Euclidian and Riemann Geometry

Date: 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/