Stuart Chapman wrote:
OK. My understanding of (one variety of) four-dimensional Euclidean
space is, that if we head off any direction, after an arbitrarily long
time we will arrive at our starting position. Reduce all dimensions by
one, and you have the 'ant on a balloon'.

However, I always understood this to be an analogy just used to aid in
visualisation, so that the centre of the balloon was simply an artifact
of the geometry of the analogy, and not a 'place' in any meaningful
sense of the term. Therefore, when we describe our universe as
'four-dimensional', all we are really saying is, that if we travel in a
straight line for long enough, we eventually end up at the starting
position. We are not making any claims on whatever higher geometry there
may be.

That's right--that is the same as saying that the universe need not be
actually embedded in a four-dimensional space. Only the topology and
the metric are "real."

My question is: If the universe is described as four-dimensional, does
that mean that it necessarily has a geometrical centre. That is to
ask, is there a place that is an equal distance from every place that we
can observe, (and travel to).

If it is an ordinary hypersphere embedded in 4-space, then yes, it has a
geometrical center.

--
Brian Tung
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