Triangle groups and the Schwarz condition





Ph.D. Students

Math Pics



Institut Fourier

Catalogue Bibliothèque

Triangle tilings

This applet attempts to draw tilings with basic tile a triangle in the hyperbolic plane, the Euclidean plane or the sphere. The angles are chosen to be rational multiples of π (and you can adjust those rational numbers at will - left click on them to increase their value, right click to decrease).

You can adjust the number of triangles that get drawn (see the parameters 'n' and 'max').

If the numerators are all equal to 1, then you should see a beautiful tiling (the applet is initialized with 1/2-1/3-1/7). Otherwise the picture is still quite pretty, but usually it is much messier (most of the time it is not a tiling). In some cases you get a tiling without having numerators one (try 2/7-1/4-1/4, or 2/7-1/3-1/7). Can you find all such exceptional cases? (the list is due to Knapp in the hyperbolic case, and to Schwarz in the spherical case).

Created by Martin Deraux (Last update: 15/01/2006)