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In the previous exercise you took a motif and repeated it according to certain instructions. The result was a plane pattern.

There is a piece of mathematical software called TesselMania! that is designed for exactly this purpose. For each pattern you produce using this software, ask yourself the following questions:

• What is the orbifold of this wallpaper pattern?
• How is it described in the orbifold notation?
• How does allowing the motif to change color affect the symmetry of the wallpaper pattern?
Another interesting question is:
• Is the list of regular tilings of the plane the same as the list of seventeen wallpaper patterns?

Start the program TesselMania! If you are at the Geometry Center, this program can be found in the Math Apps folder. This piece of software is produced by MECC, and can be purchased from Key Curriculum Press.

Read the Instructions. Pay particular attention to the discussion of which transformations the program supports. What's missing?

Start the program (click on "New") and play with it a while. Make sure each group member has the opportunity to satisfy their creative urge.

Choose New from the File menu. Find a tile you like and click on "Show Me". The program shows how each edge of the tile is created from another edge. Could you use this information to design a hands-on activity introducing your students to Escher-type tilings? (My friends in Seattle who have tried this find that their students can always make a tiling into a tiling by animals once they've drawn an eye on their critter!)

Create a simple tile, or reload a tiling you saved earlier.

Clicking on the magic wand with single tile button at the top of the screen repeats the "show me" information you just saw.

Clicking the next button over (magic wand with several tiles) shows how the tiling is generated from the single tile. As you know from the last exercise, this is very important information!

The third button shows how the original tile was transformed into the final result. This very interesting to students of tilings, but less so to those studying the wallpaper patterns that appear in a tiling.

If you haven't done so already, it's time to take a break. After the break, we'll watch the video Shape of Space and see how what we've studied today is related to astrophysics!

Next: Shape of Space Questions