Introducing Hyperbolic Geometry

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Introduction

We have looked at two geometries up to this point: Euclidean and Spherical Geometry. The question is: Are there any more?

We have looked at the axioms:

Existence of lines
Extensions of line segments
Existence of circles
Congruence of right angles
Parallel Axiom

We noticed that the first four axioms were very similar and that it's the 5th axiom that truly determines the nature of geometry. We show there is a third geometry where given a line and a point not on the line, there are infinitely many parallel lines. This geometry is called hyperbolic geometry.

We draw Euclidean geometry on the plane, we draw spherical geometry on a sphere, so how do we draw this third geometry? We chose to exclusively use the Poincare disk model of hyperbolic space. The program Non-Euclid allows for work in the upper half space model, but we decded not to include this information in our course. One consideration is that Escher's prints are all based on the Poincare model.

One of the first things we need to do is explain to the students that the hyperbolic "universe" consists of the interior of a disk and that geodesics are segments through the center and semi-circles perpendicular to the circle at infinity.

Using Escher's Prints

Escher's Circle Limit Prints are wonderful for introducing hyperbolic geometry. All four prints are using the Poincare disk model. It should be noted that Circle Limit III is a little misleading. The white curves in the print look like they may be geodesics, but they are not. Some careful measuring would show that the curves do not all meet the boundary at 90 degrees.

Another way to see that the curves cannot be geodesics is to look at any of the "triangles". The angles measure 60 degrees, but 60-60-60 triangles do not exist in hyperbolic geometry. The sum of the angles in a triangle is strictly less than 180 in hyperbolic space.