"Ernst W. Mayer" wrote:
>
> Joth Tupper writes:
>
> >the Bolzano-Tarski theorem proved (what, back in the 1920's?)
> >that you could cut a solid 3D sphere into finitely many chunks,
> >then rearrange the chunks to make another solid (no holes or gaps)
> >3D sphere with _twice_ the volume. Pretty spooky, I always felt.
>
> I don't know about spooky, but 'twould seem to violate conservation
> of mass (or mass/energy, if you're a postmodern relativist :),
> 'twouldn't it?
You can map a line segment of "length" 1 onto two line segments of
length 1 with a simple mapping function f(x)=2x+1 for instance.
This maps all the points from 0 to 0.5 onto the line segment from
1 to 2 and the points from 0.5 to 1 onto 2 to 3.
One "chunk."
a "two chunk" mapping of the same points might look like this:
f(x)={
x < 0.25 && return 3+4x
x >= 0.25 && return 4+4x/3
}
They're dealing with geometric volume, not fluid volume.
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