Re: naive questions regarding isometries

From: Keith A. Lewis (lewis_at_PROBE.MITRE.ORG)
Date: 11/23/04 

Date: Tue, 23 Nov 2004 00:51:12 +0000 (UTC)

sean_mcilroy@yahoo.com (Sean McIlroy) writes in article <e5063e96.0411221610.10a8f6c0@posting.google.com> dated 22 Nov 2004 16:10:23 -0800:
>Can anybody point me toward answers to the following questions?

This stuff was covered pretty well in a textbook I used in high school:
_Geometry: A Transformational Approach_.

>*) Suppose F is a figure, R1 and R2 are reflections, and G1,G2 are the
>respective images of F under R1,R2. In two dimensions G1 and G2 are
>necessarily congruent. What about the three-dimensional case?

A mirror image switches the "handedness" of a figure, but doing it a second
time switches it back.

>*) What conditions determine whether a two dimensional figure can be
>rotated and translated back to itself after being subjected to a
>reflection?

A rotation is equivalent to a sequence of reflections over 2 intersecting
lines. A translation is equivalent to a sequence of reflections over 2
parallel lines. Neither of those switch handedness, so the figure would
need to be bilaterally symetric (congruent with at least one reflection of
itself).

--Keith Lewis klewis {at} mitre.org
The above may not (yet) represent the opinions of my employer.

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