Trimotio: Hey Fineas, tell me this: why do mirrors reverse left and right but not up and down?
Fineas: When the plane of the mirror is horizontal, the mirror does reverse up and down. Just think of the upside down landscape you can see in the surface of a lake.
Trimotio: That’s not what I mean. I have a mole on the right side of my upper lip. However, when I look into the mirror, then no matter how it’s placed, the guy in the mirror has a mole on the left side of his upper lip. It does not jump to his lower lip, though. So in the mirror image, left and right are reversed while up and down stay the same. The mirror has to know the difference between these two directions somehow–even if I am looking right into it, when they are both parallel to its surface. This does not change if I lay my head to the side or if I take the mirror off the wall and lay it on the floor.
Fineas: Okay, so let’s place the mirror in front of you, the way they’re used normally. Then the simple answer is that they don’t reverse left and right. Look in a mirror and wave your right hand. On which side of the mirror is the hand that waved? The right side, of course. Better still: go and buy a couple of those big chocolate letters, an “R” and an “L”. Hold the R in your right hand so that it’s the correct way around for you to recognise it. Do the same for the L, holding that in your left hand. Now ask what you see in the mirror. The R is still on the right hand side of the mirror!
Trimotio: Yes, but it will be reversed in the mirror.
Fineas: Will it? It still says “R” in the mirror–you can recognise it straight away. It looks exactly the same as the R you are holding. And the same goes for L.
Trimotio: Okay, but what about your reflection, the guy in the mirror? He is now holding the R in his left hand, not his right.
Fineas: But he’s not a real person. To say that he’s holding it in his left hand means that you have mentally placed yourself in his position.
Trimotio: Sure, why not?
Fineas: You can do that, but in so doing you have uncovered the basic truth: you had to turn around. Imagine holding an arrow in your hand as you look in the mirror. If you point it up, it will point up in the mirror. If you point it to the left, it will point to the left in the mirror. But if you point it toward the mirror, it will point right back at you. Mirrors do reverse in and out–or to be more precise, they reverse the direction perpendicular to the plane of the mirror. Or, think of it like this: make the direction perpendicular to that plane point in the north/south direction. Now face north and look into the mirror. Point your hand east. Your image also points his hand east–not west! But point your hand north and your image points his hand south.
Trimotio: Okay, but if they don’t reverse any directions in the plane of the glass, but they do reverse in and out, why does the guy in the mirror hold up his left hand when I hold up my right?
Fineas: That’s due to the different ways the directions are defined. By asking why the mirror reverses left/right but not up/down, you are not comparing apples with apples. “Up” is a straightforward direction to define. It is simply representable by an arrow pointing in some chosen direction, such as away from the earth’s center. Or if we are floating far from Earth, we might take “up” to be the direction from our feet to our head. The same goes for in/out: we can take the “in” or “front” direction to be the direction pointing away from the front of our body. If you wanted to tell a Martian which way is “up” and which way is “front”, you could do it easily.
Trimotio: But “right” is also simply a direction that can be represented by an arrow pointing into the right direction.
Fineas: No it’s not. “Left” and “right” are not so easily defined, and you’d have a much harder time telling the Martian what you mean by these. That’s because “left” and “right” are relative: they depend on our deciding on an “up” and a “front” first. So comparing “right” with “up” is not a fair comparison. Ask two people facing each other to point “up”–they’ll both point in the same direction. Now ask them where “right” is. They’ll point in opposite directions. You at least have to agree on one additional direction or arrow–let’s say “front”–before you can specify where the arrow towards “right” is pointing. And because those two people have chosen opposite directions for their “front”, this has also altered their definition of “right”.
Trimotio: In our everyday life, I think “left” and “right” have more to do with the bilateral symmetry of our body.
Fineas: So look at the mirror with one eye closed. It makes no difference to what you see. But the bilateral symmetry of our body probably has promoted the definition of the words “left” and “right”. We have all kinds of words for one-dimensionally defined directions: up/down, front/back, north/south/east/west. They are just fine for explaining where things are. They all, of course, need a certain reference (e.g. the earth, or our body) for their definition, but they are all still one-dimensional. But trying to explain which hand to reach out for shaking hands with somebody–the “right” hand in most parts of the world–will only work when the directions “up” and “front” are already defined. We have invented the words “left” and “right” because we needed them quite often to distinguish between the two orientations–of course not only referring to the two sides of our body, but also all kinds of other things with two sides. It’s not a coincidence that mathematicians also use these words for distinguishing between three-dimensional co-ordinate systems being “left handed” or “right handed” in their orientation. There are only these two different possibilities for joining three arrows together to form a perpendicular set of axes. No matter how you rotate them in space, they can never “cover” each other, since they are mirror images of one another, just like our hands are.
Trimotio: What about the aliens on the planet Torsor, who have no symmetry at all: do they see “left” and “right” reverse in a mirror?
Fineas: Tell them to turn left and they shuffle round to the left while their mirror image shuffles round to its right. They would agree that the mirror reverses left and right even though they have no symmetry. In fact any rotation one observes as clockwise is seen as a counter-clockwise rotation of the mirrored object by the observer in the mirror–and vice versa.
Trimotio: And, what about the beings who spend their lives floating within the waters of the ocean planet of Skyron? They are symmetrical around two axes. They are X-shaped, with four similar arms sticking out and one eye in the center. They have no natural “left” or “right” direction, nor any natural “up” or “down” direction.
Fineas: Perhaps they do look into the mirror and see things reversed for either axes, because each time they place themselves mentally into the mirror, they are reversing the direction of their “front”; so that regardless of where “up” is, “right” will change too. But maybe if they don’t really distinguish “up/down” from “left/right”, then they will not perceive anything to be reversed in a mirror after all. Of course if their intelligence is sufficient for performing “mirror experiments” then they will mark their arms (e.g. with different colors), and fully understand the reversal. They’ll probably laugh if they ever read this some day. (link)
Even if I voluntarily don’t take part in this campaign, 
