Illuminations: Symmetries III

Symmetries III

This investigation will help you to understand how translations work and what happens when two or more translations are applied one after the other.

If students are familiar with vectors, they can use them in this context to define a translation in the plane. All band ornaments have translational symmetry, and all wallpaper patterns have translational symmetry in at least two directions.

Learning Objectives
 

Students will be able to:

  • Understand how translations work
  • Understand what happens when two or more translations are applied one after the other

Materials
 
  • Computer and Internet connection

Instructional Plan

Think About...Translations

Take a shape, move it from here to there, and that's a translation. Translations are the simplest of the symmetry transformations and they are also the most important. Artists use translations to create band ornaments and wallpaper patterns.

1. What information must be given in order to define a translation?

2. What does a pattern look like that has been created using translations?

3. What is the result when a shape is translated twice?


Describing Translations

One way to describe a translation is to give a translation vector. In the diagram below, the blue triangle is the translated image of the green triangle with translation vector PQ. You can drag point Q to change the translation vector.

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4. A vector is a mathematical object that has direction and magnitude. Why are vectors useful for defining translations?

5. Can you think of any other ways to define a translation?

6. What is the result when a shape is translated first by one vector and then by another vector?

Creating Patterns Using Translations

If a design is translated over and over using the same translation vector, you get a type of pattern that is called a band ornament, strip pattern, or frieze pattern.

7. The following patterns were created using translation. Identify the initial design and the translation vector that were used to create each of them.

 

a.

 

b.

8. Which of the following patterns were created using translation? For those that were created using translation, identify the initial design and the translation vector.

a.

 

b.

9. Make your own patterns using translation.

10. Find patterns around you that were created using translation.

11. Find examples of translation in nature.

Combining Symmetries

Sometimes a symmetric design is translated repeatedly to create a new pattern.

12. If a design with bilateral symmetry is translated, will the resulting pattern also have bilateral symmetry?

13. If a design with rotational symmetry is translated, will the resulting pattern also have rotational symmetry?

 

Infinite Designs

Imagine a design that has been translated infinitely many times to the left and infinitely many times to the right. The resulting pattern will look the same after it has been translated, and we say that it has translational symmetry. Patterns with translational symmetry in one direction are called band ornaments or strip patterns, and patterns with translational symmetry in two directions are called wallpaper patterns.

14. If an infinite pattern was created using a translation vector to translate a design infinitely many times, what translations will leave the infinite pattern apparently unchanged?

15. If an infinite pattern was created using a design with point symmetry, what kind of symmetry will the infinite pattern have?

16. If an infinite pattern was created using a design with bilateral symmetry, what kind of symmetry will the infinite pattern have?

Taking Stock...Translations

1. What information must be given to define a translation?

2. Describe how to create a pattern using translation.

3. How can you identify a pattern that was created using translation?

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

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™.
  
1 period   

NCTM Resources

Navigating through Geometry in 9‑12

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