The sum of the angles of a triangle in curved space is greater than 180 degrees, and the circumference of a circle is less than 2 R. Any two parallel lines will converge at a point (like lines of longitude on the Earth, which are parallel at the equator, but come to a point at the North Pole). In a positively curved Universe, the rules of geometry are different than the ones you learned in grade school and high school. (Think about it this way-if there is no boundary, there CAN'T be a center-you define the center of a room by looking at where the walls are. It has no boundary (you can go around and around on a ball with out ever running into a wall), and has no center. This positively curved surface has several properties. A person is small compared to the size of the Earth.) They thought the Earth was flat, because it looked that way. (This is what happened to all those 'flat-Earth' people in Columbus' time. If the balloon is sufficiently large, and you are sufficiently small, it looks flat. If you are confined to the surface of the balloon, so that you are a balloon creature, living in 2-D, you can go up-down, or right-left, but in-out of the balloon makes NO SENSE to you.
This is very hard, and will hurt your head! After that, you must try to 'imagine them' into three dimensions. We'll look at each of these separately, and try to understand them by thinking about 2-dimensional analogies to each of these options.
There are three options for the curvature of space: You will see that the curvature of space is difficult to imagine, and the curvature of time is absurdly difficult to imagine, and NOBODY is really capable of it.).
(Why not?!) Today, we are going to talk about these different possible curvatures of space (and time, but we won't really worry about time. But it's also true that the Universe as a whole could be curved. You remember from all of our discussion of black holes, that space is curved, locally by the presence of mass, and that gravity can be thought of simply as 'bent space'.
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