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Originally Posted by uniqueuponhim
I just finished reading that article, and I do have a few questions about it:
First of all, why does it have to be either a cube or a hexagonal prism? Why not a dodecahedron or an isocahedron in which the opposite faces are glued together? That would allow for rotations of 72 or 144 degrees for the dodecahedron and 120 for the isocahedron. For that matter, why not a tetrahedron, the simplest of shapes, or a sphere in which every pair of opposite points are glued together, or a cylinder with the same properties for the sides of it as the sphere, and with the two ends simply glued together, or glued together with a certain rotation (and with any possible rotation, since they are circles.) Also, with the cubes, why is it that only one pair of faces could be glued together with a certain rotation? Why not two, or all of the pairs of faces? With that situation in there with two cubes on top of one-another, why could we not add more cubes and make it even more complicated?
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You'll have to excuse me if this sounds a little rusty... it's been a while...
Anyway, you don't use other shapes because it doesn't make sense topologically. That is to say, other shapes reduce to one of the 18 characterized 3-manifolds, or else are non-Euclidean. It's the "Euclidean" qualifier that really limits how you connect things.
Take your cylinder example, minus gluing the opposite sides. No matter how much you twist it before connecting the ends, because it is continuous, it still ends up equivalent to an ordinary old torus. Gluing opposite sides together creates a different topology altogether. Now there are an infinite number of shortest paths from one point to another. It completely ruins the space, and isn't at all what we observe the universe to be like.
The particulars of how you glue things together normally make no difference. In math, it's the topological invariance that is important. Can you turn one shape into another only through stretching and bending, but without punching or patching holes? This article has added the "no stretching edges" rule to satisfy the constraint that the universe looks essentially Euclidean.
With the cubes, they did glue them together with varying rotations. That's why there are 18 such manifolds. I strongly suspect (though I haven't verified it) that stacking more than two cubes reduces to the two cube case.
In summary, um... it's just the way the math works out.
Reading the papers by Nowacki and by Hanschze and Wendt is probably the best thing to do.
As an aside, I'm not so sure about the truth of the universe having a constant topology. It seems to me like the formation or dissolution of a black hole changes the topological character of the universe. Of course, that's just my understanding of what black holes are thought to be....
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Originally Posted by uniqueuponhim
Also, why are we automatically assuming that we will be able to see ourselves through one of those glued-together sides? After all, on earth, you cannot get a good telescope and look in one direction and see the back of your own head, and neither could you see the entire surface of the earth from any vantage point; an ant on a torus would not be able to see itself, or even the entire torus from any position.
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Because the Earth is not a good analogy, because it is not a topological space (if it had black hole-like gravity, and you were right on the event horizon, it would be!). An ant on a torus constructed in our space would not be able to see himself, but an ant in a space that was topologically a torus
would be able to see himself. Interestingly, because it would take time for light to propagate around the torus, he'd be able to see himself as he was in the past. Doubly interesting, if he could see himself and count the number of occurrences he saw, he'd be able to determine the minimum age of his torus (unless it underwent a hyper-inflationary period, in which case some light might not have reached him yet). This is kind of like measuring the cosmic microwave background. In fact, he might be able to extrapolate and very accurately determine the age and expansion properties of his universe at present.
Actually, because of the way a torus is built, the number of copies of himself he sees should vary with the angle he looks at, and the distribution of those copies would tell him both the present size of his universe and also how it grew in the past. (Of course, the problem could end up being numerically unstable if the expansion is weird.)
These are very good questions! They provide a lot of food for thought.
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Originally Posted by uniqueuponhim
How do we know that the curvature of the universe into 5th+ dimensions that causes this shape to arise doesn't also limit our perception of it, in the same way that the curvature of the earth into the third dimension limits our perception of its surface?
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The assumption is that the universe is three dimensional. Or, at least, the visible space component of it is. Looking around and not seeing a fourth dimension, I'd say it's a decent place to start.