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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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