Totally Tessellated: Symmetry and Transformations, page 1/4

Symmetry and Transformations (1/4)

1. General Information
2. Translation
3. Rotation
4. Reflection
5. Glide Reflection
6. Hands-On Activities

General Information
Symmetry is the preservation of form and configuration across an point, a line, or a plane. In informal terms, symmetry is the ability to take a shape and match it exactly to another shape. The techniques that are used to "take a shape and match it exactly to another" are called transformations and include translations, reflections, rotations, and glide reflections.

There are several different types of symmetry, but in each type of symmetry, characteristics such as angles, side lengths, distances, shapes, and sizes are maintained. Each of the transformations mentioned above produce a different type of symmetry. We will now discuss each transformation and its associated symmetry.

1. General Information
2. Translation
3. Rotation
4. Reflection
5. Glide Reflection
6. Hands-On Activities

Translations and Translational Symmetry
The most simple type of symmetry is translational symmetry which results from the transformation called translation.

Translation is just a fancy term for "move." When a shape is moved, two specifications are needed: a direction and magnitude. Direction can be measured in degrees (e.g., 30 degrees north of east), while magnitude can be measured in inches (e.g., 2 inches) or some other unit of length.

Example of Translation

A simple translation of a point (red) to form another point (blue). Two specifications are needed: direction and magnitude

Here are two examples of translations applied to entire shapes, not just a single point. The original shape and its translated copies are said to have translational symmetry.

Examples of Translations

Here are two examples of translation. As you can see, translation is nothing more than making a copy and then moving it

Try an interactive Java applet that allows you to interactively translate a simple polygon.

Finally, all tessellations have translational symmetry by definition. To say that a tessellation has translational symmetry is to say that it is made of some repeated pattern, and all tessellations are repeated pattern of some sort. Here is an example:

Tessellation with Translational Symmetry

This tessellation has translational symmetry; after moving a copy in a certain direction and with a certain magnitude, you find that the copy matches exactly the original

Try an interactive Java applet that allows you to interactively explore the translational symmetry of a tessellation.

Real examples of translational symmetry:

After trying the hands-on activity, use the templates on the templates page to see why every regular, semiregular, and demiregular tessellations has translational symmetry. Use the technique that was explained in the hands-on activity.

Web Links
Symmetry (ideas and exhibits)

Symmetry (advanced)

The Geometry Junkyard: Symmetry and Group Theory (advanced)

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