Homework: Combining Symmetries

1. Using The Geometer's Sketchpad, draw a motif (for example, construct a polygon interior); we will call this motif F. Draw two lines, R1 and R2. Look at R1 R2 (F) = R1(R2(F)) and R2 R1 (F) = R2(R1(F)), where R(F) is the figure generated by reflecting F across line R. Does R1 R2 equal R2 R1? How do you know?

2. Find three or fewer reflections that transform the motif into the image shown in this Sketchpad sketch [GSP Help]. Check your work as suggested. You may wish to refer to the theorems mentioned in the previous section.

   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.

3. a) Find, draw, or construct a picture of a finite object (snowflake, house, flower, pinwheel, quilt square, polygon) on a piece of paper. (If your computer can run Java programs, check out this neat snowflake drawer, written here at the Center!) What symmetries does your object have? Discuss your findings with your classmates.

   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.

   • Three groups of triangles
   • >Images from the movie "Outside In"

   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.)

4. Using the Geometer's Sketchpad, construct three or four lines so that they meet at angles of 30-60-90, or 45-45-90. Construct a motif inside the triangle bounded by the mirrors.

   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?

   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.)

   Sketchpad Hints:

   • Try not to create too many overlapping copies of your motif. The picture won't look any different, but it may take a very long time to print or save if you create duplicate copies of each image.
   • The Define Transform option allows you to select an object and its transformed image and add a menu item that will perform that transformation on a selected object.
   • Try not to reflect your mirror lines along with your motif and image. Extra mirror lines will make your final picture more difficult to interpret.

5. 1. Construct two lines that intersect at an angle of 360/n degrees, for some choice of n. Create a motif in the angle between the two lines. Construct the n-1 images generated by reflecting the motif in the mirrors, then reflecting its image, then reflecting the images' images...

   2. Now imagine repeating the process by reflecting a picture in three dimensional space with three mirrors. Kaleidotile shows some of the patterns you might get.

      Build a model of one of the solids or the plane 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.

6. Think about teaching from the materials you just read. Are the instructions clear enough? Why or why not? When would you print them out? When would you have students read them on the computer? Why? What would you change to make them more useful to you?

The Geometry Center Home Page

Author: Chaim Goodman-Strauss, revised and edited by Heidi Burgiel
Comments to: webmaster@geom.umn.edu
Created: Dec 7 1995 --- Last modified: Jul 31 1996
Copyright © 1995-1996 by The Geometry Center All rights reserved.