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Homework: Combining Symmetries

  1. Part of the purpose of the next three units is to explore the mathematical concept of "classification" of mathematical objects into types. The goal of this exercise is to invent your own classification system for the pictures shown on the Patterns Sampler web page.

    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 < IMG 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.

  2. 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?

  3. 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.

  4. 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.

    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.)

  5. Using the Geometer's Sketchpad, construct three lines that 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? 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.)

    Sketchpad Hints:

  6. KaleidoTile provides a illustration of an object created by reflection in three mirrors at fixed angles to each other. Unfortunately, you cannot change the object that is being reflected.

    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.

    1. Construct two lines that intersect at an angle of 180/n degrees, for some n between 2 and 5. Create a motif in the angle between the two lines. Reflect the motif in the mirrors, then reflect its image, then reflect the images' images until you cannot generate any new images. How many copies of the motif (including the motif) can you get? Describe the number of copies possible as a function of n and explain why your formula is correct.

    2. 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 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.

  7. The worksheet on Creating Polyhedra With Three Mirrors is a paper handout describing a hands-on activity. The Introduction to KaleidoTile Software was a web page. What are the advantages and disadvantages of the two formats? Which activity would you rather use in your classroom? Why? When teaching, when would you use handouts and when would you have students read instructions on the computer? Why?

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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: Tue May 6 19:57:32 1997
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