Euclidean and Riemann GeometryDate: 12/29/2003 at 01:33:36 From: Ming Subject: Riemann Geometry I'm a bit confused about the basic premises of Riemann Geometry. I want to write a paper comparing it to the basic premises of Euclidean Geometry and do a comparison on how they differ, and how that subsequently affects our understanding of geometry and the natural world. What I've done so far is discuss the first 4 axioms of Euclidean Geometry, then talked about why the fifth axiom is more a statement of fact than it is an axiom, and why people have tried to prove it but failed. Then I move on to Riemann's work. Here is where the problem arises--I understand that he changed it, saying that parallel lines always meet. But how did he come up with this, and why is it true? Date: 12/30/2003 at 07:46:56 From: Doctor Edwin Subject: Re: Riemann Geometry Hi, Ming. This is a fun topic. I like it because it illustrates a lot about how math relates to the world. Really, geometry isn't about the world. It's about geometry. It's a nice, closed system. But of course the entities in geometry are a lot like things in the real world, aren't they? And that's what makes it useful. Euclid built a closed system that was similar to the way things behave in the real world. But as you pointed out, there were lots of things that were true in the real world that couldn't be derived in geometry with just the four postulates. So Euclid took a thing that seemed true in the real world and added it to his system as a fifth postulate. Then he was able to have a pretty complete model of how shapes worked in the real world. Many people (including, I seem to recall reading, Euclid) were unhappy with the fifth postulate and tried to get rid of it. If you could derive it from the other four postulates, you could have it as a theorem, and be back to the original four. One way to prove the fifth postulate would be by negation. If I assume the opposite of the fifth postulate, and add that to the system, and I can find a way that it contradicts one of the other four postulates, then I have proven that the fifth postulate is true, using the other four. There are two ways to negate the fifth postulate. You can either assume that there are NO lines through a point not on line AB that are parallel to AB, or that there are an infinite number of them. Riemann's geometry assumes that there are no parallel lines--that all lines must intersect. However, when this was done, no contradiction was found. You could generate theorems using the negation of the fifth postulate along with the other four from now until the cows come home, and you'd have nothing but a perfectly self-consistent closed system that is about itself. You'd have an alternate geometry. Okay, here's the cool part. Just like Euclid's geometry models the way shapes work on a plane, Riemann's geometry models the way shapes work in a space that curves back on itself, like on the surface of a sphere. Now here's the really cool part. Einstein said that the universe actually fits Riemann's geometry--that the three-dimensional universe we perceive actually curves back on itself in four dimensions like the two-dimensional surface of a sphere does in three dimensions. Does that help answer your question? - Doctor Edwin, The Math Forum http://mathforum.org/dr.math/ |
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