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Re: mirrors: two or more?



From: Glenn A. Carlson <gcarlson@MAIL.WIN.ORG>

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).
[snip]
I follow your argument below but you assume the property of
involution (A*A = I) which John Denker has chosen to ignore. The physical
mirror with which we are all familiar has this property of course, but
needless to say the whole exercise is precisely a hypothesis on the
(mathematical) existence of others.
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.
Enantiomorph prop. 1 : Enantiomorphs in n dimensions are are identical in
n+1 dimensions. Please show exactly how this is dependent on mirrors. As I
said in my previous post, IMO an independent definition is possible, because
of statements like the one above.




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