Parallel Lines: Euclidean and Non-Euclidean Geometry

Date: 4/25/96 at 12:25:57
From: Anonymous
Subject: Parallel lines

If two lines are parallel, do they intersect?

Thanks.

Dennis

Date: 4/26/96 at 8:15:29
From: Doctor Steven
Subject: Re: Parallel lines

Well, it really depends on what kind of geometry you are looking at.

In planar geometry (geometry on a flat surface, also called Euclidean 
geometry), no.  The definition of parallel lines is that they extend 
infinitely long in both direction without intersection.  The fifth 
postulate of Euclid says that if a straight line falling on two 
straight lines makes the angles on one side of the line less than 180 
degrees, then the two straight lines if extended will intersect at 
some point on the side of the other straight line where there angles 
added to less than 180 degrees.

The fifth postulate was a source of contention for many mathematicians 
in ancient times and even in relatively modern times.  The thought was 
that it could be proven from the other four postulates that Euclid 
gave:

  1) From any two points a straight line can be drawn.
  2) Any straight line can be extended indefinitely.
  3) A circle can be drawn of any radius at any point.
  4) All right angles are equal to one another.

Many mathematicians spent their time trying to prove this postulate 
using the others; if they could it meant that the fifth postualte 
should not be a postulate but rather a theorem.  Probably Euclid would 
like to have proved it, so parallel lines could be proved to intersect 
a falling straight line with interior angles equal to 180 degrees.  
Unfortunately for him he could neither prove or disprove the fact, so 
he added it as a postulate.  

Now other mathematicians also thought it should be provable from the 
other four postulates and so they tried - for hundreds of years!  
Finally two mathematicians decided that if the 5th postulate was 
provable by the other four they should be able to "replace" the 5th 
postulate with a contrary one, and they would get a contradiction 
somewhere.  Well they took out the 5th postulate and added a new 
contrary one and they got crazy statements like: Every triangle must 
have strictly greater than 180 degrees as the sum of the interior 
angles.  But no contradictions.  They had invented a new system of 
geometry, of course they called this Non-Euclidean geometry. This new 
geometry they created perfectly described what we call spherical 
geometry.

After their discovery other mathematicians decided to see what they 
could get by adding different contrary postulate in the 5th postulates 
place.

In a Non-Euclidean geometry such as spherical geometry, two lines can 
be parallel and still intersect.  To see this think of the globe and 
two lines of longitude.  Look at the lines of longitude, say 30 
degrees W and 40 degrees W, note that at the equator these lines are 
parallel, no look at either of the poles and you will see that they 
intersect.

Such geometries as this are called non-Euclidean, and there are many.  
To learn more on some of these you can read up on G. Bolyai, 
Lobachevsky, Riemann, and Felix Klein.

-Doctor Steven,  The Math Forum
     (with some help from Dr. Patrick)
 Check out our web site!  http://mathforum.org/dr.math/