Why does a mirror reverse left and right, but not top and bottom?

If you stand in front of a mirror and raise your right hand, it appears in the mirror that you raise your left hand. Why does a mirror reverse left and right, but not top and bottom?mirror

For some of you, this may seem like a silly or trivial question. However, “It’s just the way it is” is certainly not a satisfactory answer: experience is not the same as reason (just like you can’t explain why things fall downward instead of upward “by experience”). Let me try to convince you this question is non-trivial. Ideally, a mirror is a plane, which is two-dimensional: there are both left-right direction and top-bottom direction. But for a plane, no single direction is special: the mirror doesn’t know which way is top! So why then, is only left and right reversed, but not top and bottom?

Does the mirror have the magical power to know which way is left and which is right? Of course not! I am going to argue that the effect is actually biological/psychological rather than physical.

So, first of all, what does it mean by the mirror reversing left and right? “Easy!”, you may claim, “if you raise your right hand, the mirror shows you raise your left hand!”. Fair enough. (If you’re not convinced, try this now!) But then what is “left” and what is “right”? It is less obvious, perhaps an answer would be something like “the right-hand side is the side where the right hand is!” (Do you have a better answer?).

But wait! The “right-hand side” is relative to our body. If you are in a tilted position, your “right-hand side” now is not what your “right-hand side” was! Say, if you are a bit tilted to the left, then your “right-hand side” is pointing a bit upward. What is worse, if two people are facing each other, then their “right-hand side” are opposite. Lesson learned: “right-hand side” is relative to our body position and orientation. Another thing is that if we rotate ourselves by 180 degrees, then our “right-hand side” is reversed.

Now, this is where the problem lies in. We were a bit vague about “rotating ourselves by 180 degrees”. To talk about rotation, we need an axis of rotation. Say, you are asked to rotate a (Rubik’s) cube by 180 degrees, you should first ask what the axis of rotation is: there are (at least) three ways to rotate a cube by 180 degrees, of which two will make it upside down. Actually, for each of the three “standard rotations” of the cube, there is a pair of opposite faces which keeps facing the same direction after the rotation, and for each of the two remaining pairs (of opposite faces), the two opposite faces are switched. (Mathematically, there are of course infinitely many choices of rotation axes, and so infinite choices of rotations.)

Three ways to rotate a cube.

Three ways to rotate a cube.

Rotating a cube about the z-axis.

Rotating a cube about the z-axis.

Rotating ourselves is just like rotating a cube. Imagine your right hand pointing into the positive x-direction, your eyes looking in the positive y-direction, and your head pointing in the positive z-direction. What we normally mean by rotating 180 degrees is actually a rotation about the z-axis, since this is the easiest rotation we can do. Just like the cubic case, by rotating about the z-axis by 180 degrees, your “right-hand side” and your “left-hand side” are switched, and your “front” and your “back” are also switched, but your feet are still on the ground and your head pointing upward! For the other two rotations (about the x- and y-axis respectively), you have to stand upside down, which many of us don’t find very comfortable doing, unless you are a gymnast. (Unfortunately, I can’t do it.)xyz

Now back to the mirror. If you look at yourself in the mirror, the image is always facing you, as if you have rotated yourself by 180 degrees… about the z-axis (in our convention)! But this is not the whole story. What you see is yourself with the left and right reversed – but only after rotating about the z-axis by 180 degrees (i.e. what we usually mean by “turning around”). This is what we actually mean by the mirror reversing left and right. What the mirror effectively does to you is first rotating yourself about the {z}-axis by 180 degrees and then reversing your left and right. (Remark: We intelligent human beings are so used to “counteract” the effect of “turning around” that when we are facing someone, we instantly know where his right side is, although it apparently is on our left.)

Rotating about the z-axis by 180 degrees

Rotating about the z-axis by 180 degrees

At this point, I think it’s fair to work out the (easy, for many of you) mathematics. You can ignore this part if you don’t know linear algebra. Suppose the mirror is placed at the {x-z} plane, then the mirror effecting swaps {e_2} to {-e_2} and do nothing else to {e_1} and {e_3}. So it can be represented by the matrix {M=\begin{pmatrix} 1 & 0 &0\\ 0 & -1& 0\\ 0& 0 &1 \end{pmatrix}.} On the other hand, the rotation about the {z}-axis by {180^\circ} sends {e_1\mapsto -e_1}, {e_2\mapsto-e_2}, and {e_3} remains unchanged, so it is represented by {R=\begin{pmatrix} -1 & 0 &0\\ 0 & -1& 0\\ 0& 0 &1 \end{pmatrix}}. And finally, imagine you are facing the {x-z} plane so that your “right-hand” side is the positive {x}-axis, then the reversal of the left and right is represented by the matrix {L=\begin{pmatrix} -1 & 0 &0\\ 0 & 1& 0\\ 0& 0 &1 \end{pmatrix}}. Our previous discussion can then be nicely summarized by the equation {M=RL=LR}, meaning that the mirror image is exactly what we obtain by rotating ourselves (about the z-axis) by 180 degrees first and then reversing the left and right (or reversing left and right first and then rotating).

Enough linear algebra for now. Let’s go back to the mirror. There’s nothing special about the z-axis (the “top” direction), except for the biological fact that we are more comfortable standing upright than standing upside down, thereby privileging a certain axis of rotation.

Now, again, imagine your right hand is the positive x-axis. Suppose your favorite rotation is about the x-axis by 180 degrees, i.e. doing a {180^\circ} backflip. You like it so much that “turning around” (by {180^\circ}) means turning yourself upside down (to look at your back) to you. For instance, when someone at your back says hi to you, instead of turning around in the normal sense (as what 99.999% of us do), you prefer to do a {180^\circ} backflip to respond, which looks pretty cool, huh? Okay! Now, stand still (don’t turn yourself upside down), and look at the mirror. What would you think about your image? Time for a thought experiment… You now have 10 seconds.

A very cool rotation about the x-axis by 180 degrees!

A very cool rotation about the x-axis by 180 degrees!

Okay, time’s up. You should think that the mirror reverses the top and bottom! Why? It’s because the mirror image is actually obtained by reversing the top and bottom of yourself after “turning around”. The image is facing you, as if you have “turned around” (your favorite rotation). But no, it’s not exactly the same! If you have “turned around”, your head should point downward (towards the ground). But now your head is pointing upward, and you conclude that the mirror “reverses the top and bottom” (ignoring, or counteracting the effect that it has also “turned you around”)!upsidedown

It is of course also easy to explain this mathematically. The rotation about the {x}-axis by {180^\circ} (doing a {180^\circ} backflip) is represented by the matrix { \widetilde R= \begin{pmatrix} 1 &0 &0\\ 0 &-1 &0\\ 0& 0& -1 \end{pmatrix}}, and the action of reversing the top and bottom is represented by { T= \begin{pmatrix} 1 &0 &0\\ 0 &1 &0\\ 0& 0& -1 \end{pmatrix} }, and as before the action of the mirror (assume placing the mirror at the {x-z} plane) is {M= \begin{pmatrix} 1 &0 &0\\ 0 &-1 &0\\ 0& 0& 1 \end{pmatrix} }. Then {M= T \widetilde R= \widetilde R T}, which succinctly summarizes the above discussion.

This entry was posted in Geometry, Linear Algebra, Miscellaneous. Bookmark the permalink.

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