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I would like to learn how scientists go about measuring the large-scale curvature of the universe to determine if the universe is closed 'i.e. spherical', flat, or open 'i.e. saddle shaped'.

My simplistic thought is that you could measure the corner angles of a really large triangle and see if they add up to <180, 180, or >180 degrees. However I can't imagine how you would do that in practice, (not without owning a Ningi anyway).

Ref. Hitchhiker's Guide to the Galaxy: The 'Triganic Pu' is a Monetary unit. Its exchange rate of eight Ningis to one Pu is simple enough, but since a Ningi is a triangular rubber coin six thousand eight hundred miles along each side, no one has ever collected enough to own one Pu. Ningis are not negotiable currency, because the Galactibanks refuse to deal in fiddling small change.

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I highly recommend reading the book "Sphereland" by Dionys Burger. It is a "sequel" to the much older book "Flatland" by Edwin Abbott that discusses this issue from the standpoint of going down a dimension -- so if we lived in 2D, how would we measure curvature of our 2D universe in higher dimensions when we can't experience them. –  Stuart Robbins Jul 30 '11 at 5:46
    
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There's an excellent talk by Lawrence Krauss on precisely this subject. I can't recommend watching it highly enough, you should start watching it before even reading the remainder of this post.

In summary, we can model the matter just after the big bang at the time we see the cosmic microwave background and determine the characteristic distance scales of the "lumpiness" of the Universe at that point. We can view the lumpiness of the Universe then by observing the cosmic microwave background radiation at high resolution. Now we have something that we can compare the expected visual size of to the apparent visual size, giving us information about the shape of the Universe in between.

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In astronomy, we measure distances by one of two ways: 1) The luminosity of an object decreases as the inverse square of the distance or 2) The angular size of an object is inversely proportional to the distance. In a closed universe, the luminosity or angular size will decrease more slowly with distance (parallel lines will eventually converge) , and, in an open universe, the luminosity or angular size will decrease more rapidly with distance (parallel lines will eventually diverge). You can visualize this with the two dimensional analogs such as the surfaces of a sphere or a saddle.

In practice, the evolution and expansion of the universe makes these kinds of tests impossible and we need to rely on the mathematical modelling.

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