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