# A mirror flips left and right, but not up and down

Why is it that when you look in the mirror left and right directions appear flipped, but not the up and down?

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On which plane are your (2) eyes oriented? – user4577 Jul 21 '11 at 19:00
It COULD flip you upside down, depending on the type of mirror (flat vs concave vs convex) – MGZero Jul 22 '11 at 3:11

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And yet a lens turns the image upside down... – Digikata Jul 22 '11 at 4:11
The issue with a normal mirror is an interesting puzzle, but I think the issue with a (round) double-convex lens makes the asymmetry even more obvious and striking. With the flat mirror you can kind of convince yourself that it is a bias of the left-right orientation of your human eyes, but with a round convex lens mirror there's (seemingly) nothing to explain the left-right swap, even though the lens itself is symmetrical... – Mark J Sep 4 '12 at 20:04

## 14 Answers

Here's a video of physicist Richard Feynman discussing this question.

Imagine a blue dot and a red dot. They are in front of you, and the blue dot is on the right. Behind them is a mirror, and you can see their image in the mirror. The image of the blue dot is still on the right in the mirror.

What's different is that in the mirror, there's also a reflection of you. From that reflection's point of view, the blue dot is on the left.

What the mirror really does is flip the order of things in the direction perpendicular to its surface. Going on a line from behind you to in front of you, the order in real space is

3. Dots
4. Mirror

The order in the image space is

1. Mirror
2. Dots

Although left and right are not reversed, the blue dot, which in reality is lined up with your right eye, is lined up with your left eye in the image.

The key is that you are roughly left/right symmetric. The eye the blue dot is lined up with is still your right eye, even in the image. Imagine instead that Two-Face was looking in the mirror. (This is a fictional character whose left and right side of his face look different. His image on Wikipedia looks like this:)

If two-face looked in the mirror, he would instantly see that it was not himself looking back! If he had an identical twin and looked right at the identical twin, the "normal" sides of their face would be opposite each other. Two-face's good side is the right. When he looked at his twin, the twin's good side would be to the original two-face's left.

Instead, the mirror Two-face's good side is also to the right. Here is an illustration:

So two-face would not be confused by the dots. If the blue dot is lined up with Two-Face's good side, it is still lined up with his good side in the mirror. Here it is with the dots:

Two-face would recognize that left and right haven't been flipped so much as forward and backward, creating a different version of himself that cannot be rotated around to fit on top the original.

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– Lucas Jul 22 '11 at 2:43

Because they don't flip left with right (or up with down), they flip the 3D space you're standing in "inside out", so far from the mirror becomes far away inside the mirror and vice versa. A hand 1 meter from the mirror seems like it's 1 meter on the other side of the mirror but in the same spot with regards to left/right so nothing is flipped.

Wiggle your left hand - you'll see the hand which is to the left in the mirror wiggle. Wiggle your toes and the toes in the mirror image wiggle etc.

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I was asked this question by a fellow Junior Fellow at Harvard who was convinced that it had to be the most important open question in early 21st century physics. ;-) By the way, there is one helpful twist. If you put a mirror on the ceiling above you, or the floor beneath you, it will reflect up and down, not left and right. Mirrors always reflect/exchange "in front of the mirror" and "behind the mirror". If we stand vertically, we just imagine that the person behind the mirror (the real image) is a real man and we rotate him by 180 degrees around the vertical axis. – Luboš Motl Apr 7 '11 at 20:48

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Once we rotate the man in the mirror by 180 degrees around the vertical axis. he looks like a normal man ready to shake our hand. But the rotation by 180 degrees was artificially added. If $z$ is up-down and $x$ is left-right and $y$ is front/rear of the mirror, then the mirror which is in the $xz$ plane maps $y\to -y$ and keeps $x,z$. However, we can easily imagine rotations around vertical axes to make the mirror man look like the original man. Such a rotation changes the signs of $x,y$, so the sign flip of $y$ is replaced by a sign flip of $x$ - but the real transform only affects $y$! – Luboš Motl Apr 7 '11 at 20:50

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I created/associated my account just to say that the first part of your explanation example (mirror on the ceiling) really clarifies well, you should post it as an answer! The second part is a little more unclear, actually. – Kzqai Apr 8 '11 at 13:34
You don't answer the question. – Kris Van Bael Oct 9 '11 at 7:15

This common confusion stems from our familiarity with photographs. We forget that we rotate them to face ourselves.

Take a picture of yourself and hold it up in front of you. Probably you are holding it so that you can see your image. If so, you "flipped" the image of yourself when you rotated it 180 degrees around the vertical axis. When you look to the left side of the photo, you are looking over the right shoulder of your image. These directions are flipped!

Now look in a mirror. When you look to the left, you are looking over the left shoulder of your image. These directions are not flipped!

Now pick up the picture again and turn it so it's facing the same direction you are facing. You have removed the 180 degree rotation so that you and your image are "looking" in the same direction. The left side of your image is again to your left. If the picture is transparent enough that you can see your image, you'll see not the back of your head, but your eyes, giving you the impression that you're looking back at yourself. A mirror image! But again, left and right are not flipped.

When you say the mirror "flips" left and right, you are speaking from the frame of reference of one who is used to the 180 degree rotation that you apply to view an opaque photograph. But that's what we all do because we consider photographs, rotated 180 degrees to face ourselves, as being the "correct" left-right orientation.

What a mirror really flips is the depth dimension. That which is behind you appears to be in front of you.

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Great answer! I just imagine a photon emitting from my right eye, bouncing off the mirror, and then returning to my right eye. Nothing "flipped" there. The eye in the photograph that's directly across (or closest to) my right eye is actually my left eye, because I've flipped the photograph 180 degrees from the original direction of the camera, just as if the photons on the mirror were somehow "frozen", and someone rotated the mirror 180 degrees to face them. – webXL Jul 21 '11 at 20:55
The simple example: Write a few words on a transparent piece of plastic of some sort, then hold it up in front of yourself while looking in a mirror. You can read the text just fine in front of you, or in the mirror. – fennec Jul 22 '11 at 1:12

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Exactly. A mirror does not flip left and right. It is we who imagine another person in the mirror and change our frame of reference by "flipping" it. When I first read about mirrors flipping things (I was probably four or five), I found it confusing because I looked at the mirror and nothing was flipped. (It was only later that I learned to do the flipping with photographs and other people -- before that, I'd use to speak of the right hand of someone facing me as the left hand, because it was indeed to my left.) – ShreevatsaR Dec 27 '11 at 6:56

Take a picture and look at it. Now turn the picture to face the mirror. Question one: who flipped the picture? Answer: you did. Now, face the picture back to you, and walk to the nearest refrigerator. Turn the picture to face the refrigerator. Wow! Refrigerators flip images too! Don't believe me? Take your flipped page and hold it up to a bright light. The image is flipped; no mirror required.

Now, most people will turn a page around the vertial axis when they want to face it away from themselves. However, you could flip the page around any axis you choose, as long as it's in the plane of the page. You could easily, for example, flip the page around the horizontal axis. If you still believe that mirrors flip images, you'll notice that you've now tricked the mirror into flipping the image top-to-bottom, not left-to-right. Flip the page around a diagonal axis, and you'll get a very different result.

Bottom line: mirrors don't flip images; people do.

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First, lets separate the concepts; there is nothing that is "flipped" in the mirror image regarding one orientation more than others. the full group of transformations $O(3)$ includes transformations where $det(R) = -1$. You can consider the following transformations examples of this:

1) they have one random direction flipped in sign, or

2) for the special case where $D= 2n+1$, you can flip all direction signs

moreover, all these rotations are equivalent to each other, that is, any of them is equivalent to any other by a normal rotation $det(R) = 1$

explained in other way, the mirror image is also equivalent to your image with up and down flipped. You only think in the equivalent with the right-left flipped version because its easier to build in your mind. That might be related with the fact that our bodies are almost right-left symmetrical, but very poorly up-down symmetrical

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It's easier with images... The mirror doesn't flip left and right as you can see in the upper image. The so-called flip occurs when somebody in the real world rotates 180 degrees about the vertical axis to see you face to face, as can be seen in the lower image.

Regards Hans

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Imagine a mirror placed on the floor. Your face, looking at it, will be pointed downwards; but the face "in the mirror" is "looking" up.

The other answers here are great — I just wanted to make sure we discard the assumption that the mirror is placed on a wall!

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a lake is much better, natural example, in this case. – arivero Aug 3 '11 at 16:33

Think about where a point above, below, left, and right of your point of view are in the reflection. Your head is still on top, your feet still on the bottom in the mirror. Likewise, your left hand is still to the left and your right hand to the right. It seems flipped because, to look behind you, you are used to turning around (which swaps left/right), rather than flipping (which swaps top/bottom). If we flipped upside down instead of turning around to see behind us, you would experience the opposite effect.

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Here's how I explain it (this is pretty similar to most of the other answers).

Assume the mirror is hanging vertically on a wall, and you're standing upright and facing it, looking at your own reflection. (Just to make the assumptions explicit.) And let's assume that you're facing north, and wearing a watch on your left wrist.

The mirror doesn't flip left and right; it flips forward and back. Your front is farther north than your back, but your reflection's front is farther south than its back. Your feet and the reflection's feet both face down. The wrist with the watch is on the west side, both for you and your reflection.

So why do we think that the mirror flips left and right? I think it's because we mentally map the (mathematically simple but physically awkward) transformation to something that makes physical sense. The "simple" front-to-back reversal is mentally mapped to (a) a 180-degree rotation (something that people do all the time), and (b) a left-to-right reversal (which people don't do, but since we're pretty much bilaterally symmetric, it doesn't seem as strange). So the person you see in the mirror looks like a normal person who happens to wear his watch on his "right" wrist and part his hair on the other side.

If we weren't bilaterally symmetric, we might see it differently.

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Because your eyes are set left and right and how you relate to the mirror.

When you turn a picture to reflect in a mirror, you usually turn it left or right. If you turned it vertically, the up and down reflection will be opposite, but left and right will still be the same.

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It also works with one eye. (although I agree with the second part of your answer) – Kris Van Bael Oct 9 '11 at 7:10

Left and right is relative to you only but not to the rest of your environment (including the mirror), but up and down is static for the whole environment including you and the mirror.

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All you have to do is imagine the line y=x+1 on the cartesian plane. If you reflect the line across a mirror in the vertical direction (across the y-axis), then the line becomes y=-x+1. Therefore, a vertical mirror flips the line's horizontal coordinates.

However, if you reflect the lines across a mirror is in the horizontal direction (across the x-axis), then the line becomes y=-x-1. A horizontal mirror flips the line's vertical coordinates.

The mirror just flips the axis perpendicular to its orientation!

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I can't resist: Here are two real experiments I came up with to explore the assertion that mirrors invert left-to-right but not up-to-down. Fans of the scientific method (and of Douglas Adams) know that the first step in any good experiment is to figure out exactly what the question (or hypothesis) really is. So, if you like having "obvious" assumptions shaken up a bit, please read on!

1. On an index card, write your name or some text in large block letters. Next, use one hand to hold it just in front of you as if it were a name badge at a meeting. With your other hand, hold a mirror in front of you so that you can read your "name badge." The text will look inverted left-to-right, as expected, while up-and-down looks normal, also as expected. Next, move the mirror upwards until it is straight above you, while tilting the bottom of the card out just enough so you can read it in the mirror. Result: You will still see the card text reversed left-to-right, as before. But think for a second: You are also seeing yourself standing on your head. That's top-to-bottom inversion, so you are getting both types of inversion at once! So, is the lack of vertical inversion really inherent in the mirror question, or have you just been holding the mirror in the wrong place all these years? And if so, why does the choice of where to place the mirror give one inversion in one case, and two in the other?

2. In the second experiment, hold the same card at arms length in front of you. Have it facing towards you this time so you an easily read it. Next, use your other arm to position a small mirror to the left or right of your line of sight of the card, and adjust the mirror until you can see the card in it. You will see the card text reversed left-to-right, just as with a face-on mirror. Now start moving the mirror in a circular arc until it is above the line of sight instead of to the left or right, keeping the reflected image of the card visible as you do so. When the mirror reaches the top, what do you see? Probably not what you expected: The text now has normal left-to-right order, but each letter is flipped upside down! In other words, the missing "vertical inversion" that everyone worries about in mirror experiments is right there in front of you. If you don't believe me, try it; there's no ambiguity in the effect. And if you want something more to ponder, try to figure out exactly when and how the vertical-is-normal, left-to-right-is-flipped image of the text transforms into the vertical-is-flipped, left-to-right-is-normal image during the rotation of the mirror. You can watch every step of the process as it occurs!

So, with these two experiments, it's worth asking: Is it really true that mirrors always reverse left-to-right? The earlier answer that references Feynman's video discusses this same point nicely, so be sure to look at that if you have not already. I'll likely comment more myself at a later time. But for now, I just wanted to provide a couple of easy, fast experiments you can try yourself, ones that may make you ponder just what the question really is. (The answer of course is 42 -- that much we already know!)

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The mirror is not inverting the image in either direction but rather is acting like a rubber stamp doing a 1:1 transfer from surface to surface. If you wrote a word on a rubber stamp with fresh ink, then pressed the stamp to paper, the word on the paper would be backwards but not upside down. It's just a 1:1 transfer.

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