Transformational geometry used to be an eccentric preoccupation of mine, a big part of Space since the early 90's. (Space is one of my off-the-highway-to-calculus post-Algebra-2 elective courses.) The Common Core State Standards for Math (CCSSM) pushed the topic into the mainstream, more or less, a development I welcome.

The CCSSM mention three

*rigid motions*(aka

*isometries*), and suggest some basic assumptions about them. The three are reflections, translations, and rotations, and they are indeed fundamental. This post is about a fourth isometry, the

*glide reflection*. I do not think it would have made sense to include it in the CCSSM, because those standards are already overstuffed. (See my analysis here.) Quite the opposite: in an ideal world (ha!) the CCSSM would be pruned so as to really be

*core*standards. In such a world, there would be time to teach

*actually-core*standards well, and to complement them with interesting mathematical side trips, not the same side trips in every school or classroom.

Glide reflections belong in such an ideal-world side trip. 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.

But enough introduction. What is a glide reflection? It is the composition of a reflection, and a translation in

*a vector parallel to the line of reflection*. This image (from Richard Brown's excellent but dated

*Transformational Geometry*) gets the idea across: