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## Hyperbolic tessellations

The applet draws a triangle with three given angles a_{1}&pi/a_{2}, b_{1}&pi/b_{2} and c_{1}&pi/c_{2}, and then some of its images under the group G generated by the reflections in its side (so far the applet doesn't support cases with one vertex on the boundary, in other words you need a_{1}, b_{1}, c_{1} to be nonzero).

You can choose the angles of the original triangle, and modify how far the program goes in reflecting it (unless you are very patient, I suggest to keep the "depth" parameter under 5)

Note that if all the angles are integer parts of pi (i.e. a_{1}=b_{1}=c_{1}=1), the group G is discrete. If not, G is sometimes discrete, sometimes not. Try a_{1}=2, a_{2}=7, b_{1}=1, b_{2}=7, c_{1}=1, c_{2}=3 (the group sould be discrete) and then change b_{1} to 2 (the group is not discrete anymore).

When the group is not discrete, try also modifying the appearance. You can choose to print only the vertices and not the sides of the triangles.

Loading the applet may take a while --- please be patient...
When the applet is running, you'll see a button labeled
'Start.' Simply click on the button in order to launch the
applet.

Created by Martin Deraux

The computer-geometric framework used here to handle such objects as hyperbolic isometries and geodesic arcs
was developed by Peter Brinkmann. The GUI (graphical user interface) is based on his code as well. I cannot thank him enough for letting me access his code and providing constant help in tweeking it to produce the applet above.