Homework -- Searching for Symmetries

1. Following the directions from the in class exercise, write down the symbols in the orbifold notation that describe the patterns shown in figures 1, 4, 5, 6, 7, and 10 on the introduction and definitions pages of this section. Support your answer by describing the features of the pattern corresponding to each character in the notation.

   Example: The orbifold notation for the pattern shown above is 22*. The first sort of two-fold gyration point is in the center of the loop where the two S's meet. The second type lies between the top of one S and the bottom of one above it. The vertical columns of S's are separated by parallel lines of mirror symmetry.

2. Read the American Mathematical Monthly article on "The Plane Symmetry Groups" by Doris Schattschneider. Identify some passage in the article that was hard for you to understand. How did it make you feel? How did you deal with it? Did you skip past that section, puzzle over it, or refer to some other reference material? Try to relate your experience to that of a student reading a geometry text.

3. Using Kali or some other method, create wallpaper patterns that illustrate each of the terms defined on the definitions page, except for "handle/wonder ring" and "cross cap/miracle". Can you find a way to label each gyration point, distinguishing the different types? What about mirror strings?

4. Design a pattern whose symmetry group has orbifold notation 2*22. You know from the in class exercise that it will have some lines of mirror reflection and that those lines will meet in pairs in two different ways. You also know that it has to have a gyration point at which it's symmetric under rotations of 2/2 radians (180 degrees), and that every gyration point and kaliedoscopic point in the pattern will be one of those mentioned above.

   Use The Geometer's Sketchpad to generate a picture of your pattern. Clearly label the two different types of kaleidoscopic point and the one type of gyration point.

5. Write a half- to one- page outline for a pre-college class in which you would use Kali or The Geometric Golfer. What activities would you do to prepare your students for this class? What are the advantages of using computers in your class, as opposed to doing hands-on activities with mirrors or printed patterns? What are the disadvantages?

6. Today's exercise made use of a Web page that wasn't written specifically for the course. The advantages of this are clear -- we avoided "re-inventing the wheel" by recycling someone else's work. The disadvantages are (hopefully) not as clear. Other peoples' Web pages can move, be deleted, or contain obscenities or copyright violations. Also, other peoples' pages generally aren't written for the precise grade level and topic that you want to teach.

   The ability to connect to and reference pages of others' work is one of the great strengths of hypertext documents. How would you deal with the weaknesses discussed above when producing your own Web pages? What other strengths and weaknesses of hypertext can you think of, and how might you exploit or avoid them in your own work? How can you configure your own Web pages to make them more useful to others who might want to refer to them? You may find yourself needing the answers to these questions when working on your final project!

The Geometry Center Home Page

Author: Heidi Burgiel, adapted from notes used in Math 5337, Spring 1995 by Chaim Goodman-Strauss.
Comments to: webmaster@geom.umn.edu
Created: Dec 21 1995 --- Last modified: Jun 11 1996
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