceivably complicated. Of course, the Banach-Tarski Paradox is a mathematical theorem, and is not true in the real physical world. Or is it? The reason usually given for its counter-intuitive nature is that it relies on an axiom of mathematics called the Axiom of Choice. But it has been pointed out that there are other such paradoxical theorems which do not depend on the Axiom of Choice. Another argument is that the pieces are so complicated that they do not have a definite volume as we usually define it in mathematics (called Lebesgue Measure).

Whatever the truth of the matter, though, this is a surprising and fascinating piece of mathematics which raises deep questions about our understanding of the world.