Glide Reflections and Glide Reflective Symmetry

    A last type of symmetry is glide reflective symmetry which results from the transformation called glide reflection.

    A glide reflection is actually a combination of a reflection and a translation. Whether the reflection happens first or second does not matter. The figure that results after a reflection and translation is simply called the glide reflection of the original figure. (Notice how "glide" refers to the translation part of the combination.)

    The same terms that apply to reflections and translations apply to glide reflections. An axis is needed to perform the reflection and a magnitude and direction are needed to perform the translation.

Here are two examples of glide reflections applied to some other shapes. The original shape together with its reflection is said to have glide reflective symmetry.

What does it mean for a tessellation to have glide reflective symmetry? If we can perform a glide reflection to a tessellation that such that the result is the same as the original tessellation, then the tessellation has glide reflectional symmetry. An example is as follows:

(One interesting note for further explorations: Every tessellation that has reflective and translational symmetry also has glide reflectional symmetry. Do you see why? After performing the appropriate reflection, we can perform the appropriate translation. The overall transformation demonstrates that the tessellation also has glide reflectional symmetry. However, not all tessellations with glide reflection symmetry have reflective symmetry and not all have translational symmetry.)

HERE, THE LETTER F (and the blank space around it) is taken as our fundamental region to demonstrate the four basic symmetry operations or rigid motions: