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Re: mirrors: two or more?



{Resending with proper subject line. GAC]

Subject: Re: mirrors: two or more?
Date: Sat, 5 Aug 2000 13:49:21 +0530
From: Abhishek Roy <fingerslip@YAHOO.COM>

I cannot speak for John S. Denker, but I don't think he is trying to
describe a "standard flat mirror". His hypothetical mirrors are as real the
familiar ones if you can't prove that they are non-existent (mathematically
not physically).

By definition, the reflection operator R transforms as x -> -x.

Now, by the properties of real numbers, R has the property that R*R^-1
= R*R = I, i.e., the reflection operator is its own unique inverse.

Decker 1: Suppose we had two kinds of mirrors; let's call them V and
W
respectively. We could then have the following properties:
two V-reflections makes a W-reflection (V^2 = W)
two W-reflections makes a V-reflection (W^2 = V)
V^3 = W^3 = VW = WV = I

Since the reflection operator is its own inverse, there are no
reflection operators V and W which have the property in Decker 1. In
particular, for any reflection operators V and W, VV != W and WW != V.

Decker 2: Suppose we have two kinds of mirrors; this time let's call
them V and C
respectively. We could have
V^2 = C^2 = I
VC = CV
VC != I

Since the reflection operator is its own unique inverse, there are no
reflection operators V and C which have the property in Decker 2. In
particular, for any reflection operators V and C, VC = CV = I.

If we abandon the definition of the reflection operator above, then
what definition are we using?

Since the properties of enantiomorphs depend on the definition of the
reflection operator above, if we abandon the definition, we are
talking about a entirely different problem. And, until we define what
this alternative reflection operator is, there is no problem to talk
about.

Glenn A. Carlson, P.E.
St. Charles County Community College
St. Peters, MO
gcarlson@mail.win.org