Transformations in the 2D Plane as Composition of Reflections
Date: 06/27/2005 at 19:35:27 From: Michele Subject: Symmetry- Reflections, rotations Transformations I am teaching Geometry--reflection, rotation, transformations--and this is not my area of knowledge. (It is a summer school objective, not one I regularly teach.) Considering the 4 symmetry transformations--reflection, rotation, translations, and glide reflections, is it possible to express any transformation as a composition of at most three reflections?
Date: 06/29/2005 at 12:29:40 From: Doctor Douglas Subject: Re: Symmetry- Reflections, rotations Transformations Hi, Michelle. This sounds like the "geometric transformations" of a 2D plane (at least the ones that preserve scale--no magnification/shrinking is involved). Yes, every plane motion is the composition of three or fewer reflections. Let me give you some questions and actvities that you can work on with your students to help think about why that is the case: A reflection is of course a composition of one reflection --itself. It can also be a composition of three reflections, but not two (ask your students: why not?) It may help to investigate what a series of N reflections does to say a capital letter "R". A translation is a composition of two reflections. Ask your students: "Why not one and why not three?" Again, it may help to imagine a capital letter "R". A further topic is to ask your students to see if they can IDENTIFY or CONSTRUCT two reflections that combine to make any given translation (for example "6 meters northwest"). If they are comfortable with the Cartesian coordinate plane this will be easier, but it can be done just by drawing reflection axes. This type of thinking is good geometric training. Is the set of two reflections unique or is there more than one possible set that works? For the set(s) that do work, do you notice anything curious about the directions of the reflection axes, relative to each other and relative to the translation direction? Can you relate the distance between the reflection axes and the distance of the translation? Any glide reflection is of course a combination of a translation with a reflection, so if you can make a translation with two reflections, then you can make a glide reflection with three reflections. Can your students exhibit three reflections that work? Any rotation is the combination of two reflections. You might guess that it is two because of the way it transforms the letter "R". Can your students find two reflections that work? Where do the reflection axes intersect and what is the measure of the angle between them? There's still more work to be done to fully establish that any plane transformation that preserves scale is the composition of at most three reflections. For example, you would have to prove that the combination of a translation with a rotation is a (different) rotation. - Doctor Douglas, The Math Forum http://mathforum.org/dr.math/
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