"There is no one way"

Wednesday, January 6, 2016

Glide Reflections and Symmetry

In my previous post, I introduced glide reflections, and explained their importance from the point of view of congruent figures: in the most general cases, given two congruent figures in the plane, one is the image of the other in a rotation or a glide reflection. (In some special cases, one is the image of the other in a translation or a reflection.) Another way to state the same thing, as commenter Paul Hartzer pointed out, is that if the composition of two of the well-known rigid transformations (rotation, translation, reflection) is not one of those three, then it is a glide reflection.

In this post, I will give an additional argument in defense of the glide reflection: its importance in analyzing symmetric figures. The Common Core does not have much to say about symmetry (see my analysis.) This is unfortunate, because symmetry provides us with connections to art and design, as well as to abstract algebra, and is very interesting to students.

Symmetry is deeply connected to rigid transformations, and can be defined in terms of those: a figure is symmetric if it is invariant under an isometry. (In other words, if it is its own image in an isometry.) In the most familiar example, bilateral symmetry, the isometry in question is a line reflection. Another well-known symmetry is rotational symmetry. In these examples (from my Geometry Labs), the stick figure is its own reflection in the red line, and the recycling symbol is its own image in a 120° rotation around its center:
   
Therefore the stick figure is line symmetric, and the recycling symbol is rotationally symmetric. These are example of symmetries for finite figures. They are known as rosette symmetries.

But what if we have a figure that is its own image under a translation? That is the case for this infinite row (or frieze) of evenly spaced L's. It is its own image under a translation to the left or to the right by a whole number of spaces:
...L L L L L L L L L L L L...
A frieze can be thought of as an infinitely wide rectangle, with a repeating pattern. A symmetry group is the set of isometries that keep a figure invariant. As it turns out, there are only seven possible frieze symmetry groups. In the example above, translation is the only isometry that keeps the group unchanged. But look at this one:
 
It is invariant under the composition of a horizontal translation and a reflection in a horizontal mirror. In other words, a glide reflection. If you want to analyze frieze symmetry, the glide reflection is absolutely necessary. 

Likewise, if you want to analyze wallpaper symmetry. In this Escher design, for example, the light-colored birds are images of the dark ones in a glide reflection (the reflection lines and translation vectors are vertical.)


Do all students need to know this? Probably not. But to some of us, this is a lot more interesting than many of the "real world" applications of math I have the opportunity to present.

--Henri

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