Tim Attwood <tim### [at] comcastnet> wrote:
> Well, the volume is half for the new spheres, but there is a 1:1 mapping
> from each of the new spheres to the original.

  But the thing is, the Banach-Tarski paradox states that you can slice
the sphere into a finite amount of pieces and then, using translations
and rotations only you can construct two spheres of the same size.
Everything in cartesian coordinates. No topology changes.

  Besides, your "1:1 mapping" doesn't double the volume of the sphere.
The Banach-Tarski paradox does.

> >  Besides, if what you say was true, the same trick would work with a
> > 2-dimensional circle, but it has been proven that it doesn't.

> The same slicing should work with circles, what did who
> disprove where?

  Just read the theorem.

-- 
                                                          - Warp

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