Non-Euclidean Geometry for 9th Graders

Date: 23 Dec 1994 13:10:38 -0500
From: Cerreta, Pat
Subject: Non-Euclidean geometry

Dear Dr. Math,

I would to know if there is non-euclidean geometry that would be 
appropriate in difficulty for ninth graders to study.

Thanks for your patience.

Pat Cerreta at Byram Hills

Date: 2 Jan 1995 13:54:00 -0500
From: Dr. Ken
Subject: Re: Non-Euclidean geometry

Hello there!

Euclidean geometry is the kind of geometry that assumes the Euclidean
parallel postulate.  This states that given any line and any point not on
that line, there is exactly one line through that point which is parallel to
the given line.  It is important to remember that "parallel" always means
"lines that never intersect", i.e. "lines that share no common point", 
NOT "lines that are the same distance apart everywhere."

Well, non-euclidean geometry is the geometry that you get when you 
assume the negation of the Euclidean parallel postulate: there is some 
line and some point on that line such that either (a) there is no line 
through that point which is parallel to the given line, or (b) there is more 
than one line through that point which is parallel to the given point.

As it turns out, you can show that if case (a) happens somewhere in 
your geometry, then there are no parallel lines _anywhere_ in your
_whole_geometry_.  This is pretty amazing.  It means that every pair 
of lines in this geometry will intersect somewhere.  This kind of 
geometry is called "Spherical" or "Elliptical" geometry.  You can 
model it on the surface of a sphere, which I'll talk about later.

As a consequence, you can show that if case (b) happens anywhere in 
your geometry, then it happens _everywhere_, too.  We call this 
geometry "Hyperbolic" geometry.  This is geometry in which, given 
any line and any point not on that line, there is more than one line 
through that point which is parallel to the given line.  As it turns out, 
you can then show that there are _infinitely_many_ such parallel lines 
through your given point and your given line.

Anyway, here is a model for Elliptical geometry.  Take the upper half 
of a sphere as the plane you're working in, and let lines in this model 
be arcs of great circles on the sphere (great circles are circles [on the 
sphere] whose plane contains the center of the sphere.  Examples on 
Earth include the equator and the meridians, but not the tropic of cancer 
or the arctic circle).  Points are just normal points.  Then any two "lines" 
you draw are going to intersect in some point.

Actually, it's a little trickier than that.  See, there's a problem at the
boundary of the hemisphere.  You have to include only half of the 
boundary, because you don't want lines to intersect in two different 
points, and if you included the whole boundary, some would.  If you 
included none of the boundary, you'd have some lines that don't intersect 
anywhere (basically, you want to start out with a whole sphere and get 
rid of points until there are no antipodal points left - antipodal points are 
points directly across from each other on the sphere, like the north pole 
and the south pole).

So that's Elliptical geometry.  Here's a model for Hyperbolic geometry.  
Let the plane you're working in be the upper half of the x-y plane, without 
the x-axis.  In other words, the set of all points where y>0.  We call this 
the open upper half-plane.  There are two kinds of lines in this model.  
One kind will be half-circles whose center is on the x-axis, and the other 
kind will be lines that are perpendicular to the x-axis.  Points, again, are 
just normal points in the plane.

So draw a couple of "lines" in this model, and note that at best they'll
intersect at only one point.  Then pick one of the lines you've drawn and
take some point not on that line, and see how many lines you can draw
through that point that are parallel to (don't intersect) the given line.  A
bunch, right?  So this is Hyperbolic geometry.

I hope you can do some neat things with these models.  There's a great 
book that deals with this stuff.  It's called "Euclidean and Non-Euclidean
Geometry", and it's by a guy named Greenberg.  I forget his first name.  
The book may be a little advanced for ninth graders to use as a textbook 
(I used it in college), but maybe you could get some ideas from the book 
to show them, anywyay.  There is also a great deal of wonderfully 
interesting historical material on the development of this stuff in there.

Anyway, I hope this helps you out!  Fortunately, our school is on break 
now, so I had plenty of time to write your response.  Write back if you 
have more questions!

-Ken "Dr." Math