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Re: A question about mirrors



At 02:07 PM 8/2/00 +0530, Abhishek Roy wrote:
Along the same line, how do you show that apart from L and R forms there
is
not a third - X form of the hand which is 'opposite' to both the others.

Wow, that's a deep question.

The answer has to do with the relationship between the reflection operator
and the rotation operator. Actually there are several possible answers:

1) In a world where rotation is a symmetry, you can pick a single
"standard" reflection (such as the one that takes X to -X) and then show
that
-- any odd number of reflections is equivalent to one standard
reflection
plus a rotation
-- any even number of reflections is equivalent to a rotation.
Thank you for your answer. Yes, I am familiar with the proof. But I was
looking for some way to describe a reversal of 'handedness' without using
mirrors. For example- two books I looked at started by saying that all
isometries could be descibed as products of reflections. Then you show that
any even number of reflections may be reduced to a rotation and an odd
number of reflections into a (reflection+rotation). Alright so far, but I
don't understand how you now classify the latter as 'orientation-reversing'.
It seemed to me that there must be more fundamental way of describing
orientation (not the specific L or R, just handedness), which you could use
to see that ther were only two forms and *then* apply that to a mirror
transformation.
Abhishek Roy






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