Introduction to Hyperbolic and Spherical Geometry

Date: 06/01/2004 at 19:27:14
From: Alisha
Subject: non-euclidian geometries

I know that in hyperbolic geometry, the sum of the angle measures of a
triangle is less than 180 degrees.  But I don't understand why that
is.  Can you explain it in a way that a 9th grader can understand?

Date: 06/01/2004 at 21:05:54
From: Doctor Tom
Subject: Re: non-euclidian geometries

Hello Alisha,

I'll take a shot at explaining it to you.

One way to think of hyperbolic geometry is as the geometry that you
would see if you lived on a hyperbolic surface.  Without being
mathematically exact, think about the surface of a saddle for a horse:
it slopes up in front and behind you and down where your legs go.  The
surface of a saddle is rougly the same as a hyperbolic surface.

Now imagine that you are an ant on the saddle, and you want to walk
between two points on the surface by the shortest possible route. 
Although we humans can see that the ant is not moving in a perfectly
Euclidean straight line, it is moving along a curved path that is the
shortest for it.  That path will be called a "straight line" in
hyperbolic geometry.  This makes sense, since one way to think of a
straight line is as the shortest distance between two points.

Now imagine three points on the saddle A, B and C and imagine that the
ant walks in hyperbolic straight lines from A to B, then from B to C
and finally, from C back to A.  This path will trace out a triangle in
hyperbolic geometry.

But if we as humans look at the path, to us, it will seem to be like a
triangle with the middle parts of the lines bent in toward the center.
 Thus the angles at the tips will be less than what they are for our
Euclidean triangles.  Since each will be less than the Euclidean
angle, if you add all three, the sum will be less than the sum on a
Euclidean triangle, or less than 180 degrees.

If you have trouble visualizing paths like this on a saddle, try doing
it on a sphere (like the surface of the earth.  This will be spherical
geometry, which is sort of the opposite of hyperbolic geometry.  In
spherical geometry, the sum of the angles of a triangle will be more
than 180 degrees.

To see why, imagine the following route:  Start at the north pole, and
go south along the prime meridian through Greenwich, England down to
the equator in or near Africa.  Now take a 90 degree turn and go 1/4
of the way around the earth, about to the Galapagos islands off the
coast of Ecuador.  Turn 90 degrees and go due north to the north pole.
 You will arrive there and your arrival and departure paths from
the pole will make a 90 degree angle.  Thus you've traced out a
"triangle" of three shortest paths, and each angle of that triangle is
90 degrees, for a total of 270 -- much more than 180 degrees.  By
choosing triangles carefully on the surface of the earth, you can make
them have a sum of angles of any size less than 540 degrees total.

On a hyperbolic surface, the sum of the three angles of a triangle can
similarly be made quite small (with a very big saddle, of course).

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