definiton and example of glide reflection

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. Here is an additional argument in defense of the glide reflection: its importance in analyzing symmetric figures. From the point of view of congruent figures, one can argue that glide reflections are as important as rotations.Īs for #3: what do I mean by symmetries beyond the rosette? In fact, what do I mean by rosette, frieze, and wallpaper symmetry? Read on. Note that the activity is more fun if you go a little further: if it's a reflection, where is the mirror line? If it's a rotation, where is the center? (The latter question is hard for most beginners.) Note that reflections and translations require a fairly special setup, while rotations and glide reflections are very general. The 'cheat' button will likely not be necessary, but it is there just in case. (Along the way, we prove #1.)Īs one way to experience this at an intuitive level, you can have your students guess the mystery isometry, in various cases, starting here. See the epic proof of this result, abundantly and interactively illustrated with the help of GeoGebra, here. What this theorem tells us is that if there is such a sequence, then it is equal to a single rotation, reflection, translation, or glide reflection. Think about it: in the CCSSM, two figures are deemed congruent if one is the image of the other in 'a sequence of rotations, reflections, and translations'. #2 is truly a fundamental and phenomenal theorem. Glide reflections are essential to an analysis of symmetries, as soon as you go beyond rosette symmetry to frieze and wallpaper symmetry.Given any two congruent figures in the plane, one is the image of the other in one of the three CCSSM isometries, or in a glide reflection.

This turns out to be super-convenient, and is not the case with the other cases mentioned above.

In a glide reflection, the two component transformations can be performed in either order.

(The figure is from Richard Brown's excellent but dated and out-of-print Transformational Geometry) Glide Reflections and CongruenceĪ glide reflection is the composition of a reflection, and a translation in a vector parallel to the line of reflection. In the less-than-ideal world we live in, they can provide some good exercises and problems in grades 8-10, serve as an enrichment topic in precalculus, or be part of a full-fledged unit in a Year 4 advanced geometry elective.īut enough introduction. Glide reflections belong in such an ideal-world side trip. Of course, the side trips would not be the same in every school or every classroom. In such a world, there would be time to teach actually-core standards well, and to complement them with interesting mathematical side trips. (See my analysis here.) Quite the opposite: in an ideal world (ha!) the CCSSM would be pruned so as to really consist of core standards only. I do not think it would have made sense to include it in the CCSSM, because those standards are already overstuffed. This post is about a fourth isometry, the glide reflection. The three are reflections, translations, and rotations, and they are indeed fundamental. The CCSSM mention three rigid motions (aka isometries), and suggest some basic assumptions about them. On this page, I explain the significance and importance of the glide reflection, an important rigid motion that is not mentioned in the CCSSM.

But the Common Core State Standards for Math (CCSSM) pushed the topic into the mainstream, more or less, a development I welcome. It included a fair amount about transformational geometry, a part of math which at the time was a somewhat eccentric preoccupation of mine. Space is a course I taught from 1991 to my retirement in 2013.

Definiton and example of glide reflection