Hyperbolic Tessellations Exploration

From EscherMath

There are two good applets on the web for drawing hyperbolic tessellations:

• HyperbolicApplet by Don Hatch.
• PoincareApplet by David Joyce.

Play with both applets.

1. In the HyperbolicApplet, the white lines are the tessellation. What are the blue lines?
2. In the PoincareApplet, what are n and k?

The Schläfli symbol for a regular tessellation is written {n,k}, where n is the number of sides on each face and k is the number of faces coming together at a vertex.

3. Use an applet to draw the {3,10} tessellation. What are the corner angles of the triangles? What is the defect of each triangle?
4. Use an applet to draw the {4,5} tessellation. What is the Schläfli symbol of its dual?
5. When the polygon has three sides (n = 3), what values of k (the number meeting at a vertex) allow a tessellation to be drawn? (Easier to answer: What values for k don't work when n = 3?) What about with n = 4? 5? 6?
6. What is the reason for the results of the previous question?

Other things to try:

• Look at Santiago el Grande, by Salvador Dalí. What is the background?
• Experiment with the hyperbolic symmetry group in Kaliedotile.