Tim Attwood <tim### [at] comcastnet> wrote:
> 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.

  That doesn't work. The volume of the sphere cannot be modified by a
simple change in topology. You cannot simply change the topology and
then calculate the volume as if you hadn't. You have to calculate the
volume using the *new* topology, not the old one.

  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.

  And besides, the original setup happens in regular cartesian coordinates,
without any change in topology.

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

  You clearly haven't understood the theorem.

-- 
                                                          - Warp

Post a reply to this message