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When we draw a triangle on a flat piece of paper and measure the angles using a protractor, the sum of the angles is ##180^\circ##. So we conclude that the universe is locally flat. Suppose we draw a very big triangle that spans across galaxies (say, using lasers and mirrors) and measure its angles. If the sum of the angles exceeds ##180^\circ##, then the universe is closed. If it equals ##180^\circ##, then the universe is flat. If it is less than ##180^\circ##, then the universe is open.

Confusion 1:

There seems to be a circular reasoning in the definition of "flatness".

Suppose Alice lives in a universe that is both locally and globally closed. Since Alice's universe is locally closed, she would measure the sum of the angles of a triangle she draws on a "flat" piece of paper to be more than ##180^\circ##, say ##290^\circ##. Since the paper she uses is "flat", she concludes that the sum of the angles of a triangle in Euclidean or flat geometry is ##290^\circ##. Then, she draws a very big triangle that spans across galaxies and finds its sum of angles to be ##290^\circ##. She concludes her universe is both locally and globally flat, when in fact, it is not.

That means that even though we measure the sum of angles to be ##180^\circ## locally, our universe may not be locally flat. To Alice, our universe is locally open, since ##180^\circ<290^\circ##.

Confusion 2:

It seems that the sum of angles in a triangle is always ##180^\circ## regardless of the local geometry.

If the local geometry is curved, then when we measure the angles of a triangle on a "flat" piece of paper, don't the lines of the protractor we are using bend accordingly by the curved local geometry such that the angle measured is still the same as the one measured in a flat geometry? In other words, since the protractor we are using exists in the curved local geometry, its lines (or scale lines or graduations) are bent by the curved local geometry so that the angle measured is still the same.

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# What does it mean by a flat universe?

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