1. Start up Geometry Explorer. Once the main Geometry Explorer window is open, go to the File menu and click on New. A dialog box will pop up asking you to specify which type of geometry environment you wish to work in. Choose "Hyperbolic" and hit okay.  A new window will pop up with a circular disk inside.  This disk is called the "Poincare Disk" in honor of Henri Poincare, one of the greatest mathematicians of the 20th century.  You should think of this disk as a little "geometric universe", just as you think of the plane as a universe to do normal geometry.  This disk is a model for a geometry called Hyperbolic Geometry.  In this project we will explore what it is like to "live" in this strange geometric universe. What is similar to living in a Euclidean universe and what is different?

2.  In the first part of this project we will look at triangles in the Poincare disk.  Using the segment tool in the Create Panel, create a triangle ABC  and measure its interior angles.  (Remember to be careful as to angle orientation). Then, use the Calculator to find the sum of the interior angles of triangle ABC.

Note that the sum of the angles is not 180 degrees, as in Euclidean Geometry. In fact, the angle sum is less than 180 degrees. In the first part of this lab I want you to speculate on the behavior of triangles in hyperbolic space. Without doing anything on the computer think about and then write down what you think will happen in the following situations:

1) The three vertices of the triangle approach the boundary circle.
2) The three vertices of the triangle shrink down to almost a single point.

3. In the second part of this project we will investigate another very strange property of hyperbolic geometry having to do with moving about in the geometry.  Clear the window and use the circle and segment tools to construct a hyper-person as shown.

4.  Now, use the select tool to draw a box around the person, thus selecting all the objects making up the figure.  Then, drag the person in a circle and return to your original spot.  You will notice that our little being does cartwheels as you drag him/her around the disk.  Also, when you return back to where you started the figure will not be in the original position!!  Shown below is the result of doing this.

Now, you may think that our hyper-being has also changed size as we moved around the disk. But, he is actually always the same size!  If you want to convince yourself of this measure the lengths and circumferences that make up the figure and see how they remain constant as we move around.

5. The property of changing orientation as you move around is generally true of spaces that are "curved".  By this we mean that the space is not flat like Euclidean space is.  In a flat space if we just walk around in a circle and return to where we started, we always end up in the same position.  But, this does not happen in a curved space, like Hyperbolic Space.

6. What other differences can you discover between Hyperbolic and Euclidean Geometry?  Are there parallel lines? perpendicular lines? Does the Pythagorean Theorem hold true?  Experiment with this Geometry and find two things (other than the two illustrated above) that are different between Hyperbolic and Euclidean geometry.

7. As Janos Bolyai (one of the discoverers of non-Euclidean geometry) said "I have discovered such wonderful things that I was amazed ... out of nothing I have created a strange new universe."  Hyperbolic geometry is indeed a very weird place.  However, it has proven to be a very useful geometry for modern theories of the geometry of the universe.  Extra Credit: Research the idea that the universe may be curved.  Write-up a short report of your findings.