## Hyperbolic Tessellations

### An applet to display your own

If you see this image, your browser isn't Java-enabled. This Java applet creates figures like these of specified size, and they change with time.

#### Controls

Some of the controls determine which tessellation is displayed. Others control how it is displayed. After you've set the controls the way you want them, click on draw; changing the controls isn't enough to request that the new tessellation be drawn.
• Regular/Quasi: You can select whether you get a regular {n,k}-tessellation, or a quasiregular {n|k}-tessellation.
• n: The number of sides on a polygon. The value of n must be at least 3, but no greater than 20.
• k: If reguar, the number of polygons that meet at each vertex; if quasiregular, the number of sides on the other polygon. The value of k must be at least 3, but no greater than 20. Note that certain combinations of small values of n and k are not allowed since the tessellations they give aren't hyperbolic.
• skip: If this parameter is 1, then regular polygons are drawn. If greater than 1 then vertices are skipped and stars are drawn instead. This value has to be between 1 and (n–1)/2.
• layers: The number of layers of polygons around the central polygon to be displayed. If 0, then only the central polygon is displayed. The maximum is 4 or 5 layers
• Random/Alternating: If random, then the colors are selected randomly. If k is even, then the alternating value can be chosen. Then two colors are randomly selected, and the polygons are alternately drawn in these two colors.
Don't forget to click on draw when you've changed the control parameters. If you click draw without changing parameters, new colors will be selected.

#### Larger images

If you would like to create a larger image than the one above which is 450 pixels high, choose one of these larger ones instead. Of course, your screen should be taller than the number of pixels you choose or you won't be able to see the whole image.

Dec 2002
David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

The address of this file is http://aleph0.clarku.edu/~djoyce/poincare/poincare.html