Connecting Geometry©

Chapter 2

Symmetry and Transformations

Perhaps you have heard of the word Symmetry before: we often talk about the symmetry of a design in art, and we are all familiar with the symmetry of the human face and body: a face usually appears to be approximately the same on the left as on the right. (Source for this image: Clark Kidder; Madison Daniels. Marilyn Monroe unCovers. Edmonton, Alberta, Canada: Quon Editions, 1994.)

This is called Reflection Symmetry. The left side of the image is approximately the same as the right side, but reflected, as in a mirror, across the line of symmetry. The line of symmetry is often called a mirror.

There are other kinds of symmetry. The stained glass window below, from Chartres Cathedral in France, has Rotation Symmetry:

Rose Window, Chartres Cathedral, France

For more information about Chartres Cathedral, click on the link below:


If a tracing of a figure can be rotated less than 180° around a central point, and matches up with the original, then we say the figure has Rotational Symmetry. In the picture below, you can see that the Rose Window has 90° rotational symmetry:


The project for this chapter is to create a symmetrical geometric design, such as the examples below. You can create a design with Rotation Symmetry, such as the first one on the left below, or with Reflection Symmety, such as the one on the right below. Begin by constructing a simple geometric shape (a triangle, for example) and then use the Transformations menu to repeat the shape in a symmetrical pattern. a Sketchpad example is shown below:

Then you can color your graphic using a computer painting program, or colored pens and pencils. If you do the design using Geometry software (such as the Geometer's SketchPad) it will look better if you Hide the points: first. (To do this, click on the Point tool, choose Edit . . . Select All Points, then Display . . . Hide)

This beautiful geometric image was created by a high school geometry student, using SketchPad and then a painting program:

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