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Exploration of Isometries

Draw a motif on a piece of paper. Darken it so that you can see it when a thin sheet of paper is laid over it.

Trace the motif on a second, overlaid, sheet of paper, translate the paper, and trace again. You will have drawn a new motif and a translated image of it. (How would you draw a rotation of the image or a reflected copy of the image?)

Notice that you can't tell from your tracing which is the motif and which is the image. The translation taking the first object to to the second has an inverse, which takes the second object back to the first. In fact, every isometry has an inverse. The inverse of a translation is a translation. Is the inverse of a rotation another rotation? What is the inverse of reflection?

Have one member of your group secretly draw a motif and its image under rotation on a fresh piece of paper. Can the rest of the group determine what the rotation is? How, or why not? What is its inverse?

Once you have guessed the angle of rotation and its center, indicate the center of rotation with a dot. Draw an arc with an arrow to indicate the angle of rotation. How could you construct the precise center of rotation using ruler and compass? How would you find the angle of rotation?

Create a motif and its reflected image. Check your work by placing a real mirror along the line of mirror symmetry; you should see an image of the original object exactly where you constructed its image to be. What is the inverse of the reflection?

There are many exciting theorems about isometries to discover, but let's leave them for now and experiment with compositions of reflections.

[pattern formed by parallel reflections]

With real mirrors, it's easy to study the product of two reflections. Put a second mirror on the desk near your first. How many reflected copies of your motif do you see? How are they alike? How are they different? Holding the mirrors perpendicular to the desk, adjust the angle between them and watch how the image changes.

Use a protractor to measure the angle between the mirrors (you might want to trace the lines where the mirrors meet the desk and measure the angle between them -- the angle between two planes can be defined to be the angle between the lines they make when intersected with a perpendicular plane.) Can you find a relationship between the number of images you see in the mirrors and the angle between the mirrors? What happens when the mirrors are parallel?

Now see what you can do with three mirrors. Start out with the mirrors standing perpendicular to the desk and forming a triangle around your motif. What sorts of different patterns can you make this way? How do the angles between the mirrors relate to the patterns formed?

An example of a pattern formed by three mirrors is found in a kaleidoscope. Adjust the mirrors to form a pleasing pattern of your own design and record the angles between the mirrors. Compare your choice of angles and your pattern with those of the other groups in the class.

We've just seen how combinations of mirrors can be used to make patterns in the plane. All of what we've done could be reproduced using The Geometer's Sketchpad, working strictly in the plane of the computer screen. (You may wish to try this). What can we do with mirrors if we use all three dimensions of the space we live in?

If you tilt one of your mirrors so that it's no longer perpendicular to the desk, what happens to the image of the surface of the desk? What happens to the copies of your motif? Can you guess which angles of tilt will give interesting patterns? If you have the worksheet "Creating Polyhedra with Three Mirrors", you may start working on it now.

Next, please work as far as time permits through the KaleidoTile activity sheets. These worksheets were written by teachers at the Geometry Center during the summer of 1995. As you work, think about how you might use them in your classroom. What aspects would work well in class? What might not work well? What would you change if it were up to you?

If you finish early, you can read Evelyn Sander's WorldWideWeb press release announcing the first version of KaleidoTile, or see if you can prove some of the theorems about isometries that were mentioned above.

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Author: Heidi Burgiel, adapted from notes used in Math 5337, Spring 1995 by Chaim Goodman-Strauss.
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Created: Dec 7 1995 --- Last modified: Jun 11 1996
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