Reflections on The Mirror Question

When looking in your rear-view mirror, objects in front of the mirror 
(as seen in the image in the mirror) are reversed left and right, but 
not up and down.  Why are left and right reversed, but up and down 
are not?  To answer this commonly-asked question, we need to first
answer the question: "Reversed relative to what?"  From the context 
of the original question we can infer that it refers to the image in
the rear-view mirror being reversed relative to the image we would 
see if we turned around and looked directly out the back window.  If 
so, then the answer depends on HOW we would turn around.  Most likely
we would rotate our field of view about a vertical axis, so the image
we see will be reversed laterally (left-to-right).  Of course, if we 
turn about a horizontal axis (difficult to do in a car, but not 
impossible), the image we see would be reversed top-to-bottom.

Another answer that is often given to this question is that mirrors 
reverse neither left-to-right nor top-to-bottom, they reverse front-
to-back.  This is actually the answer to a slightly different question
than the one asked above, but it's such a cute answer that many people
can't resist giving it whenever anyone asks anything about mirror 
images.

The reason there are consistently conflicting answers to "the mirror 
question" is partly due to the fact that the question is often poorly
expressed, so that it can be construed in several different ways.  
The ambiguity is two-fold, because, first, the question often doesn't 
explicitly identify the two things that are posited to be "reversals" 
of each other, and second, the question often doesn't define the 
intended sense of "left" and "right", i.e., as relative directions 
or as designations of "handedness".

Thus, if someone asks "Why do mirrors reverse left-right?" we could
imagine that he's asking 

  (0) Why, as I look at a mirror, do objects that are actually on
      my left appear to be on my right, and vice versa?

In other words, he's assuming a fixed set of relative spatial directions 
based on his current orientation as he looks at the mirror, and he's 
asking how the directions of the actual objects compare with the 
directions of their reflected images.  Of course, on this basis we're 
forced to conclude that the PREMISE of the question is totally erroneous, 
because it's obvious that objects to the left of the viewer appear to 
the his left in the mirror.  Thus, if we assume this interpretation we 
have the fun of telling him that he's so stupid, even the PREMISE of 
his question is wrong, i.e., he's asking for an explanation of why 
something occurs when in fact no such thing occurs!  Hah!  It's always 
deeply satisfying when we can slam someone like this, thereby 
demonstrating the clear superiority of our thought processes.

However, when answering questions - especially informally-posed
questions in a non-adversarial context - there is a certain class
imperative to at least consider the possibility that the question
has a valid basis, and to seek a construal of the question that 
is consistent with such a basis.  For example, in the case of "the 
mirror question" we must ask ourselves whether there is ANY sense 
in which mirrors could be said to effect a left-right reversal.
Given how often "the mirror question" is asked by sincere people,
it is perhaps not surprising to find that there IS a way of
construing the "left-right reversal" premise such that it isn't
totally idiotic.  In fact, there are TWO such ways, both of which 
assume the "reversal" in the mirror is relative to the image that 
would appear if the same objects were viewed directly.

First, if some asks "why do mirrors reverse left-right?", it's 
possible to interpret the question in terms of left-right 'handedness',
i.e., the chirality of real or apparent objects in real or imagined 
space.  From this standpoint we would map the question to something 
like

  (1) My friend Wally is right-handed, but when I see him
      reflected in a mirror, he appears to be left-handed.
      What gives?

This is a perfectly valid observation, and it's worth noting that
imaginary-Wally is left-handed regardless of how we orient the mirror, 
the real Wally, or ourselves.  Even if we arrange things so that 
imaginary-Wally appears upside down in the mirror, or so that his 
left hand is on our right side, he will still appear to be left-handed.  
This is because we're not using the words "left" and "right" here as 
relative directions, we're using them as designations of left and 
right-handed chirality of the viewed objects (real and imagined).  
When the question is interpreted as something like (1), the answer 
is that the imaginary "objects" contained in the fictitious space on 
the "other side" of the mirror are symmetrically reflected vis-a-vis 
the actual objects about the plane surface of the mirror.  

Consequently we have a parity reversal that switches the 'handedness' 
of all the imaginary objects relative to their real counterparts.  
Thus the "left-right" handedness reversal of apparent objects in the 
mirror relative to the actual objects is a real effect on imaginary
objects, i.e., a reversal of parity of all "objects" in the imaginary 
space "behind" the mirror.  It isn't quite as much fun to give this 
answer as to give the "your-premise-is-wrong" answer, because here we 
have to admit that there is at least some rational basis for the 
question.  However, it's still kind of fun, because we can phrase the 
answer so that it comes out sounding like "The left-right [handedness] 
reversal you've observed is actually the result of a front-back 
[directional] reversal.  Hah!"  Thus, although we can't make the 
questioner sound like a complete idiot, we can still make him sound 
very confused indeed.

Paraphrases (0) and (1) represent the two "fun" interpretations, but 
in order to present them plausibly it was necessary to truncate the 
question relative to how it is normally presented.  Almost invariably 
when someone asks "the mirror question" he asks "Why do mirrors reverse 
left-right" and then adds the phrase "RATHER THAN UP-DOWN?"  Hmmm...  

In paraphrase (1) we assumed the term "left-right" referred to 
handedness (chirality), but it's hard to maintain that assumption if 
"left-right" is contrasted with "up-down", because the alternative 
to a left-right handedness reversal is not an up-down handedness 
reversal, but simply no handedness reversal at all.  So if someone 
contrasts left-right with up-down, we need to consider the possibility 
that they are referring to left and right not as designations of 
chirality but as relative directions in the plane of their visual 
images.  This suggests that we should map their question to something 
like

  (2) If I look in the rear view mirror of my car, the image I see
      is typically reversed left-to-right relative to what I would
      see if I looked directly out the back window.  (For example,
      the license plates read backwards in the mirror.)  But surely
      a plane mirror is symmetrical about all directions in the 
      plane, so why do I see backward images rather than upside-down
      images?  In other words, why is there a left-right reversal 
      rather than an up-down reversal, or even some diagonal 
      reversal?

Note the similarity between this and the question posed at the start
of this note.  The answer to this question clearly has to do with the 
axis of rotation of our field of vision.  When the person turns his 
field of vision from the mirror to the back window he is turning about 
some axis, and the relation between the two visual images depends on 
that axis.  If, as is most likely, he turns his head about a vertical 
axis, then the image he sees out the back window will indeed be reversed 
left-to-right relative to the image he saw in the mirror.  In other 
words, things that were on his left are now on his right, and vice 
versa.  However, if he rotated his field of vision about, say, a 
horizontal axis (difficult but not impossible in a car) the two visual 
images at his retina would be related to each other by being "flipped" 
about a horizontal line, i.e., the images would be upside-down relative 
to each other.  In general, the direct and reflected visual images are 
"flipped" relative to each other about a line parallel to the viewer's 
axis of rotation.  

(Needless to say, our driver has probably looked directly out the rear 
window enough times that he can imagine how it will look even without 
turning, but the point is that when he talks about image reversals in
the mirror he means relative to the image he *would* see IF he turned 
around in some specific way, even though he may never have cogitated 
on the fact that his preferred choice of physical rotation axis is what 
determines the axis of reversal of the mirror image.)

Notice that although none of the above answers is based on "social 
expectations", they do involve consideration not only of "what mirrors 
do" but of what WE do when we look in a mirror.  In both cases (0) and
(1), if we say mirrors "really" reverse front-to-back, we base this 
claim on the observer's (mis)interpretation of the visual image coming 
from the flat surface of the mirror as if he was looking into a three-
dimensional region of space, and making inferences about the depth or
chirality of imaginary objects (although case (0) is somewhat problematic
considering that both the real and the reflected object may be in
"front" of us, even without imaginary "depth" inferences).  In case 
(2) our answer is based on the raw visual images, without any 
interpretation, but those images are understood to be functions of 
the observer's field of vision and relative directions, which change 
as he turns (or imagines turning) from the direct to the corresponding 
reflected image.  Thus, none of these construals of the question allows 
us to avoid dealing with the observer as an active element in our 
description.  

Without the observer, the only answer we could give to the question 
"What do mirrors do?" is that mirrors reflect rays of light symmetrically 
about the normal to the surface at the point of incidence, but this 
clearly is not responsive to a question about perceived image reversal.
Furthermore, there's no valid reason to shy away from phenomena that 
involve the observer in some essential way.  There was such a prejudice
among physicists at one time, but quantum mechanics has taught us that 
physics can't always clearly separate the measurement process from the 
thing being measured, nor the observer from the observed.  

As for which of our two non-idiot interpretations is more likely to be 
correct, I think the fact that people almost invariably contrast left-
right with up-down when asking the question clearly implies that they 
have in mind something like (2) rather than (1) (and certainly rather
than (0)).

For related discussions, see
What Mirrors Do
A Mirror to Physics