Reflection, Rotation, Translation and Glide Reflection

Date: 06/27/2005 at 19:35:27
From: Michele -- Teacher
Subject: Symmetry- Reflections, rotations Transformations

Considering the four symmetry transformations--reflection, rotation, 
translation, and glide reflection--is it possible to express 
transformations in the two-dimensional plane as a composition of at 
most three reflections?

We are working on Geometry--reflection, rotation, transformations-- 
and this is not my area of knowledge.  It is a summer school 
objective, not one I regularly teach.

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 with no magnification/shrinking
involved).

Yes, every plane motion is the composition of three or fewer
reflections:

  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 are there more than one
  possible set that work?  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/