Hyperbolic Geometry Exploration

From EscherMath
Jump to: navigation, search


Time-25.svg

Objective: Learn about distortion, geodesics, and triangles in the Poincaré disk model.

NonEuclid is a Java program, written by Joel Castellanos, for doing geometry in hyperbolic space. Go to the NonEuclid page and run the applet. It should display a large black circle - that's the Poincaré disk.

The program opens with a sample drawing. To clear the disk, choose "New" under the File menu. This will leave you in "Draw Line Segment" mode, as shown in the upper left corner. Draw some line segments.

  1. What do lines that go through the center of the disk look like?

Clear your drawing, and draw a triangle. On the Edit menu, choose "Move Point". Now you can drag the corners of your triangle and see how it changes.

On the Measurements menu, select "Measure Triangle", and click the three corners of your triangle. You should see the side and angle measurements of the triangle in the box on the left. Go back to "Move Point" mode.

Move your triangle around to get a feel for the lengths of its sides and the sum of its angles.

  1. Draw a triangle which appears large but really has sides of length under 4. Draw a triangle which appears small but really has sides of length over 10.
  2. Try to make the angle sum 180°. What do you have to do?
  3. Try to make all three sides of the triangle large. What happens to the angle sum?

Draw an infinite geodesic ("Draw Line" on the Constructions menu). Now use "Reflect" to make the reflection of your triangle across your infinite geodesic. Move the geodesic around, and notice the position and size of the triangle's reflection. The reflected triangle is the same size and shape as the original, it just appears different.

  1. On your answer sheet, make a sketch showing the two reflected triangles and the reflection line.

If you have time, try these other web sites:

  • Hyperbolic Applet by Paul Garrett. (Use "draw" to create a figure with click-and-drag; then use "move" with click-and-drag to move the entire Poincare disk, via hyperbolic translations, much like moving the sphere around in Spherical Easel.)
  • Hyperbolic animations. by Jos Leys. (See animations of hyperbolic translations applied to various tessellations of the Poincare disk.)

Handin: A sheet with answers to all questions.