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Re: handedness without artifacts



Abhishek Roy <fingerslip@YAHOO.COM> wrote:

>Actually my primary
>query was regarding a definition of 'handedness' independent of both
>specific examples and mirrors.

Then at 02:15 PM 8/5/00 -0500, Glenn A. Carlson wrote:
It is impossible to define "right-handedness" and "left-handedness"
without reference to a specific example.

True.

And, given the constraints of your query, it also appears impossible
to define "handedness" since you seem to exclude any definition
referencing examples from geometry, chemistry, astronomy, particle
physics, etc.

I disagree. It is quite possible to say that the special unitary group
SU(3) differs from the unitary group U(3) without reference to any
particular real-world example.

And if one does choose to cite an example, it doesn't need to be a special
example. Practically any object of reasonable size is unlike its mirror
image. Grab a potato off the shelf. I bet you $100 it's macroscopically
chiral.

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)

Abhishek Roy correctly used the word "isometry". Perpendicular, and other
needful concepts, can be defined in terms of distances between points.

The idea here is that given any two collections of points,
identically described in terms of distances between points, we find they
are identical except for a translation, a rotation, and possibly a mirror
image. This is not a trivial statement. It is not silly to ask
-- how much of this statement rests on theoretical guarantees, and
-- how much is based on observation (or, rather, the lack of observed
violations to date).

Other ways of formalizing the issues include questions such as
-- Does U(3) describe the symmetries of Euclidean space?
-- Does U(3) describe the symmetries of the real space we live in?
-- What are the continuous subgroups of U(3)?
-- What are the discrete subgroups of U(3)?
--- How do we know there can't be any others?

Some people consider these legitimate questions. Some people even consider
them interesting questions. Sophus Lie certainly did.