Concerning the recent discussion about the shape of the Universe - whether it is spherical, ellipsoidal, or, more likely, having the shape of Christie Brinkley's celestial body (which I suspect is closer to the truth) - I neglected to raise an interesting point. Ever since reading Roger Pink's original post, I've had this vague, gnawing feeling in my gut that any discussion of the shape of the Universe is essentially meaningless.
Here's why: One essential feature that spheres, ellipsoids, and even Christie's body all share in common is a surface. In each case, the surface serves the essential purpose of dividing the interior from the exterior. In the current context, this surface would, of course, curve in four dimensions - if, in fact, the Universe has a surface. The problem with any discussion of shape in the context the Universe arises when one considers that the Universe has no exterior. There is no Outside, no Something Beyond, No Anything Beyond the Universe from which a hypothetical surface might partition the interior. And since there is no exterior, the whole concept of an interior is meaningless as well.
OMG, Such Heresy!
Allow me to point out that the concept of an interior is meaningful only in the context of an exterior and an intervening surface. As neither of these requirements are met in the case of our own Universe, the concept of an interior exists in a complete vacuum (you decide if this is a pun or not).
One might speculate that there is a Something Out There Beyond The Universe (deities aside) in which the Universe is embedded somehow, but allow me to point out, however, that such a Something must either extend the Universe - in which case we're back in the same boat as before - or the Something's topological properties must be so completely distinct from our own four dimensional spacetime as to be utterly alien. This latter requirement is necessary, again, if any distinction is to be made between the Universe and What Lies Beyond. If it is not, then it becomes merely an extension once again of our four-dimensional Universe - and we find ourselves once again bailing water.
Finally, assuming the latter scenario is true, this Something must not be so distinctly distinct from the topological properties of our own Universe that the concept of shape - or even the concept of embeddedness itself - has no meaning. We seem to be in something of a quandry!
I submit, therefore, that the Universe has no shape. Not only does it not have a shape, but the whole idea of 'shape,' itself, in this context, has no meaning whatsoever.
Sorry, Christie. Maybe next time.
--Europium
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