Mirrors

[gfish]

I was standing at a bus stop the other day when what seemed like a trivial question occurred to me: why do mirrors reverse about the vertical axis? The idea that gravity or binocular vision affects it is easily dismissed, but it took longer than I expected to think it through. It turns out to be a pretty interesting problem.

Step 1: Mirrors don't actually reverse, they just reflect light. The best way to visualize what is happening is to imagine a transparency with some text (or other non-symmetric pattern) on it. Place it on the wall behind your head so it's normally readable if you turn around. Now move it forward to in front of your head without rotating it. Bingo, the text is reversed left-right! That's all a mirror does, bouncing the light right back at you.



Step 2: Rotations in three dimensions are not commutative. If you rotate an object first about X and then Y, you don't end up with the same orientation as if you rotated Y and then X. (This is why you start using quaternions and other weird maths when you'd working with 3D orientations on a regular basis.) In simpler terms, what an object looks like depends on how you turn to look at it.



Conclusion: We see mirrors as reversing about a single axis because there's only one way we rotate (and therefore visualize our environment). Go back to the situation in step 1. What if, instead of turning your head to look at the text behind you, you did a half somersault? (That is, rotate about X instead of Y.) Compare your view of the text now with what you saw in the mirror. According to this view, the mirror had reversed the text about the horizontal axis. It would have been flipped upside-down instead of left-right.



The important thing is both of these perceptions are equally valid. What the text "actually looks like" when you're facing away is entirely a question of how you turn to face it. The mirror isn't flipping anything, so our view of what has happened depends entirely on how we orient ourselves in 3-space. It's another symptom of our limited 2.5 dimensional view of the world. We simply can't get over gravity. People raised in freefall might have a very different understanding of the term 'mirror image'. (Though even then, it's always going to be easier and less disorienting to rotate about our long axis.)

Comments

[tfabris.livejournal.com]

In high school, I spent a lot of time thinking about that very question. Possibly because I saw it posed in the "Games" column of Omni magazine once. I didn't come up with an explanation as elegant as that.

[tithonium.livejournal.com]

I always just thought of it as changing front and back, which made it all make more sense.

[randomdreams.livejournal.com]

The physics teacher I worked for, doing science demos, always said "you're reversed, not the mirror", which worked pretty well for me.

[bigbumble.livejournal.com]

A mirror does reverse the image, but it does it front to back, not left to right. As you pointed out, it is flipping the paper around the vertical axis that reverses left and right.

A quit experiment is to write on transparent material and hold the transparency so that you can read it in front of a mirror. Both the transparency and the mirror image look "right". Now put a sticker on the transparency that is blue (or any other color) on one side and white on the other. Both images are still readable, but one has a blue sticker, the other image a white sticker, showing the in-out dimension is reversed.

[ilmarinen.livejournal.com]

I totally didn't follow this at all.

-B.

[ilmarinen.livejournal.com]

I don't get the cube rotations--shouldn't new faces be showing in the two rotated-twice views? How about if you showed the intermediate states (first rotation) then the second?

-B.

[ilmarinen.livejournal.com]

But a pinhole camera (and other similar optics) does invert an image--does it also pervert it (mirror) it? it seems it does, but we don't perceive it as such?

[memegarden.livejournal.com]

I think Douglas Hofstadter had a similar explanation (about which axis you think of yourself as turning along to "match" the reflection) in one of his books, but I don't know which one. Presumably either Godel, Escher, Bach or Metamagical Themas. *restrains herself from going and looking right now, knowing that she will then stay up for hours reading*

[ubiquity.livejournal.com]

Neat!