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Why Mirrors Reverse Left and Right but Not Up and Down

October 26, 2015

Why Do Mirrors Reverse Left and Right but Not Up and Down?

http://www.slate.com/blogs/bad_astronomy/2015/02/17/physics_girl_mirrors_flipping_images.html

Saw this the other day, and tried to think about it, but was too busy on the typological stuff. Saw it again today, and watched the video

To simplify it in my own terms, it’s true, as she says, that in the mirror image, left is still left and right is still right. When I thought of this, even before I mentioned, I started by imagining lines, from my right to the “right” side of the mirror image, and from the left to the left, and of course, top and bottom as well. I at the time just figured that is what reverses the image, but didn’t really think of why horizontally (width) and not vertically (height). But the key is the fact that it’s not a left-right switch, but back and forth (depth).

In any rotation, two dimensions are involved. We can “left/right face”, and with 180° (a full “about face”), then back and forth are reversed, but then so are right and left. We can also rotate head forward 180°, and then right and left will be the same, but back and forth and up and down will be opposite. And we can rotate sideways, and then left and right will be opposite again, along with up and down; and back/forth will be the same.

What we’re interested in is comparing the first two, where back/forth are reversed.

The reason there is a difference between width and height, and width seems to be the one that changes, is because height is the dimension in which we are anchored to the ground by gravity, and width then becomes an extra “free” dimension (while depth is the basic dimension you look “forward” in). So when we want to reverse “back and forth”, the dimension we turn in is always right or left. As the video mentions, we have a symmetry in this free dimension, but not in height, where our feet and legs are on the bottom to stand on, and our heads are on top. We could “turn around” the second way (like a half somersault), but that would mean standing on our heads (or hands).

With a mirror, the light from left, right up and down heads straight from you, to the mirror, and then bounces back to you, in the same position. This flips back/forth without flipping any of the other two dimensions. So we end up with a negative image, meaning an image with one dimension reversed. Like multiplying positive or negative humbers, an odd number of negatives is negative, but an even number is positive.
Notice, when you add another mirror (or look in a horizontally bent mirror, like in a funhouse), you get a reflection of the reflection, which is then a positive image (the words on your shirt will read as normal), but rotated 180°; facing you, so that width is now reversed (your head now nods in different directions), and depth is still reversed as well (And height still normal). The double negative is a positive. If the mirror is bent vertically, then right and left will be normal (your head nods the same way again), but up and down will now be reversed. The words on your shirt will be both backwards and upside down, which will be a positive image that when rotated 180°, will read as normal. A completely concave round mirror (bent in both dimensions); then all three dimensions are reversed, and you get a negative image rotated 180°. Every move you make will be reversed by the image. The word on your shirt will read forward, but upside down.

The way to think of it, again, is that in the mirror image, what’s to your left is still left, and right is still right, but that’s just your left and right. Most people looking at you (who will read you shirt normally) are going to be looking toward you, which is actually the other direction, and what’s left to you, is right to them, and vice versa.
If you want to read what is on your shirt normally (like another person would), you must take it off and rotate it horizontally (or we can imagine the shirt staying where it is, and you have to step forward and then rotate horizontally). So you are flipping both width and depth. The mirror flips you in depth only, which reverses the image you are looking at. You are now looking at yourself as an outside observer. However, what other observers of you see as left (where a word begins when we read left to right) is now right, and what they would see as right, where the word ends, is now left.

This again, is because the way we rotate is horizontally, while the vertical dimension is what we are anchored to, and is basically only for holding ourselves up, not for rotating, normally. Up and down are the same for everyone, where left and right (and back and forth) are not. And as the only other dimension we don’t look “through”, like depth, and can see things parallel to our line of sight, besides height, is width, we are totally geared around width, like in reading and writing.

As the video says if our bodies had vertical symmetry; let’s say, we were living “X’s”, with two other legs instead of arms, and our heads in the middle, AND also, if we lived in zero gravity, with no ground, the “vertical” dimension would not be the same for everyone, and we would be using it too, as another way to “about face”. When others saw us, facing us, but upside down, they would see the words on your shirt upside down and backwards. But is’ important to remember you wouldn’t really think of it as “upside down”; it’s the way you’re used to see the image when looking at someone else. But it is still a vertically flipped; you just don’t realize it.

If, instead of zero gravity, you lived on the side of a vertical surface, where everyone agreed on one of the horizontal directions. Let’s say the surface is to the east (facing west), and then everyone would be pressed to the east, and then the subjectively oriented “right/left” would not be used, and everyone would have the same fixation of one side of their body to the east. But if they had freedom in the vertical dimension; maybe there was still gravity (so that “down” would still be felt as “down”), but you had a way of clinging onto this surface and moving around it, like insects. So looking in a mirror, we would still see the same negative image, but expecting it to be both upside down and backwards, we would only see it as backwards, so that width seems normal, but height seems flipped, (that is, from the way we normally see it).

This is where mirror dynamics resemble the fourth dimension, as you would read in Rudy Rucker. In a 4D rotation, the two dimensions could be the forth dimension itself (called ana and kata) plus back and forth (or left-right or up-down). That rotation would involves only one of our familiar dimensions, so you would get a negative version of the object without using a mirror!

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