Invent a system for categorizing pictures and describe which category each of the pictures on the sample page falls into. (You may wish to assign each of the five parts of example 3 to different categories. You may find that some of the pictures don't fit any of your categories. You may create categories that none of the pictures shown fit into.)
One system of categorization could be to group the designs into horizontal rectangles, vertical rectangles, and almost square pictures.
When would it be useful to organize pictures according to your system of classification? When wouldn't it be useful? Could you extend it to be more useful? How, or why not?
Example: The sample categorization by rectangle shape might be useful for deciding which picture will hang on a door and which will hang above the couch, or which orientation will be used when the picture is sent to a printer. It would not be useful to someone interested in the contents of the picture. It would be more useful if it also included information on the size of the rectangle, so you could tell a tapestry from a postcard.
Hint: including the line
SRC = "http://www.geom.umn.edu/~math5337/Wallpaper/rotations.gif"
> in the text of a web page will cause picture number 9 to
appear when that page is loaded by Netscape.
Your answer should take the form of a Sketchpad sketch showing mirror lines for all the mirrors you used labeled R1, R2, R3, etc. and with the intermediate images labeled R1(motif), R2 R1(motif), etc. Feel free to change the size and position of the motif, and to hide the text surrounding it.
b) Identify the different symmetries illustrated in the following sample images. Each image consists of more than one part -- each object or group of linked objects should be considered separately. The images should be considered as pictures on a plane, not 3D ojbects. In your answer, state whether or not you're considering different colored objects to be similar.
What different sorts of symmetry did you find? What different sorts of symmetry do you think a finite planar picture can have?
(Once you know the answer to this question, you can categorize all finite planar pictures. Although this may not be the best classification system for museum curators, it could be useful to mathematicians, biologists, or chemists. Our goal in these chapters is to come up with a similar classification system for infinite planar patterns.)
Reflect that motif across one of the mirrors, then reflect the motif and its image across another mirror. Reflect all four images across a third mirror. Continue to reflect copies of the motif across the mirrors until you have a distinguishable pattern formed by at least ten images, without any gaps in it. Examine this pattern -- can you find any rotational symmetries in it? Move your motif until copies of it form a ring around one of the corner points of your triangle. How many copies appear in that ring? Repeat this experiment with the other two corners of the triangle.
The angles between the mirrors were carefully chosen so that you would get a neat tiling of the plane. What properties must be satisfied for a triangle to generate such a nice pattern? Can you think of other polygons that might also generate such a pattern?
(This is meant to be an open ended question. Some things to think about are: the relationship between the number of copies of an image seen in two mirrors and the angle between the mirrors, the pattern that results when you use the interior of your polygon as your motif, the reflected images of the mirrors, and the plane patterns shown by KaleidoTile.)
Use KaleidoTile to display an object in the Tiling window which matches a figure you can make by aranging mirrors at different angles on the surface of a desk. What are the settings in the Symmetry Group and Basepoint windows?
Imagine you could draw in the Basepoint window the same way you can draw on the paper on your desk. Describe three properties you would expect to see if KaleidoTile could faithfully repeat your motif in its tiling.
Suggestions: If you have access to three mirrors, try this experiment yourself. Read the next question, which is closely related to this one.
Build a model of one of the solids shown by KaleidoTile. Draw your own motif on the base triangle. The step command under the help menu shows what happens when you reflect the base triangle across the planes of the mirrors. What happens when you reflect your pattern across those mirror lines? Draw the resulting pattern on your model.
Author: Heidi Burgiel, adapted from notes
used in Math 5337, Spring 1995 by Chaim
Comments to: email@example.com
Created: Dec 7 1995 --- Last modified: Tue May 6 19:57:32 1997
Copyright © 1995-1996 by The Geometry Center All rights reserved.