"There is no one way"

Sunday, January 3, 2016

Glide Reflections and Congruence

If you are a regular reader of this blog, you know I am very interested in transformational geometry, and have written a fair amount about it: some curricular materials for students, and some mathematical background for teachers and curriculum developers. See the links on this page of my Web site.

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: