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# Kali Exercises and Orbifold Notation

This lesson requires the software package Kali which was developed here at the Geometry Center. If you do not have a copy of this program, it is available from the Geometry Center software archive. The archive includes a page of instructions on using the software.

Choose one of the first twelve wallpaper symmetry groups shown on the Kali Wallpaper palette. Write down your choice and design a pattern with that symmetry.

Print out a copy of your pattern and give it to another group in the class or hide the Wallpaper palette window and trade computers with another group.

Find the lines of mirror symmetry, the mirror strings, the kaleidoscopic points, and the gyration points of the pattern you are given by the other group. (You may refer back to the definitions page for help.) When you're done, move on to the next section.

#### Orbifold Notation

Now you're ready to find the orbifold notation for the symmetry group of your mystery wallpaper pattern! In other words, you're going to find out where those weird labels under Kali's buttons come from.

Get out your notebook and a pencil, and look at the mystery pattern given you by the other group. If you found lines of mirror symmetry, write down a "*". Leave plenty of space before and after the "*" for numbers.

Remember that a mirror string is a bunch of mirrors bounding a box, like the sides of a polygon. Pick a point on a mirror string (preferably a corner point) in your pattern and memorize what it looks like. Follow the mirror string around the smallest region it encloses until it reaches a point that looks exactly like the point you started at. (The second point could be a reflected, rotated, or translated copy of the first. What's important is that those two points are related by some transformation that is a symmetry of the entire pattern.)

Look back at the section of the mirror string you covered between the two identical points. After the "*" in your notebook, write down the number of mirrors meeting (if any) at each kaleidoscopic point on that section of mirror string. (If you started at a corner, record either the starting point or the end point, but not both.) For example, if you passed two points where four mirrors meet and then a point where two mirrors meet, you should write *442. (It is conventional to write the largest numbers first.)

Now look at the gyration points in your pattern. How many different kinds are there? In other words, how many different gyration points can you find that aren't related by some translational, rotational, or reflectional symmetry of the wallpaper pattern? Write down the orders of the different gyration points before the "*" in your notebook. (If you don't have a "*", just write down the numbers.)

Hints:

If there are lines of mirror symmetry in your pattern, you only need to look on one side of them for gyration points. The gyration points on the other side of the mirror will be related to these by reflection.

If there are no lines of mirror symmetry in your pattern, there's another way to restrict your search for gyration points to a finite region. Look for two translational symmetries of your pattern. Imagine the translation vectors of those symmetries as the edges of a parallelogram. You can recreate the entire pattern from this parallelogram by translating it around. (The parallelogram acts as a sort of mega-motif for your pattern.) So, any gyration points outside of this parallelogram will be copies of some gyration point inside the parallelogram. You can restrict your search for gyration points to the inside of the parallelogram.

Check your work: Does the symbol in your notebook correspond to one of the first twelve Kali wallpaper patterns? If so, try to reproduce the pattern using Kali. (This may not be easy, even if you've got the right pattern!) If not, what went wrong? If you have too many numbers on the page, check that the gyration points and kaleidoscopic points you describe in your notebook really are all different. If you don't have enough numbers in your notebook, look for more symmetry in your pattern. If there's a "5" in your orbifold notation, it's time to ask for help!

When your time is up, or when you're sure you've got the right answer, ask the other groups what their pattern was. If you have time left over, repeat the exercise with another mystery pattern.

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