Hyperbolic Tessellations Exploration

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Time-20.svg

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 <math>n</math> and <math>k</math>?

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

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

Other things to try:

  • Look at Santiago el Grande, by Salvador Dalí. What is the background?
  • Experiment with the hyperbolic symmetry group Kt-sym7.svg 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.