Re: A question about mirrors

[Abhishek Roy <fingerslip@YAHOO.COM>]

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

> > Actually my primary
> > query was regarding a definition of 'handedness' independent of both
> > specific examples and mirrors.
>
> It is impossible to define "right-handedness" and "left-handedness"
> without reference to a specific example.
        My objection is not to examples per se. We would never get off the
ground without them. But does using a specific example actually help in
describing the general case with complete rigour?
[snip]
> In particular, the "triplet of directions" definition seems too
> specific an example even though it only assumes knowledge of what
> "perpendicular" means (and a three-dimensional Euclidean space) and
> does not assume anything about a mirror.  You can't get any simpler
> (i.e., general) than that.
        I don't get why you equate simple with general. I most certainly
agree that the triplet is the simplest asymetric object you can describe.
But for argument's sake consider the following solid (which I had described
earlier in a different way) which may be constructed as follows : Take four
indentical cubes. Paste three of them together to form a 'L' shape, lying
flat on its side. Now stick the fourth cube on top of the cube forming the
short arm of the 'L'. You have just got yourself another 'example' which
requires only a little more Euclidean geometry to construct, does not assume
anything about a mirror, and has the advantage of being more fun to play
around with (atleast I think so). Take it in your hand, sit front of the
mirror and convince yourself that it is distinct from its mirror image.
Finally take four new blocks and show that no third solid may be constructed
in an analogous manner.
        Before you start to express doubts about my sanity, the point of all
this is that we have achieved the same results with both the above objects.
Taken together they provide a very useful guide to the property of
handedness. BUT neither of them - as far as I can see - provide any clue to
a complete and rigorous definition of the same - which brings us to

> If even this triplet of directions is excluded as "too specific," then
> the answer to your query is "No, there is no way to define or explain
> handedness independent of even one specific example or reference to
> mirrors."
        This is where I disagree emphatically. I am convinced of the
converse, and this was the very reason for my asking the question on this
list. The fact that enantiomorphs in n dimensions are identical in n+1
dimensions and indeed may be interchanged by a turn, is a pointer to how it
might be done.



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