A closed universe

To visualize what a closed universe is like, we should go back
to the example in Chapter 1 of the 2D’ers living on a spherical
surface. Their universe is also closed, and is thus not infinite in
size since the surface has a certain area. A sphere is said to have
positive curvature, since if you were to move along two paths on
the surface at right angles to each other, both would curve round
in the same direction. Such a closed universe most certainly does
not have an edge since the 2D’ers could travel anywhere they liked
in the surface without reaching an edge. In fact, if a 2D’er were
to head off in a rocket and travel in what is to him a straight line
he would eventually come back to where he started from. That
is exactly what would happen if we lived in a positively curved,
closed universe; we would eventually come back to where we
started from.
Remember also that for the 2D’ers, the inside of the sphere
(and the outside) does not even exist. It is outside their two
dimensions. If our Universe is closed then the simplest shape
it can have is the surface of a four-dimensional ball known as a
hypersphere. This is the equivalent of the 2D’ers’ surface of a 3D
ball, only it has one dimension more and is impossible for us to
visualize. We should therefore spend a little more time thinking
about the 2D’ers’ universe since that is what ours would be like if
we were to throw away one of our dimensions of space.
The following example is the standard way of explaining the
concept of the Big Bang. Imagine the 2D’ers’ universe is the surface
of a balloon that is being blown up. The expansion of this universe
is now exactly the same as the expanding flat sheet of rubber I
discussed earlier. Every point on the surface of the balloon will
be moving away from every other point. Now it becomes clear
that the Big Bang is not somewhere on the surface of the balloon.
It is more correct to think of it as the centre of the balloon itself,
since not only is every point on the surface moving away from
all others, they are all also moving away from the centre of the
balloon. Even this picture is not quite right, however, since the
interior of the balloon does not even need to exist. You see I have
used the analogy of a balloon which is a 3D object, so that we can picture its 2D surface. After all, you would think that it makes no
sense to talk about a sphere without imagining it containing an
interior volume. But that is just for our own convenience. Such a
closed 2D universe is able to exist without being imbedded in 3D
space and we would say that its big bang happened everywhere
on the surface at once, and since the whole surface was squeezed
down to a point anyway, we do not need to specify where that
point was situated within three-dimensional space. It is simply a
handy way for our brains to visualize things.
To recap, if the Universe contains enough matter it will one
day stop expanding and start collapsing. It would be a closed,
finite universe which has positive curvature, and which will not
have an edge just as the surface of a sphere does not have an
edge. It might be helpful to think of it as expanding into a higher
dimension, but this is really only an aid and the higher dimension
does not necessarily have to really exist. In terms of where the Big
Bang happened, we can say that it happened everywhere at once
since the whole Universe would have grown out of a point and
everywhere would have been confined to the same place. Whether
that point was floating in higher dimensional space we do not
know.