A TRANSLATION is the symmetry operation that "slides" or "shifts" a point or design element a certain distance along a certain line. A translation is always done without rotation. Connecting any point and its translated image is a translation vector that shows the specific distance and specific direction that defines that specific translation.

Applying any transformation to a point gets you to an image point. Repeatedly applying a translation vector to a design element and each new image generates an infinite set of equivalent elements. This is different than either a reflection or a rotation, in which a design element generates a specific number of equivalent elements.

What would happen if you applied more than one transformation to a design element? This is referred to as a composition or combination of operations, and the most important composition is a glide reflection. Footprints in the sand suggest why this is a very natural type of pattern.

A GLIDE REFLECTION is the symmetry operation that combines a reflection and a translation. Under a glide reflection, you find the image of a point or design element by doing a "flip and slide". This requires both a glide mirror and a glide vector, which are always parallel. We will refer to the tool for a glide reflection simply as a glide mirror/vector. In diagrams, we will show reflection mirrors and translation vectors with solid lines or rays, and show glide mirrors and glide vectors as dashed lines or rays.

In a glide reflection, it would not matter whether you did the translation or reflection first, as long as it is immediately followed by the second operation. In a glide reflection, a design element and its mirror-image both show, but they are not directly across the mirror from each other.

Action

1. Working with a partner or two, use two blocks to create a simple asymmetrical shape. Using copies of this shape to act as an original and its image(s), illustrate a few examples of translations. Identify the vectors for translations. Illustrate how moving a vector changes the location of the image(s).

2. Which of the pattern block shapes can be used to make a translation in which identical neighboring blocks fit flush? Show each translation and its vector.

3. Use a single pattern block shape to illustrate a glide reflection. Trace the shape and show the glide mirror/vector as a dashed line and dashed ray. Use a piece of transparent film to act out how to get from one design element (an individual block) to its neighboring image.

Now use two blocks to create a simple asymmetrical shape. Using copies of this shape to act as an original and its image(s), illustrate a few examples of glide reflections. Identify the glide vectors and glide mirrors for each. Illustrate how moving a glide vector or mirror changes the location of the image(s).

4. Can a design that exhibits simple bilateral symmetry also have glide reflective symmetry? Explain.

5. In a group, create an interesting glide reflective pattern that uses at least two different pattern block shapes and has matching edges. Use the templates to make an illustration of the pattern, and show the glide mirror/vector clearly with a colored marker.

6. We will adopt the following notation to describe compositions of transformations. We will use "R" to represent a rotation, "M" for a mirror reflection, "T" for a translations and "G" for glide reflection. A glide reflection could be expressed as "TcM", read "T composition M", tells you to apply both operations, and that they are applied as listed from right to left. (This is contrary to the convention of reading in English, but correct for mathematical notation.) For some combinations, the order will not make a difference in the pattern that results. For example, we know that order wouldn't make a difference for a glide reflection, so it could be symbolized as "TcM" or "McT". However, for many other combinations the order makes a big difference in the resulting pattern.

7. As a warm up for performing compositions, show how to generate each of the following starting with only a segment or a triangle:

• the hexagon, using a combination of rotation
• the square as a combination of a rotation and a reflection
• each of the other blocks as a combination of two operations

8. Dihedral symmetry is a composition of a reflection and a rotation. Specifically, describe D₆ as a example of such a composition.

9. Give a few examples of a composition of two reflections through perpendicular mirrors.

10. Give a few examples of a composition of two reflections through parallel mirrors.

11. Can you find at least one design that exhibits all four types of symmetry: rotational, reflective, translational and glide reflective?