John Denker described some alternative types of mirrors, and asked how we
know they do not exist. That might be a fun thing to think about, but it
does not seem relevant to the question at hand. It seems the question at
hand pertains to a "standard flat mirror." I do not know if we will
discover, or someone will invent, other kinds of mirrors. But I certainly
assumed Abhishek Roy was talking about a standard mirror. If that is not
correct, he will have to tell us.
What does a standard mirror do? First, it defines a plane, i.e. it is a
plane. Next, images observed in the mirror are "perverted." This means
they are reversed "front to back." This means the reversal occurs in the
axis (of a 3-D mutually perpendicular coordinate system) that is
perpendicular to the mirror. For example, if the plane of the mirror
contains the x and z axes, then the y axis of the image is reversed from the
y-axis of the object. As you face the mirror, z-axis up, x-axis to your
right, the y axis for real objects points away from you, i.e. in front of
you. The y-axis for image objects points toward you (to the back of you).
Hence the mirror performs front/back reversal. In the old days this was
called "perversion." Today it seems perversion has a negative connotation,
so authors/editors are tending to call this "mirror reversed" in current
text books. I still prefer the word perverted.
Note, a mirror does not reverse left/right directions as some respected
textbooks claim... that is simply wrong. Stand in front of a mirror. Hold
a sharpened pencil. Point the sharp end of the pencil to your right and the
pencil's image also points to your right. Point the pencil up, and the
image also points up. However, point the pencil away from you, and the
pencil's image points toward you. That is front/back reversal... that is
perversion. Unfortunately, "left/right reversal" can be both correct and
incorrect for what a mirror does. That's because left/right can refer to
directions; or left/right can refer to handedness. A mirror does not switch
left/right directions; a mirror does switch left/right handedness... that's
because the way a left hand and right hand differ from each other is by
front/back reversal, where front/back is defined as perpendicular to the
plane separating the two hands.
Anyway, all the flat standard mirror does is reverse front-to-back. The
image axis that is perpendicular to the mirror points the opposite direction
of the real axis that is perpendicular to the mirror. If front/back
reversal is all a mirror does, then it is easy to show there is only one
mirror image...
Choose a physically-real 3-D object. That means it consists of real atoms.
We would like to construct a new physically-real object that is the same as
the mirror image of the original object. How do we do this? First, write
down all the coordinates of all the atoms of the original object. Gather up
a new set of the appropriate atoms to construct a second object, but as you
assemble the second object, change the +/- sign of every y-coordinate. (Of
course it does not matter if you change the sign of every x-coordinate or
every z-coordinate as long as you only change the signs for one and only one
axis. I chose the y-coordinates because I assumed the mirror is in the x-z
plane.)
If the formula for constructing the physically-real mirror image of a
physically-real original object is to put identical atoms at all the same
coordinate locations, but for each atom change the sign of one and only one
axis coordinate (for the same axis for all atoms)... there is only one way
to do this. There is a unique one-to-one correspondence between the
location of each atom in the first object and the location of each atom in
the second object. There is only one mirror image. And I maintain this is
indeed the correct formula/process for constructing the mirror image. John
or others can dream up any other process they want, but it will not describe
a standard flat mirror.
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