# How can geometry be different in space?

I was talking Saturday with a woman who had taken some Astronomy courses recently. It seems there can be more than 360 degrees between objects in space? How does one measure this but more importantly for a layman like myself is there an explanation of this which I might understand? Hopefully I have phrased the question correctly, but not completely understanding what she was refering to may make the question a little bizarre.

It's true that geometry on large scales can be very different from Euclidian. This is because space is curved. What your friend is probably referring to is the commonly stated fact that triangles in a closed universe can subtend more than the usual 180 degrees. The concept is really not that unusual if you think about it; we deal with curved geometry on Earth all the time. Take a look at a globe, and imagine a triangle with one vertex at the north pole, and two others on the equator 90 degrees apart in longitude. You've just constructed a triangle with three ninety dergee angles, made possible by the curvature of the surface of the Earth! Similar things are true over large areas in space, because of the curved geometry.

In practice, we can measure this my measuring the sizes objects apear to be on the sky. Here, one apex of the triangle is our telescope, and the two others are the sides of the object we observe. Nearby, things get smaller in angular size just as you'd expect, but beyond a redshift of about 2 or so, angular size is relatively constant, independant of distance from you, and even farther away than that, things appear bigger the farther away they are! Another way to think about this is to imagine a meter stick hanging around the universe when the universe was about a meter across. Such a meter stick was the size of the universe, so if we were to observe the light from it now, it would be extremely distant, but we would see it across the entire sky.

You can see this effect with the globe as well. Imagine yourself standing at the North Pole, and that you could see along each line of longitude all the way to the south pole. Held just in front of your face, a meter stick subtends 180 degrees in longitude. Near the equator, it subtends perhaps an arcsecond, but going south from there, it starts getting bigger in angular size again. At the south pole it is extremely distant, but once again subtends 180 degrees. Clearly, in actuality you can't see around the Earth this way, but that is the way we see around the universe.

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