Hyperbolic Tessellations Exploration

Objective: Find examples of hyperbolic tessellations.

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 $k$ allow a tessellation to be drawn? What about with $n=4$? 5? 6?

Other things to try:

• Look at Santiago el Grande, by Salvador Dalí. What is the background?
• Experiment with the hyperbolic symmetry group in Kaliedotile. On the View menu, choose Symmetries and see if you can determine which choices give hyperbolic tessellations.

Handin: A sheet with answers to all questions.