Introduction to Hyperbolic Geometry

First review Dr. Sarah's Notes on Gauss. Notice that if Playfair's postulate is false (and it is not always possible to draw exactly one line through the given point parallel to the line) then there are 2 possibilities:
  1. spherical geometry - There is some point and line so that there are no parallels.

    Notice that number 1. holds in spherical geometry because any two "lines" (ie great circles) intersect at 2 points - at opposite points on the sphere. Hence I can't find any non-intersecting great circles on the sphere and so there are no parallels on the sphere (recall that non-equator latitudes are not shortest distance paths so they are not considered to be "lines".)

  2. hyperbolic geometry - There is some point and line so that there are at least 2 parallels.

    We are going to work our way towards number 2. Next week we will see real-life applications of hyperbolic geometry.

For the remainder of lab today, you will begin to explore hyperbolic geometry. The following picture shows a model of hyperbolic geometry called the Poincare disk model. Imagine yourself at the center of the white disk and imagine this as a bowl that curves away from you. The blue circle that encloses the disk is actually supposed to be infinitely far away from you. Hence, while this model looks like it is a flat disk, it really is not, and so the geometry is different too. We see three points, G, H and I. To the left of the model, I've measured the sum of the angles of the resulting hyperbolic triangle. We see that this sum is 87.485 degrees!

The following file is an interactive version of the model. Drag the points H, G and I around to see what happens to the sum of the angles in the resulting hyperbolic triangle.
Interactive Poincare disk angle sum

Question 1 How large can the sum of the angles of a hyperbolic triangle get?

Question 2 What kind of triangle results in a large angle sum? Explain why this makes sense by using the visualization of this model (above) and an argument similar to the one we made to explain why small spherical triangles have angle sums close to 180 degrees.






Question 3 How small can the sum of the angles of a hyperbolic triangle get?

The following picture shows the Poincare disk model with three points X, Y and Z. I have measured angle XYZ and m[2] shows me that the measure of this angle is about 90 degrees. Hence XYZ forms a right triangle with XZ as the hypotenuse. I then calculated XY2 + YZ2 and compared it to XZ 2 to see whether the Pythagorean theorem holds in this model. We see that for this triangle XY2 + YZ2 - XZ2 = -.335 and so we see that XY2 + YZ2 < XZ2 by .335. Hence the Pythagorean theorem does not hold for this triangle in this model.

The following file is an interactive version of the model. Interactive Poincare Disk Pythagorean theorem

Question 1: Drag the points to make a small right triangle. Be sure that the points don't touch and be sure that the angle (m[2]) is about 90 degrees.
Part a: What is the measure of the angle?
Part b: What is XY2 + YZ2 - XZ2?

Question 2: Drag the points to make a large right triangle and be sure that the angle (m[2]) is about 90 degrees.
Part a: What is the measure of the angle?
Part b: What is XY2 + YZ2 - XZ2?

Question 3: Given this model, why does it make sense that for small right hyperbolic triangles the calculation XY2 + YZ2 - XZ2 is closer to 0 than is the same calculation for large right hyperbolic triangles?