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July 1999
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Pat Murphy & Paul Doherty
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by Pat Murphy & Paul Doherty

Mirror, Mirror, On the Wall

Mirrors have a reputation for trickiness. Magicians use them to fool your eyes and make rabbits come out of hats. Break a mirror, and you get seven years bad luck. In science fiction and fantasy literature, mirrors have often played an important role. In Through the Looking Glass, Alice travels through a mirror to a world where a talking egg advises her on the meaning of words. In the fairy tale of "Snow White," a magic mirror tells the wicked queen who is the fairest, thus getting poor Snow White in no end of trouble. In Greek myth, Perseus uses a polished shield as a mirror so that he can lop off the head of the Medusa, a Gorgon with snakes for hair and other unattractive attributes. (Looking directly at the Medusa would have turned Perseus to stone.)

Now there are some interesting cultural issues here-like why did Perseus have to go hounding this poor Gorgon anyway, since she lived on a remote island and didn't seem to be bothering anyone. And why was the wicked queen so obsessed with her beauty, rather than her wit, her charm, her bravery, or any of a number of other attributes? Pat may take up these issues in future stories, but they aren't the topic of this column.

Here, we are dealing with the scientific side of mirrors. There's no need to venture into fantasy or gender politics to recognize the trickiness of mirrors. All you need is a mirror or two and the time to make a few observations.

A Tricky Question

But before we get to that, here's a question to ponder. Suppose you're in a Twilight Zone story in which the mirror image of your best friend steps out of the mirror, a flesh-and-blood doppelganger that tries to take over your friend's life. The doppelganger is your friend's mirror image-it looks just like her. Or does it? How can you tell the original friend from the mirror-image doppelganger?

Take a minute and think about it. We'll tell you the answer in the next paragraph. The answer is simple-you probably knew it immediately-but as one learns around the Exploratorium, simple answers can be as tricky as mirrors.

So how do you tell the doppelganger from the original? Well, let's say your friend was right-handed. You toss a pencil to one version of your friend and then to the other. One catches the pencil in her right hand; the other uses her left hand. Ah, ha! If your friend was right-handed, the mirror-image doppelganger will be left-handed. You've identified the doppelganger; it's the sinister lefty.

Take Another Look

You can confirm that the doppelganger of a right-handed person will be left-handed by taking a look in a mirror. Reach up and touch your right ear with your right hand. Your mirror image doppelganger--that is, your mirror image--will reach up and touch its left ear with its left hand. So you know that mirrors reverse right and left.

Yeah, yeah, yeah, you're saying. This is old news. Tell me something I didn't know.

Hold on there! Not so fast! We told you that mirrors are tricky. Let's take another look at what mirrors really reverse.

First, following in the footsteps of the wicked queen in Snow White, find a mirror on a wall. For this experiment, it is important that the mirror is on the wall rather than lying flat on a table or on the ceiling over your bed (but let's not get into that). Hold this arrow so that it points to the ceiling. Look at the reflection of the arrow and yourself in the mirror. The reflected arrow also points toward the ceiling. So now you know that up and down remain the same for the arrow and the mirror image in the mirror on the wall.

Next, point the arrow to one side. We'll need to identify the direction in which the arrow is pointing, so move something to one side of you and the mirror. When Paul was doing this experiment in his office, he moved a trashcan to one side of the mirror, so let's just say you do the same. Point the arrow to the trashcan side.

What does the mirror arrow do? It also points to the trashcan side-the mirror doesn't reverse the direction of the arrow.

Now isn't that odd? Your friend's doppelganger was left-handed because mirrors reverse right and left. So why doesn't the mirror reverse the direction of the arrow?

Here's where the trickiness of mirrors because entangled with the trickiness of semantics. When discussing mirrors we have to struggle with the meaning of "right" and "left."

On the Other Hand

To explore the meanings of right and left, you'll need a pair of gloves. (We used the rubber dishwashing gloves from under Pat's sink.)

Pick up the right-handed glove and hold it with fingers up and palm toward the mirror. Take a look at its image in the mirror. Could you put your actual right hand into a glove that looked like the one you see reflected in the mirror? Place your right hand against the mirror as you compare it to the glove. Be sure to look at your hand and compare it to the reflection of the glove-if you look at the mirror image of your hand, you'll just confuse yourself even more. Your right hand won't fit in the mirror image of the right-handed glove.

Somehow the mirror doesn't reverse up-down, or trashcanward-antitrashcanward, but it does change the reflection of a right-handed glove into a left-handed one. How can this be?

To find out what the mirror really reverses, hold up the arrow on page 00 again. Look in the mirror and point the arrow away from you, toward the mirror. Look at what happens. The image of the arrow in the mirror points toward you, out of the mirror.

Ah, ha! The mirror reverses in and out. This in-out reversal is what changes a right-handed glove into a left-handed one.

What's Right, Anyway?

So how do you tell right from left? You could start with a hand print, like the ones you see in Anasazi pictographs in the American southwest or on the walls of elementary school classrooms at Thanksgiving (colored-in to make turkeys). Make a tracing of your own right hand. Put a piece of paper on a table. Place your right hand on the paper, palm down, with the thumb stuck out to the side at right angles to your fingers. Use a pencil to trace your hand. (You can, if you like, now decorate it to look like a turkey. Or not.) Try to fit your left hand into the drawing. Notice that when you place your left hand on the drawing, palm down, the thumb is on the wrong side. Show the drawing to someone who has not seen you do the tracing. Ask them if it is a right hand or a left hand. They will probably say it is the tracing of a right hand since they guessed that you traced your hand with the palm toward the paper like any sensible person would.

Ah, but that's not the only answer. Get another piece of paper. This time, put your left hand on the paper with the palm up, and trace your hand. Compare this tracing to the tracing of your right hand with the palm down. What do you know? A tracing of a left hand with the palm up looks just like a right hand with the palm down. If you showed someone the tracing of your left hand, chances are they'd guess it was a tracing of your right hand. Whether the viewer sees the trace of a right hand or a left hand depends on which way they think the palm of the hand is facing. If the palm of the traced hand points into the paper it is a right hand; if it points out, it is a left hand. From the tracing you can't tell which way the hand was facing.

So this simple tracing leads us to appreciate one important facet behind the concept of right and left. To make a right hand or a left hand you need to specify all three dimensions: you need to know which way the fingers point, which way the thumb points, and which way the palm points. Change the direction of any one of these hand parts and the hand changes from a right hand to a left hand.

This last sentence is the key to how mirrors change right hands into left hands. Remember: mirrors reverse in and out. Hold your right hand up with the palm toward the mirror. The mirror image of your hand has the palm facing out, but the fingers are still pointing up and the thumb is on the same side as it was before--if your real thumb was pointing toward the trashcan, your mirror image thumb is pointing toward the trashcan. The only thing that has changed is the direction that the palm is facing, but that's enough to change your right hand into a left hand.

What else can change a right hand into a left hand? To answer that question, take a look at that glove you were fooling with before. It's a right-handed glove. There is no way to rotate this glove in our space so that this right glove will fit onto your left hand. However, suppose glove is thin and flexible enough--like the rubber gloves used for dishwashing or like the latex gloves used by doctors for less pleasant things. Then you can turn this right glove into a left one by turning it inside out.

If you have a glove like the one we described, try this. Hold the glove with the fingers pointing up, the thumb pointing to the left, and the palm pointing away from you. Now, turn the glove inside out. The glove now fits a left hand. And it looks exactly like the mirror reflection of the original right handed glove. Position this glove so that the thumb points to your left and the palm is away from you. To do this, you'll have to point the fingers down, rather than pointing them up. Once again, you've changed one direction to convert right into left.

Physicists' Rules

All this talk about swapping one direction and changing the "handedness" of a hand got Paul thinking about how physicists use this property of hands as a convenient tool for doing vector algebra. Physicists describe the world with three dimensional coordinate systems placing x, y, and z axes at right angles to each other. There are two ways to assemble coordinate systems from these axes: one is called right handed; the other, left handed.

To see what a right-handed coordinate system looks like, hold out your right hand. If the x axis points along the extended fingers of your right hand and the y axis points out your palm, then the z axis will point along your thumb. To see what a left-handed coordinate system looks like, substitute your left hand. When Paul knows a law of physics uses a right handed coordinate system he can hold up his right hand and remember how the axes go. (Incidentally, Paul notes that physics professors tend to be biased toward right- handed coordinate systems. This bias actually benefits left-handed students during exams. The lefties can do the vector algebra with their right-hands and keep writing with their left hands.)

A Topsy Turvy World

At the Exploratorium, exhibits on the museum floor encourage people to ask questions about mirrors. After touring the exhibits, more than one visitor has ended up at Paul's office with the question he calls "the mirror question". That is, "why do mirrors reverse right left and not up down?" Paul helps them toward an answer (just as we have hopefully helped you in this column), explaining that mirrors actually reverse in and out. At the end of the talk, when the visitor thinks they understand, Paul points to his ceiling, where he has mounted a large plastic mirror. In the ceiling mirror, objects in the room are turned upside down! Look at Paul's bookshelves on the wall over his desk. In the mirror, the desk appears above the bookshelves!. When a floor or ceiling mirror reverses in and out it also reverses up and down.

For Extra Credit

Now that you have done the experiments and become an expert of right-left mirror reversal, here's another question for you. Does a mirror on the ceiling also reverse right and left? Good luck!

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