Non-Euclidean Geometry

Date: 10/22/2001 at 21:44:19
From: Yj
Subject: Non-Euclidean geometry

I have read some of the questions and answers in the archives about 
non-Euclidean geometry, but can't understand all of them. Please tell 
me, simply, What is non-Euclidean geometry?  And what two concepts  
(no-parallel geometry, many-parallel geometry) are totally different 
in non-Euclidean geometry?

Thank you for your time.

Date: 10/23/2001 at 00:20:02
From: Doctor Schwa
Subject: Re: Non-Euclidean geometry

Euclidean geometry is based on several assumptions about space,
one of which is:

   in any given plane,
   if you choose a line l, and a point P not on l,
   there is exactly one line through P that's parallel to l.

This is usually called the "parallel postulate."

Non-Euclidean geometry is based on changing one or more of the
assumptions, most commonly this parallel postulate. So you can see
that some easy flavors of non-Euclidean geometry to make involve
replacing the "exactly one" with "none" or with "more than one."

The "none" version turns out to be very useful on spherical surfaces,
where lines that start out parallel-looking eventually meet (try 
starting at the equator with two "parallel" lines headed due north, 
for instance).

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/