- This week's introductory web page
and definitions page had lots of
examples of wallpaper patterns on them. Our goal is to come up with a
system for describing such patterns.
Invent your own classification system for wallpaper patterns and
explain where each of the example patterns fits in to your scheme.
Use your imagination: you could classify patterns as "abstract" versus
"concrete", by color, or by the shapes that appear in them.
When would your system of classification of wallpaper patterns be
useful? When wouldn't it be useful? Does your system cover all
immaginable wallpaper patterns? Could you expand it to cover all
wallpaper patterns?
- 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.
- 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?
- Invent a pattern whose symmetry group has orbifold notation 2*22
(you may use Kali). 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
![[pi]](pi.gif)
/2
radians (180 degrees), and that every gyration point and kaliedoscopic
point in the pattern will be one of those mentioned above.
Using this information and anything but Kali, generate a
picture of your pattern. Clearly label the two different types of
kaleidoscopic point and the one type of gyration point.
- 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?
- 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!
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