Illuminations: Symmetries IV

Symmetries IV

Of all the different symmetries, this is the hardest for students to understand and to identify. Glide reflections are very tricky to identify in a tiling or wallpaper pattern. The “footprints” image helps, but you can still expect students to have trouble. Using tracing paper over a design may also help.

Learning Objectives
 

Students will

  • learn about glide reflection—a symmetry transformation that is made up of two other symmetry transformations, a translation and a reflection

Materials
 
  • Computer and Internet connection

Instructional Plan

Think About...Glide Reflections

Footprints are a perfect example of a glide reflection—a symmetry transformation that is made up of two other symmetry transformations, a translation and a reflection.

1. How would you describe the symmetry of a row of footprints?

2. Can you find other designs or patterns that look like a row of footprints?

3. What information must you give in order to describe a glide reflection?

4. What is the effect of performing the same glide reflection twice?


Describing Glide Reflections

A glide reflection is a symmetry transformation that consists of a translation followed by a reflection across the translation line. In the sketch below, the green polygon has been reflected across the pink line to give the pink polygon which in turn has been translated by the blue translation vector AB to give the blue polygon. Drag any of the points on the polygon to change its shape, drag point A to change the direction of the mirror line, and drag point B to change the length of the translation vector.

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5. What information must you know to describe exactly what glide reflection is being made?

6. When will the image under glide reflection look like a translation of the original shape?

7. Can a glide reflection have any fixed points?

 

Glide Reflection Designs

8. The following designs were created using glide reflections. Find the reflection line and the translation vector.

a.

b.

9. Which of the following band ornaments were made using glide reflections? For those that were, find the mirror line and the translation vector.

a.

b.

c.

10. The following wallpaper patterns were created using glide reflections. Find two different glide reflections in each pattern and give the mirror line and the translation vector for each.

a.    b.    c.

11. Find some patterns in your environment that were created using glide reflections.

12. Use glide reflections to create your own patterns.

 

Composing Translations and Reflections

In a glide reflection, the translation is always a translation along the mirror line of the reflection. What if you have a translation that is not along the mirror line of the reflection?

In the sketch given below, you see a green figure that has been reflected across a red line SR and then translated using the red translation vector PQ; the final image is shown in red. The green figure has also been reflected across the blue line AB and then translated using the blue translation vector AC; this image is shown in blue.

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13. You can change the location and slope of the blue mirror line AB by dragging points A and B and you can change the length of the translation vector AC by dragging point C. See if you can get the blue image to coincide with the red image. What is the relation of the blue mirror line AB to the red mirror line RS when the blue image coincides with the red image?

14. Why do you think mathematicians require that the mirror line and the translation vector be parallel for a glide reflection?

 

Taking Stock...Glide Reflections

1. What information do you need to give in order to define a glide reflection?

2. Describe how to make a pattern that has glide reflection symmetry. Make your own design.

3. How can you tell if a glide reflection was used in creating a pattern?

4. Why do we require that the mirror line of a glide reflection be parallel to the translation vector of the glide reflection?

NCTM Standards and Expectations
 
Geometry 9-12
  1. Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
  2. Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools.
  3. Understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices.
  4. Use various representations to help understand the effects of simple transformations and their compositions.

References
 
  • For All Practical Purposes: Introduction to Contemporary Mathematics, W.H. Freeman and Company, New York, 1997

  • Understanding Congruence, Similarity, and Symmetry Using Transformations and Interactive Figures: Visualizing Transformations, NCTM Principles and Standards for School Mathematics: E-example 6.4. http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm

  • Java Applets were created using the Geometer’s Sketchpad™ and JavaSketchpad™.
  
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