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Treebeard's Stumper Answer
24 October 2003

Looking-glass World

Ordinary things can be remarkable! Look at yourself in a mirror, like the beginning of Alice's second adventure: "That's just the same... only things go the other way." Sure, it's a mirror image. If I raise my right arm, my mirror-twin raises his left arm. But he's not upside-down. Why do ordinary mirrors reverse left and right but not up and down? Mirrors are the same on all sides, but rotating a mirror does not rotate the image. Why is there a preference for left and right? There's nothing hidden in a mirror. Can you explain what we see?

I took the photo on the left shooting straight into our bathroom mirror. I was holding the camera with my right hand up to my right eye and the word "mirror" on the card is normal. But the photo shows my mirror-twin in the looking-glass world where "things go the other way". He is holding his camera in his left hand, and the card looks very different, and he definitely has less hair than me!

The mirror-image photo above-right is what you would see standing "behind the mirror" and looking right at me as I took the picture. Right is now right, left is left, and you can read the sign. Remember the famous mirror scene in the Marx Brothers' Duck Soup? This is the photo that Harpo (in the "mirror" on the right) would take of Groucho on the left. (There is a 21 MB QuickTime movie file here.)    

Here are more looking-glass stumpers:

It seems mysterious that mirrors reverse left-right and not up-down. But mirrors really don't reverse sides at all. Mirrors only reverse front to back. Face a mirror and raise your right arm. Your mirror-twin will point in the same direction, though you are facing opposite directions. If you face your real twin, you will look and point in opposite directions. We're so accustomed to meeting real people that it seems strange when the images are not reversed! Still confused? Maybe it's like a Zen koan, and the answer is that mirrors do exactly what they should do!


It sure is hard to get clear about mirror images! Most accounts I've read on the Web (including mine) trace back to chapters 1 and 3 of Martin Gardner's The Amidextrous Universe (1964, 1990). Gardner ends his explanation by saying "This may still seem confusing. You may have to read over the last seven paragraphs several times and think everything through carefully before you grasp exactly what an ordinary mirror does." There's a discussion at the New Scientist The Last Word site where they conclude "This question has provoked a greater number of replies than any other." I'm still confused. When something so simple and literally "in your face" seems so complicated, that's a sign that we don't understand the question.

It helps to think of compass directions.

In each case, there is just one direction that reverses in the mirror, and it is always when you point towards or away from the mirror, whatever direction that happens to be. The mirror reverses only the axis at right angles to it's surface. That's normally front to back, but it can be any direction depending how you are standing.

Now think of the ways you can "turn around" without a mirror. In each case, start by standing facing north.

In each case, there are two directions that reverse and one that stays the same. But with a mirror-image, there is only one direction that reverses and two that stay the same. I could grab my real twin by the shoulders and turn him around so that we're standing front to back. I've reversed two of his directions and everything lines up. If I could grab the real tangible twin I see in the mirror and stand behind him, I will always find that I've reversed a direction too many.

I still have a nagging sense that this does not answer the stumper. Why does it seem so natural to think that the mirror "reverses left and right" even when I know it doesn't?

Graybear sent this answer:

I was thinking about the mirror-image stumper. I think the problem is that we sometimes confuse left and right (some of us more than others) but we rarely confuse up and down. Suppose you were holding a card with an arrow drawn on it instead of the word mirror, such that the arrow was horizontal. Then, holding the card the same way, you stood to the left side of the mirror and leaned to the right so that your image was now "appearing" from the left side and the arrow was pointing toward the ceiling. Most people would say that your image has flipped from 'left' to 'right', but they would agree that both the arrow and its reflection are pointed 'up'.
Graybear's first sentence got me thinking about a creature that would sometimes confuse up and down. Consider the odd stick figure critter on the left. Maybe it can fold up on the horizontal center line like a pair of headphones or earmuffs. It has up and down hands just like we have left and right hands, and its hands can lie together just like ours do when we clap.

The middle picture shows the critter's mirror-image, and the right picture shows it "standing on its head". It would be natural for such a critter to think that the mirror image is "reversed up and down" and he might confuse up and down, especially if his "hands" were really the same color like ours. We're different because we stand upright with left-right bi-lateral symmetry, so it's natural for us to think the mirrror image is "reversed left and right". The mirror is doing the same thing in both cases. It's how we are built that changes how we think about it.

Does this really answer the original stumper? You can hand me the threaded bolt in your right hand, and it will fit the nut in my hand. But the bolt I'd pluck from the mirror (if I could) would not fit the nut in my hand. Maybe my answer really is that mirrors do exactly what they should!

Mirrors always reverse one direction too many, so it's not surprising that you can make a "true mirror" that shows you as others see you by using two separate mirrors. In the photo at the right, I really am holding my camera in my right hand. A simple arrangement of two mirrors standing at right angles like a half-open book (or a curved mirror) will reverse in two directions. (See here and here.) It really does reverse left-right as well as front-back. Normal mirrors don't.

Because normal mirrors don't reverse left-right like this trick mirror does, there is no question why they prefer left-right to up-down. The question is wrong. There is no stumper. That's why the answers are so confusing. As the philosopher Wittgenstein said in the Tractatus (6.512), "The solution of the problem... is seen in the vanishing of the problem." It really is like a Zen koan.

You can have lots more fun arranging three mirrors in a triangle to make a kaleidoscope. Note that some images are reversed and some aren't. I'm sure this has to do with parity, whether the image has been reflected an odd (reversed) or even (normal) number of times. These are both photos of me taking the photo, so you can see the camera. In each photo I'm actually holding my camera with my right hand, shooting directly into a mirror. The photo on the left is a homebrew kaleidoscope I made from three small mirrors and some duct tape. Hold it in front of the camera lens and tilt it slightly to get good reflections. The other photo is inside the walk-in three-mirror kaleidoscope at the Exploratorium in San Francisco. You can see two mirrors meeting behind me. I'm shooting almost straight at the third mirror.

Here are some links for your own research about mirrors:

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