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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/ 
Associated Topics:
High School Symmetry/Tessellations

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