Geometry, Geometric Reasoning, and Visualization (1/10/2002)
Department of Mathematics
York College (CUNY)
Jamaica, NY 11451
Email: firstname.lastname@example.org (for additions, suggestions, and corrections)
Geometry is a very important branch of mathematics, yet increasingly few students in the Mathematical Sciences are electing to take courses with significant geometric content. My purpose here is to clarify via an example some of the issues in comparing and contrasting geometry, geometric reasoning, and visualization.
Suppose you are a graphic designer and you want to prepare a decoration for a scarf which would consist of taking a motif of some kind and translating along a line to generate the design. We are constructing a mathematical model here because the scarf will consist of a finite piece of the design we are thinking of as being on an infinite strip. Designs of this type are often called frieze designs or strip designs. (Not only do we sometimes see them on a scarf but also on buildings.) Thus, if we use as the motif the letters H and A we would get the two designs for a band on a scarf as shown below:
(strip based on an H:) ..... H H H H H H H.......
(strip based on an A:) ..... A A A A A A A......
The problem we wish to address is to count how many different kinds of strip designs are possible. This is a problem in geometry and it helps clarify the ideas of geometry. geometric reasoning, and visualization.
First of all we have to deal with the issue of when two patterns are different. Since there are infinitely many different motifs to choose from, in a sense there are infinitely many strip designs possible. However, suppose we classify the strip designs based on the geometric transformations which transform the infinite strip onto itself. Thus, we are looking for those geometric transformations which are called isometries (preserve distance). One can determine that these consist of translations, rotations, reflections, and glide reflections. Thus, the strip of A's admits translations and reflections in a vertical axis as isometries, while, assuming that the bar across the H is in the middle, we get translations, rotations, reflections in a horizontal axis, reflections in a vertical axis, and glide reflections.
Geometry problems often involve the kinds of issues raised here. Counting something, discussion of what concept of isomorphism underlies when two things are the same, distance, functions, etc. In attacking geometry problems we can use any of the techniques in the mathematical armory. These include ideas from combinatorics, analysis, algebra, or geometry itself. Thus, one might, in the problem here, introduce the idea of a group of transformations and use group theory (a subject with roots in both algebra and geometry) to move forward.
In thinking about how many different strip patterns there are, one might think about what rotations will transform the strip onto itself. Using geometric reasoning one sees that only the identity rotation or one through positive or negative multiples of 180 degrees will work. Such rotations are often called half-turns. However, depending on the motif we start with, a half-turn of a strip may or may not transform the design on the strip onto itself. To see what is going on requires visualization skills. Can you see in your mind's eye that when the strip of A's is given any half-turn, it will not transform onto itself while there are points about which the strip based on the H motif will transform into itself? Can you describe exactly where the points of rotation for half-turns of the strip which transform the design on the strip to itself are located?
Visualization skills involve either thinking about a geometrical object in one's mind's eye or using such skills to start with one geometrical situation and draw another. For example, using visualization one can carry out the following processes: draw what you get when the strip of A's is rotated 90 degrees counterclockwise; 90 degrees clockwise, 180 degrees. Instead of visualization problems based on the whole strip one might limit oneself to just a motif. If the motif were the letter F, one could visualize: the result of rotating the F 90 degrees clockwise about the point where the top horizontal piece meets the vertical piece of the F; about the point where the middle piece of the F meets the vertical pieces; the point in the middle of the horizontal piece. One can visualize the result of reflecting the F in the line lying along its middle piece or the line lying along its top. Visualization skills get much harder when one wants to visualize the results of carrying out operations on 3-dimensional objects in 3-dimensional space. John Conway, Bill Thurston, and Peter Doyle developed a variety of nice examples of problems in visualization for their course for liberal arts students at Princeton University that paid homage to the famous book of the same title by Hilbert and Cohn-Vossen, Geometry and The Imagination. Notes for their course can be in part found in reference  below.
Using a variety of tools from geometry, geometric thinking, and visualization one can arrive at the result that there are 7 different kinds of strip patterns. Mathematicians have extended this work to colored designs on strips and to designs and colored designs in 2-dimenions. Along the way one learns many lessons about geometric phenomena.
1. Malkevitch, J., (ed.), Geometry's Future, Consortium for Mathematics and Its Applications (COMAP), Lexington, 1992.
2. Steen, L., (ed.), On the Shoulders of Giants, National Academy Press, Washington, 1990.
3. Washburn, D. and D. Crowe, Symmetries of Culture, Washington University Press, 1988.
4. Weeks, J., Exploring the Shape of Space, Key Curriculum Press, Emeryville, 2001.
5. Zimmermann, W. and S. Cunningham, (eds.), Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington, 1991.
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