No contradiction?

You would think that there must be an error in the construction of the puzzles because it looks like we doubled a volume. The forty pieces from the ball when taken together give a volume of 4pi/3 (the volume of one ball) but also 8pi/3 (the volume of two balls). This means that the volumes of the pieces add up to 4pi/3 and to 8pi/3.

There is no error in the construction but in the argument against it: the argument contains the tacit assumption that the puzzle pieces have a volume. But it was exactly the purpose of Hausdorff, Banach and Tarski to show that that assumption is wrong. The correct conclusion is that the puzzle pieces must be so weird that you cannot assign a volume to them in any sensible way. If you were to, mentally, dip them in a glass of water their volumes would all seem to be 4pi/3, but if you were to, again mentally, try and fill them with water you wouldn't be able to get anything in.
Last modified: Wednesday 26-02-2003 at 00:13:19 (CET)