From: Rafal Smigrodzki (rms2g@virginia.edu)
Date: Wed Aug 21 2002 - 15:20:41 MDT
Lee Corbin wrote:
What meaning of *closed* is this, now? Not having "infinite lines of
sight"?
Maybe you mean by "closed" straight lines (geodesics) of infinite length.
(For example, in a 2 dimensional square where the top is identified with
the bottom, and the left side with the right side, one has two kinds of
straight lines: those that form a quick loop, i.e., start from the middle
of the left side and go straight across to the middle of the right side
and at that point intersect themselves, and those others that are at an
irrational angle, and so are truly infinite in the other meaning.)
So there seem to me to be three meanings of closed possibly
meeting here: (1) topologically closed, i.e. a set containing
all its limit points and, if it has one, containing its boundary
(2) closed as having infinite non-intersecting geodesics, and
(3) closed as in universes that eventually collapse.
### Thanks for setting things straight here. I did get too fuzzy in thinking
about this issue. I am most curious about universes with infinite space
dimensions. If I am right, these would have infinite lines of sight, that
is, light would never return to the point of origin or its vicinity. In any
manifold produced by stitching together of the borders of a finite volume,
this would not be the case. So when I speak about infinite volume, I
recognize it functionally by having infinite lines of sight, or the ability
to travel forever without ever coming back to the starting area or
intersecting one's steps.
------
> For a closed n-dimensional universe with an infinite volume I could
suggest
> an n-dimensional surface folding as a fractal in an n+1 space. I don't
have
> the oomph to think this through.
Your closed n-dimensional universe with infinite volume
would support infinitely long geodesics, no? Now I'm
again doubtful that I even understand what is meant by
"closed infinite volume" universe; I can't think of an
example.
### I agree with you - using the functional approach above, I see the
fractal wouldn't't fit the bill. After a finite time I would come back to
where I started, even if the fractal dimension slowed me down somewhat.
Since I do not have zero-thickness like a geodesic I would eventually
intersect my path.
Rafal
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