Hyperbolic Paper Exploration

Objective: Build a paper model of hyperbolic space and investigate its properties

Materials

• Triangle paper
• Scissors
• Tape
• Protractor

Hyperbolic Paper

You can make Euclidean space by gluing equilateral triangles together so that six touch at each vertex (this is just the usual tessellation by triangles).

You made an icosahedron by gluing equilateral triangles together so that five touch at each vertex. This corresponds to a tessellation of the sphere, and makes a pretty good model for an actual curved sphere. Gluing four, three, or two triangles also makes a sphere, of sorts.

Using triangle paper, cut and tape triangles together so that seven triangles meet at every vertex.

Some tips:

• The less taping you do, the better. Try to tape on chunks of five connected triangles at a time, folding along the edges for flexibility.
• To start, it's helpful to cut out an entire hexagon of six triangles, then make a slot to the center.

Keep building until you've got at least one vertex which is surrounded by two rings of triangles.

Lines

Each triangle in your model is flat, so geodesics are the usual straight lines. When a line crosses a fold in the model, flatten out the fold and continue the line in the usual straight way.

Try drawing a long straight line or two on your model. Use light pencil, or things will get cluttered later.

1. Start two "parallel" lines as shown: Continue the two lines in both directions.
   What happens? Do they stay the same distance from each other?
2. Draw a triangle on the hyperbolic paper. How does it's angle sum compare to 180°?
3. What happens to the angles of triangles when the triangles get larger?

Handin: A sheet with answers to the questions, but keep your hyperbolic paper.