MONT 108N – Mathematics Through Time Discussion 5 – A Euclidean Model of Hyperbolic Geometry December 2, 2010

Background

The final step of showing that hyperbolic geometry is as consistent as Euclidean geometry was provided by a number of different 19th century mathematicians who showed that models of hyperbolic geometry could be constructed within the Euclidean plane. If there were any contradictions, there would be contradictions in Euclidean geometry as well. (This sort of argument is known as a relative consistency proof in “the foundations of mathematics biz.”) Today, we will look at one such proof that was provided by a 19th and 20th century German mathematician named Felix Klein (1849 - 1925). In the Klein model of hyperbolic geometry:

• A point is a point of the disk K strictly inside the circle of radius 1 with center at (0, 0) in the coordinate plane.

• A line is the portion of a straight line in the plane that lies inside K. (Note: points on the boundary circle C of the disk K are not part of the Klein model.)

Discussion Questions

A. Show that in this model, two points determine one and only one line (that is, a statement corresponding to Euclid’s Postulate 1 holds).

B. Show that in this model, given a line ℓ and a point P, there are infinitely many different lines m containing P that do not meet ℓ. (These are the parallels, so the hyperbolic postulate (H) holds in this geometry.)

C. To measure distances in K, we proceed as follows: Let P, Q be points, and using the Euclidean construction, extend the (Euclidean) line P Q until it meets the boundary circle in points S, T, in the order S, P, Q, T along the (Euclidean) line. Define:

distance P Q = ln (\frac{(QS)(P T)}{(P S)(QT)}) ,

where the (QS), (P T), (P S), (QT) are the Euclidean distances.

1. Show that if P, Q, R are three points on a line in K, then

distance P Q + distance QR = distance P R.

2. Let P be fixed and let Q move along the line toward the point T on the boundary circle as in the definition of the distance. Show that distance P Q → ∞. Hence something like Euclid’s Postulate 2 holds here as well.

3. Finally, explain how you would define and construct a circle in this geometry (Postulate 3).

In case you are curious here, to define angles in this model, we must proceed by a different method from the usual Euclidean one. This is a bit too technical for us to get into. However, with the correct definition, it can be seen that all right angles are equal, so the analog of Postulate 4 holds as well!