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Re: A question about mirrors



On Sun, 6 Aug 2000, Abhishek Roy wrote:

From: Jack Uretsky <jlu@HEP.ANL.GOV>

So an answer to your question is that there are
only two "hands" because there are only two real roots of unity!
?! How does this have to do with anything? Please explain.

Let's go a step at a time. We need to agree on precisely what
you mean by "reverses". So consider a body to be a collection of points
in a space. Will you agree that "reverses" means an isometric (your word)
transformation on the collection that cannot be duplicated by a rotation
and/or a translation?
Agreed.

Good. Then, as John has been (more than broadly) hinting,
the answer to your question is found in the theory of Lie Groups.
The applicable group that describes the type of transformation
that you have agreed to is the group of rotations in 3-space -
called O(3). Matrices that represent that group satisfy the
condition M^tM=1, where M^t is the transpose of the matrix M,
and 1 is the identity matrix.
Note that the above condition -> (detM)^2 = 1 (det M
is the determinant); also, the matrices are necessarily real.
Now we get to the kicker.
Every such matrix with detM=1 can be reached by a
continuous path from the identity (a Lie Group is also a space);
these matrices represent pure rotations of objects. But you
have excluded pure rotations, so these matrices are not of
interest.
The ONLY other possibility is detM= -1. These may
be reached by a continuous path from a diagonal matrix such as
(1,1,-1), but not from the identity matrix. But the matrix
(1,1,-1) represents a reflection, since it "flips" one axis.
Also, there is only 1 such possibility because any other
placement of the minus sign is simply a rotation of my choice.
The uniqueness of the mirror image in classical physics,
then, is represented by the fact that the isometric transformations
in 3-space divide into two "discrete" subgroups that cannot be
connected by a rotation. There is no rotation that converts
the identity matrix into (1,1,-1).
The succinct answer to your question is that there are
only 2 discrete subgroups, because there are only two real solutions
to the equation x^2 = 1.

[snip]

No. I actually asked, how aliens who had never seen a
mirror
would explain both the difference and similarity between say, a
right
and
left helix.
Send the chemical formula for a substance with 2 isomers;
then keep sending your agreed symbol for the number "2".

Or (assuming they are super-intelligent) how would they, on
seeing an asymetrical object for the first time, would deduce that
it
had
one and only one counterpart (again lacking a mirror).

Since most objects are asymetrical, I don't understand
your question. There can only be one inversion in 3-space
because two co-ordinates determine a plane and the "handednedness"
of the co-ordinate system is determined by the orientation of
the 3rd co-ordinate.
Replied to by John Denker. By the alien analogy, I meant to
ask
how intelligent beings would - on first chancing upon an asymetrical
object
reason that it has a (only one) counterpart. Of course if they were
familiar
with a standard mirrors, or atleast the mathematical transformation,
they
might explain it way. But without the above, it seems to me that they
would
first - by pure cerebration - discover the property of 'handedness' (in
the
general case- not just for that object or a class of objects). And my
original question was - how?

This takes us back to your definition of "counterpart".
Which is what I am trying to determine, in the first place? Just
replace 'counterpart' with enantiomorph. See the thought experiment with the
alien above.

I think that the aliens would have discovered their own version of
O(3).
Regards,
Jack