>  Ever heard of the Banach-Tarski paradox? A solid ball in 3-dimensional
> space can be split into several non-overlapping pieces, which can then be
> put back together in a different way to yield two identical copies of
> the original ball.

I hadn't heard of that one, but it makes perfect sense in a topology 
context.
In that context "split" really means a 1:2 mapping of spaces.
Cut the ball in half, then in the two hemispheres the new equator plane
is a curve from the lip that follows a 0.5*y scaled sphere, from there
just map the curved planes back to flat to get two spheres.
Since we're talking about a mapping, not a real object, there's no
conservation of volume, a sphere is a sphere, no matter what the size.

The paradox is that the more mathematicians learn,
the less they are able to explain clearly.

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