> Mitchell Porter <mitch@thehub.com.au> writes:
> >Suppose you get a sheet of paper and roll it up from bottom to top.
> >The left side is not any closer to the right side than it was.
> >In the same way, having compactified extra dimensions does not
> >in itself mean that A is closer to B than it looks. I don't
> >entirely discount the possibility of what you're talking about,
> >but it would require geometries of a sort not presently contemplated.
>
> Not at all. Go to altavista (http://altavista.digital.com) and search
> for "space filling curves". There you will find ways to squeeze an
> infinitely long 1-dimensional curve into a finite 2-dimensional space
> (like a square). Although this doesn't put *every* point on the curve
> infinitely close to every other point, it does put infinitely many
> points at arbitrarily large distances along the curve, arbitrarily close
> to each other in the square. This is just intended to give you an
> "intuition" about how you can fold up a dimension to do the things
> described previously.
This is an interestingly bizarre idea for a cosmology, but
what I meant was "not presently contemplated by physicists".
All the Kaluza-Klein-type models of the real world that have
been proposed involve product spaces of the form M x S,
where M is a "large" 4-D spacetime and S is a compact
manifold, constructible by substituting a copy of S for
each point in M. In these models, the extra dimensions
do not offer a shortcut.
-mitch
http://www.thehub.com.au/~mitch