🚀 go-pugleaf

Matt Kellner wrote:
> ag...@connect.ab.ca (Aden Grue) wrote:
> >Actually, In Stephen Hawking's book (I can't remember the exact
> >title), he explained the theory that the universe doesn't end, but it
> >also isn't infinite;  rather, it's a sphere.  It's like, if you could
> >fit all of our 3 dimensional space onto the 2d surface of a sphere,
> >that would be the universe.  So basically, if you travel far enough in
> >one direction, you'll end up where you started.
> Here's a question for ya, though: If that's true, what's on the
> outside of this theoretical sphere?  Logic dictates that there must be
> something on the other side of the theoretical boundary of the
> universe (whatever/wherever that might be).
You've got your dimensions confused. All three of them are encapsulated
on the surface of this sphere, so to go off the surface would require
movement in the fourth dimension. 
> For that matter, if you tried to go from the center of this universe
> out to the edge and beyond, would you be able to do so?
There is no edge to the surface of a sphere. Once you'd managed to
figure out how to define the "center", you could go in any direction,
and would eventually wind up where you started. 
> I think the train of thought that I'm following here points out a
> shortcoming in the "spherical" curvature-of-space theory.  It would
> make more sense to think of the Universe as an extension of the
> Moebius loop.
That's essentially what this is, it's just harder to visualize. In the
mobius loop, your taking a two-dimensional object (the surface of a
peice of paper) and manipulating it in three dimensions so that one of
its axes is continuous (the whole "if you go far enough you'll end up
where you started again" deal). The surface of the sphere is trying to
say, okay, now take all three dimensions, move them around in at least
four dimensions to make them _all_ continuous. That's a lot harder to
visualize, and the sphere itself is really just an analogy like your
mobius strip, saying the same thing, only in a different way. With the
sphere theory, you could go in _any_ direction and come back where you
started. With the mobius strip, you would have to move parallel to the
edge of your paper.
> To construct one yourself, cut a piece of paper into a strip, hold
> both ends of the strip and flip one of them around ONCE, then tape the
> ends together.  Now, draw a line along the middle of the strip from
> start to finish.  You traverse the loop exactly TWICE in order to end
> up where you started, but you pass the starting point in the process.
> On that first time around, you end up ON THE OTHER SIDE of the strip
> of paper from the starting point.
There's a problem with that logic: space has no other side. Each side of
your piece of paper is a separate two-dimensional object. The mobius
strip is made by linking each one to the other at opposite ends, making
only one two-dimensional object, but the two places on opposite sides of
the paper _are_in_totally_different_two_dimensional_positions_.
> The analogy I'm drawing here has been thrown around in astronomical
> theory for a while - if you were to travel infinitely in a straight
> line in space, you'd eventually end up where you started, but you'd be
> inverted somehow (probably left-to-right, ie. everything now reads
> backwards) with respect to the rest of the world.
Once again, there is no "other side" to the universe that would put you
in the same place only reversed. The "other side" of our
universe-as-mobius strip would in actuality be a completly different
point in three-dimensional space. Now, if you wanted to come around to
the same point again, only reversed along one axis, you would have to
induce more than a "half-twist" to the universe. I think it would take
one-and-a-half(?) to do that on the regular mobius strip.
> I don't know much more than that, and I certainly don't know how the
> theory states that all this would happen.  I just know that this
> particular theory seems to make more sense to me than the one stating
> that the universe is a 3-dimensional environment.
Makes a lot more sense? Not to me. Perhaps it is easier to visualize,
because you aren't trying to extend it past that two-dimensional sided
piece of paper with one continuous axis. The universe would have to have
all three dimensions warped in such a way that all three of its axes
were continuous. Just imagining that two-dimensional mobius strip
connected along _both_ axes is very interesting. Imagining three
dimensions arranged on the surface of a sphere is a bit easier for me.
But I think in order to actually reverse something through one axis, you
would need more dimensions to play with (about 6?), and imagining things
that complex is just messy.

The mobius strip really doesn't become an interesting topological
artifact until you start messing around with more than just a half-twist
or with splitting it down the middle (or even someplace other than the
middle ;).

-derf