Re: mirrors: two or more?

[Abhishek Roy <fingerslip@YAHOO.COM>]

From: John S. Denker <jsd@MONMOUTH.COM>
Sent: Saturday, August 05, 2000 12:02 AM

        Thank for making the point very well. Continuing - I read that (as
first pointed out by Moebius) that an asymetric solid object may be reversed
be turning it through a higher space. So a right hand may be converted into
a left by flipping it in the 4th dimension. Similarly one may turn the
letter 'L' and the number '5' which are asymetric in 2-space. So the
mathematics behind this (which I am very eager to learn) might form a
starting point from which one could answer such questions as the ones below.
For example (I am simply guessing here), you might derive that there could
be only two froms of opposite handedness, from which using the some basic
assumptions about a mirror (isometry but *not* involution which you change
below), one may be able to see that there is only one basic kind of mirror.
        Now is the higher dimensional flip the best or only way to elegantly
define enantiomorphs? Can it be continued in the aforementioned way?
Unfortunately, I simply don't know enough maths to answer. Can anyone help?
Regards,
Abhishek Roy

> > Suppose an object had two possible mirror images.  Which image would
> > you see if you placed the object in front of a mirror?
>
> That argument is ingenious, but it does not really answer the questions
> Abhishek Roy has been asking.
>
> The argument is somewhat circular because it assumes there is only one
kind
> of mirror.  This is not a safe assumption, for the following reasons:
>
> 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
>
> How do you KNOW that no such thing exists?
>
> 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
>
> How do you KNOW that no such things exist?



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