# Hyperbolic Tessellations Exploration

Objective: Find examples of hyperbolic tessellations.

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

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 [itex]n[/itex] and [itex]k[/itex]?

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

1. Use an applet to draw the [itex]\{3,10\}[/itex] tessellation. What are the corner angles of the triangles? What is the defect of each triangle?
2. Use an applet to draw the [itex]\{4,5\}[/itex] tessellation. What is the Schläfli symbol of its dual?
3. When the polygon has three sides ([itex]n = 3[/itex]), what values of [itex]k[/itex] allow a tessellation to be drawn? What about with [itex]n = 4[/itex]? 5? 6?

Other things to try:

Handin: A sheet with answers to all questions.