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25 April 2008

Fun With Mirror Images

Ever wonder why mirrors reflect left and right but not up and down? In other words, why does the mirror image of text appear backward instead of upside down?


Let's look at an example.


Here I am proudly displaying the latest Amoeba Crunch poster, designed by Chip Kidd and available in the store for $13. Look at that reflection. It sure looks like the text (and therefore the whole mirror image) has been flipped horizontally.

However, that's only compared to how we expect it to look:

We think the text has been "flipped" around an imaginary line going from the ceiling to the floor, like this:


The problem is, how does the mirror "know" which way is up? Why didn't it just as easily do this?


(Of course, that's not quite right. The whole image would be flipped so I'd be on my head, but bear with me.)

The faulty assumption is that forward text factors into things at all. You'll notice that easily readable, forward text does not appear in the original photograph. In fact, it's backward both in front of the mirror (albeit viewed through the paper) and in the mirror image. This is easier to see if I take another photo with the paper turned around.

Now it ought to be clear that the text reads forward both in the original and the reflection, though in the reflection you're looking through the paper to text written on the other side. So in a sense the mirror hasn't flipped anything at all -- it's the same on both sides.

Another example: Consider my left hand, which has a wedding ring. To be overly pedantic, from my perspective the ring is on the hand that's to the left of my body. When I look in the mirror, as in the photo above, it also appears to the left (from my own perspective) of my body's mirror image. There is no left to right flip here. It's left and left on both sides of the glass. Likewise, the mirror hasn't changed the text. It's readable on both sides.

I can hear some of you crying foul. After all, from my mirror image's perspective he's wearing the ring on his right hand -- that's the flip. If you're thinking that, what you've done is redefine "left" and "right" in terms of his perspective. You're considering him as if here were someone in the real world. But he isn't!

Let's think about this. If in the real world a friend and I stand toe to toe facing each other, it's no stretch for us to agree her right hand is, from my perspective, to the left of her body. Without thinking, I assume the same is true for anyone facing me, including my mirror image. The trouble is, my mirror image is not in the real world. There is no guy over there! It's just me over here. The world in the mirror image is not an extension of the room I'm in. The fact is, the definitions of left and right don't change across a mirror and neither do the definitions of up and down. The left hand has the hand with the wedding ring on both sides. This can be hard to get around. We're hard wired to deal with people as real beings, and it sure looks like the guy in the mirror is a real guy standing there facing me with his right hand where I'd expect it to be. But the key is that my mirror image is just that, an image, a trick of light. The usual assumptions about people don't apply.

So if the mirror isn't swapping left to right or up to down, what is it doing? Is it showing you what you'd see if you were standing in place of the mirror? No, because if that were true the ring would be on the opposite hand (from your perspective, on the hand to the right of the body).

What a mirror does is flip things in to out. Specifically, along the imaginary line that is normal to (that is, perpendicular to) the mirror's surface. If you're facing it as I am here your mirror image is not merely you turned around, as if you stood facing the opposite way, but is you flipped through your belly to your back. (That would make a great animation. If someone has the time and expertise to put it together I'll include it.)


Notice that the unmarked side of the paper is toward the mirror in both the real world and the mirror image. Likewise, the paper is always between the mirror and me. There is not so much an axis of symmetry as a plane of symmetry, the plane of the mirror. So the reflection has nothing to do with the mirror "knowing" left from right or which way is up. The only direction it "knows" about is toward or away from its surface.

In mathematical terms, we can define a mirror's behavior with a Cartesian coordinate system having its origin on the mirror's surface and your eyeball on the positive z-axis, like so:


The negative z-axis extends into the mirror image world. For any point (x, y, z) in the real world you can find the corresponding point (x', y', z') on the far side using these equations, suitable even for the mathophobic:

x' = x
y' = y
z' = -z

This page explains mirror images in a slightly different way that you might find more clear.

5 comments:

Anonymous said...

This is correct. Mirrors DO NOT reverse the object, rather they present it to the observer, who in the act of seeing, reverses it. It is the eye's or camera optics which reverse images, not mirrors. In a mirror my right arm is still in the same relative position - on the right. My head is atill at the top. It is rather amazing actually that so few ppl seem to have noticed this.

Anonymous said...

This is an interesting article, something I always wondered about myself considering it is just a flat sheet of glass in all directions. A good explanation.

Scarlet said...

I like this, it took me a moment to realize that, in the first image, "AMOEBA CRUNCH" looks the same both on the real sheet and in the reflection. Cool!

I remember reading on Wikipedia (in the article "List of common misconceptions") that mirrors don't flip right-to-left, they flip back to front. I didn't get that until I considered the question, "Why don't mirrors flip up-to-down?" As I thought, I realized, "...Because...they flip BACK-TO-FRONT! OMG, I GET IT!" It was a total eureka moment. And the image helped me make sense of it. Thanks!

MikeO said...

The mirror reversal puzzle is not so much a matter of the science of light rays, optics, and the Physics behind reflection as it is a PEOPLE thing. I’d say it can best be solved by a science of human activities, a science that deals with how people interact with complicated or subtle things. The idea of “reversedness” in perceived images is far more subtle than the mere reversal of a photon trajectory.

When people want to compare any two nearly identical objects for some subtle difference there are two common strategies they can use. These two strategies are often useful, but they are oddly contradictory.


***


In one strategy, the two objects are lined up to face in the same direction before they are compared. For instance, if two nearly identical pens are to be compared, no one I know of would ever hold one pen horizontally, the other vertically, and then proceed to compare them. People commonly want hold them facing in the same direction for such a task.

When a person uses this strategy in the mirror situation they like to imagine themselves rotating about a vertical axis for a comparison with their image. This rotation brings them to face the same direction as their earlier image was pointing. This also requires them to mentally freeze their image as it was when they faced the mirror. When ALL this is done, the Left/Right reversal is obvious.

It can get a little complicated, but many people seem to have the mental circuitry to perform all this in a near subconscious flash of imagery. Verbalizing it is far more difficult and hardly anyone can hold on to the images long enough to do that.


***


The second common human strategy for dealing with complicated situations, is to freeze EVERYTHING, avoid disturbing the scene, and look at the situation “as is” in order to perform an analysis of the subtle differences between two objects. This is the classic Sherlock Holmes approach to a crime scene.

In the mirror setup, performing this “as is” strategy means NOT rotating anything at all before doing the comparison between image and object, and the Left/Right reversal fails to show up. Instead, a Front/Back reversal is apparent in this “as is” comparison.


***


Of the two strategies, the second is FAR simpler. The first is almost too complicated to be done mentally, and it’s often impossible to document in any way. Yet, it can happen in a flash of mental imagery, only to fade as soon as words are brought in to capture it.

Often a person, deep in the throws of the mirror riddle, will drift from one strategy to the other subconsciously. This is often described as “magical” by those who delight in that sort of thing, while others will literally complain of the headache it causes them.

When each strategy is carefully thought through, ONE AT A TIME, clarity results.


***


It’s unlikely, but it might possibly occur to a gymnast to apply the first strategy by rotating about an horizontal axis, instead of the vertical in order to face the same direction. This means performing a hand stand to face in the same direction as the earlier frozen image, and then Up and Down would be seen to be reversed. But who is prone to do this difficult action or even to imagine it? And not only is it difficult, but it risks injury and even seven years of bad luck if the mirror is broken?

So, it’s not the mirror that does any reversing of perceived images, it’s people who do that by their selection of strategies. The three strategies mentioned “reverse” all three dimensions: Left/Right, Front/Back, and Up/Down. Mirrors reverse the direction of light rays, but people decide how they are going to compare image to object.


***

I’ve been examining, collecting and analyzing answers to this riddle for many years. It’s been puzzling people for centuries. Even Nobel physicist Richard Feynman has tried his hand at it, and failed, in my opinion. Recently I set up a website to discuss this subject at length.

Please visit me at: http://mirror-reversal.proboards.com

Thanks,

MikeO

Anonymous said...

Another way I've heard it explained is that the apparent reversal depends on how you choose to turn something that's facing you, if you want it to face the mirror instead. In one of the pictures, The Amoeba Crunch sign was turned on a horizontal axis to face the mirror, and in that case, it appeared upside down in the mirror (as it did through the back of the paper). When a person turns towards towards or away from a mirror, it's much easier to turn on a vertical axis than to stand on one's head, and that choice affects what we see.