Two points here. One about communication with an "intelligent being," and
the other about whether there are more "opposites."
(1) Rather than discuss the rotations and reflections as John Denker did,
another way to approach this is from the stereochemistry of molecules. If
we assume the "intelligent being" has never seen a mirror, but knows some
chemistry, and knows some astronomy, we can define mirror images and right
and left handedness by chemistry and astronomy. I will assume we have had
sufficient dialogue with this "intelligent being" that we can discuss
chemicals... that is, the being understands what we mean when we say carbon,
hydrogen, chemical bond, etc. If not, you'll have to reach that level
first.
Take a carbon-containing molecule in which at least one carbon atom has four
other atoms bonded to it, and one (and only one) of the carbon atoms has
four "different" atoms attached. A simple example would be CHClFBr
(flourochlorobromomethane).
The geometry around this carbon is tetrahedral. This molecule is also
chiral (handed). Construct two model molecules with balls and sticks, or
whatever. Depending upon how you assembled the two models, they might be
identical (can be stacked so they are superimposed on each other)...
otherwise they will be "mirror images" of each other. Whichever they are
(identical or mirrors) one molecule can be converted to the other form by
switching the positions of any two attachments. Assemble them so they are
not superimposable on each other. We now have two objects that are "mirror
images" of each other. This could serve as one type of definition of
"mirror image."
Hold the molecule so the lowest-mass atom is pointing away from you. In the
example, the hydrogen points away from you. Now number the remaining
attached atoms using 1 for the most massive, and 3 for the least massive.
In the example Br is 1, Cl is 2, F is 3. While viewing the molecule with
the hydrogen pointing away from you, make a circle with your finger as you
trace the other atoms in the order 123123123... We know we will be doing a
clockwise trace, or a counter-clockwise trace. If it is clockwise, the
molecule is designated as R (or right handed). If it is counter-clockwise
the molecule is designated as S (or left handed).
If the "intelligent being" does not know clockwise and counter-clockwise,
this can be communicated via suitable astronomy once you ascertain you both
know of some astronomical object and you can discuss its rotation or
revolution from some common vantage point. For example, we could use the
rotations of our solar system viewed from the vicinity of some other
particular star.
(2) The reason for stating that "one an only one" carbon was chiral is
because such a molecule only has two stereoisomers... mirror images of each
other. If there are more chiral carbons then there are more stereoisomers,
some of which are mirror images, and some not.
For example, suppose there are two chiral carbons in a molecule. The
designations of the carbons (as outlined above) could be SS, SR, RS, RR
The SS and RR are mirror images of each other, and the SR and RS are mirror
images of each other. This might represent four different molecules, or
might in fact represent three different molecules. It depends upon what all
the atoms are. SS and RR are definitely mirror images and represent two
different molecules. SR and RS might represent a third and fourth molecule,
or perhaps only a third molecule. If the two carbons have identical
attachments, there is actually a plane of symmetry between the two carbons,
and the SR and RS are actually identical (super-imposable). Chemists call
these "meso stereo isomers". Tartaric acid (which could be communicated
with an "intelligent being") has three stereoisomers...RR, SS, and meso
(RS/SR).
If we would modify tartaric acid to 2-chlorotartaric acid, then there would
be four different molecules... SS, RR, SR, and RS. These would consist of
two pairs of mirrors... RR/SS = one mirror pair, and SR/RS = the other
mirror pair. Neither of the SR/RS pair have any mirror relationship to
either of the SS/RR pair as a whole, although there is a mirror relationship
at individual carbon atoms (individual chiral centers).
In this type of situation, I we have an example (and there would be
thousands of other examples) where we have "handedness" within a molecule,
but there are more handed-possibilities than just one mirror image. Of
course, once you pick a single molecule, it has one specific mirror image.
But those same atoms could be arranged into other stereoisomers (where only
the spatial orientation differs... not what is bonded to what) and we would
have reversal of some of the handedness without complete molecular reversal
of handedness.
When viewed at this level, I suspect the actual answer to Abhishek Roy's
question is that there cannot be "more than a pair" of mirror images. Any
chiral molecule with multiple chiral centers will have a mirror image that
has *all* chiral centers reversed. Therefore there is only one true
opposite or only one true mirror image of the entire molecule. However, not
using mirrors, the molecule could be arranged so that some of the chiral
centers are reversed, but not all, such that we would obtain a molecule that
differs from a set of two mirrors in somewhat of a mirror way (some centers
have been mirrored), but the overall molecule is not a mirror because not
all of the chiral centers have been mirrored. Example... SRRSS is a mirror
of RSSRR... but SRSSS is not a mirror of either of the first two, even
though its third carbon is a mirror of SRRSS. The one and only mirror of
SRSSS would be RSRRR.
I use the organic chemistry example because it is figured out, standardized,
and discussable. I believe the reduction of other objects into "chiral
centers" is also possible and would follow the same logic.
Apologies to those of you who have not taken an organic chemistry course and
do not have experience with the fascinating subject of stereochemistry. I
have found that my dual physics background and chemistry background are very
helpful in this area.
Michael D. Edmiston, Ph.D. Phone/voice-mail: 419-358-3270
Professor of Chemistry & Physics FAX: 419-358-3323
Chairman, Science Department E-Mail edmiston@bluffton.edu
Bluffton College
280 West College Avenue
Bluffton, OH 45817