Monday, August 11, 2014
The Mirror Puzzle
(This is a chapter I wrote for a children's philosophy book called The Outer Limits (now part of The Complete Philosophy Files). This chapter was thought too abstract by the editors, and was not included).
Sometimes it is the things that are most familiar to us that turn out to be the most deeply puzzling. Take mirrors, for example. How many times do you see yourself reflected in a mirror each day?
1.ILLUSTRATE: BOY LOOKING INTO A MIRROR
At least ten or twenty times, I should think. Most of us never stop to think about what we see. But, as you are about to discover, mirrors are very strange and puzzling things.
An adventure in the mirror
Aisha and Kobir are visiting Kobir’s auntie. Auntie Anaximander lives in an enormous, fusty old house deep in the moors.
2. THE BIG SPOOKY HOUSE IN THE MOORS. LIGHTENING.
It’s a wild and stormy night and the phone and power lines are down. Auntie Anaximander has gone off in her car to report the powercut leaving Kobir and Aisha all alone in the dark house.
3: ILLUSTRATE: NERVOUS KOBIR AND AISHA AT DOOR OF CREEPY COUNTRY HOUSE. STORY WEATHER. THEY ARE WAVING OFF AUNTIE A WHO IS DRIVING A MORRIS MINOR. NB KOBIR IS OLD PHILOSOPHY FILES CHARACTER WITH GOATIE BEARD HAS A RING ON HIS RIGHT HAND AND WATCH ON HIS LEFT AND A HIS-LEFT-SIDE HAIR PARTING.
They light a candle and decide to explore. After a while, Aisha and Kobir come across a huge echoing hall.
4. ILLUSTRATE: KOBIR AND AISHA ENTERING SPOOKY ROOM AS DESCRIBED, HUNG WITH OIL PAINGINGS .HOLDING CANDLE.
There’s no furniture, just lots of dusty oil paintings of Auntie Anaximander’s dead relatives, and, at the very end of the room, a huge mirror stretching from floor to ceiling.
Aisha and Kobir walk up to the mirror and look at their reflections.
5 ILLUSTRATE: KOBIR AND AISHA, HOLDING CANDLE, BEFORE THE HUGE BAROQUE MIRROR, WITH CANDLE. STARING AT THEMSELVES.
As they have nothing better to do, they decide to sit down on the carpet in front of the mirror and play cards. Outside, the wind howls and rumbles over the chimney tops. But inside it is deathly quiet. The candle casts flickering shadows up the walls. Aisha opens her notebook to keep score and starts to shuffle the cards. While she is shuffling, Kobir stares distractedly into the vast mirror. And the more he stares, the more perplexed he starts to look.
KOBIR: That’s really, really weird.
AISHA: What is?
KOBIR: Take a look at our reflection.
Kobir and Aisha both stare into the mirror.
KOBIR: Notice anything odd?
KOBIR: You can see a mirror version of yourself can’t you? And a mirror version of me.
AISHA: Of course.
KOBIR: But there’s something very, very peculiar about our mirror selves. They are reversed.
Kobir stands up in front of the mirror.
KOBIR: Yes. Suppose this wasn’t a mirror but a big sheet of glass. And suppose I was actually over there, where the mirror-version of myself appears to be standing. Then my right hand, the one with the ring on, would be where my left hand actually appears in the mirror. And my left hand, the one with the watch on, would be where my right hand appears.
6. ILLUSTRATE: KOBIR STANDING NEXT TO SEATED AISHA BEFORE MIRROR. WATCH ON LEFT HAND AND RING ON RIGHT. HAIR PARTED TO HIS LEFT. HE IS LOOKING IN THE BIG, FULL LENGTH MIRROR.
Kobir walks closer to the mirror, so that he’s staring right into the face of his mirror-self.
Kobir: And look, my left eyebrow – the one I’m raising – appears where my right eyebrow would be.
7. ILLUSTRATE: KOBIR’S FACE AND ITS REFLECTION, WITH EYEBROW RAISED AS DESCRIBED. HAIR PARTED TO HIS LEFT.
Isn’t that odd?
Kobir is right about the reversal. When we look at Kobir in the mirror we see this:
8. ILLUSTRATE: KOBIR STANDING LOOKING AT US, RING ON LEFT HAND AND WATCH ON RIGHT. HAIR PARTED TO HIS RIGHT.
But if Kobir was actually standing where his image appears to be standing, we would see this:
9. IDENTICAL IMAGE BUT WATCH ON LEFT HAND AND RING ON RIGHT. HAIR PARTED TO HIS LEFT.
See? Kobir’s left and right sides have been reversed.
Why do mirrors reverse left to right, but not top to bottom?
Aisha just wants to play cards. She’s irritated by Kobir’s question.
AISHA: Why is it odd?
KOBIR: Well, mirrors reverse us left to right. So why don’t they also reverse us top to bottom.
AISHA: Top to bottom?
KOBIR: Yes, in the mirror, what’s top and bottom is the same. Take a look: my head is still at the top and my feet are still at the bottom. But my left and right sides are swapped round.
KOBIR: So my question is: Why do mirrors reverse one way, but not the other? It’s very, very strange.
Kobir is correct. In a mirror image, what’s at the top and what’s at the bottom remains unchanged. But left and right are switched. That is why the word “ambulance” is painted in reverse on the front of an ambulance.
10. ILLUSTRATE: AMBULANCE WITH WORD “AMBULANCE” REVERSED ACROSS THE FRONT.
The word is seen by other motorists in their rear view mirrors. Because mirrors reverse left to right, and because the word “ambulance” is itself reversed left to right, a rear view mirror has the effect of turning the word the correct way round so it can easily be read.
11. ILLUSTRATE: SAME AMBULANCE SEEN IN REAR VIEW MIRROR: NOW THE WORD IS RIGHT WAY ROUND (DRIVER ON OTHER SIDE, OF COURSE)
Why do mirrors reverse left and right, but not top and bottom? Some of the world’s greatest minds – including that of the Ancient Greek philosopher Plato – have struggled with and been defeated by this infernal mystery.
12.ANCIENT GREEK IN BEARD AND ROBE LOOKING IN MIRROR AND THINKING BUBBLE: “WHY DO MIRRORS REVERSE LEFT/RIGHT BUT NOT TOP/BOTTOM?”
Can Kobir and Aisha do any better? Can you?
Perhaps. But before we try to come up with an explanation that might work, let’s take a look at a few explanations that definitely don’t work.
Aisha can now see the puzzle.
AISHA: Hmm. I’m not sure why mirrors do what they do. Now you mention it, I suppose that is rather peculiar.
KOBIR: It’s weird!
Suddenly, Aisha thinks she has the solution.
AISHA: Doesn’t it all depend on which way up we are? If we lay down, then it would be top and bottom that are reversed, not left and right.
KOBIR: I don’t think so. Let’s try.
Kobir moves the candle safely to one side. Then they both lie on their sides and stare at themselves.
13.ILLUSTRATE; T AND A LOOKING INTO MIRROR AS THEY LAY ON THEIR SIDES. KOBIR IS WIGGLING FINGERS OF RIGHT HAND (THE ONE WITH RING).
KOBIR: You see? If I was over there, my right hand – the one with the ring – would be where my left hand appears. My left and right sides are still reversed.
AISHA: True. And our tops and bottoms are still the right way round.
KOBIR: So the mirror flips things around left to right but not top to bottom no matter what way up we may happen to be in front of it.
AISHA: Hmm. I guess you’re right.
KOBIR: I am right. The reversal has nothing to do with which way up we are. Actually it doesn’t have anything to do with which way up the thing we are looking at is, or which way up the mirror is, either. Mirrors always reverse left to right but not top to bottom.
You can confirm this for yourself. Here’s the word “ambulance” reversed from left to right:
14.ILLUSTRATE: “AMBULANCE” REVERSED L/R
Try putting this page up in front of a mirror. Because the mirror also reverses left and right, the word gets switched round the right way again, doesn’t it?
15.ILLUSTRATE: MIRROR IN FRONT OF REVERSED WORD “AMBULANCE”, IN MIRROR THE WORD IS ROUND THE RIGHT WAY.
Now let’s try turning you, the observer to see if that has any effect on how the word is reversed. Hold the book upright, but turn your head sideways like this.
16.ILLUSTRATE. KID BEFORE MIRROR: HE IS HOLDING BOOK UPRIGHT WITH HEAD TILTED OVER SIDEWAYS.
See? It doesn’t matter which way up you are. The mirror still flips the letters back the right way. It still reverses them left to right to make them readable again.
Now let’s try rotating the book 90 degrees, like this.
17.ILLUSTRATE: SAME KID BEFORE MIRROR, NOW WITH HEAD UPRIGHT BUT HOLDING THE BOOK SIDEWAYS ON.
Does that have any effect on how the word appears? No. The word “ambulance” is still legible. So it doesn’t matter which way up the word is either.
Maybe you’re thinking that the solution has something to do with which way up the mirror is. But it doesn’t. Turn the mirror upside down, and the image will remain exactly the same.
So it doesn’t matter which way up we are, which we up the thing we are looking at is, or even which way up the mirror is, a mirror always reverses left to right but not top to bottom.
18.ILLUSTRATE: KOBIR AND AISHA LOOKING AT THEMSELVES IN MIRROR 9STILL IN THE SPOOKY ROOM): BIG THOUGH BUBBLE WITH QUESTION MARK IN IT COMING OUT OF THEIR HEADS. THEY LOOK PERPLEXED.
The more you grapple with this mystery, the deeper and more profound it seems to become. In fact, the more you think about mirrors, the more they seem to take on a rather spooky, almost magical quality. Just why do mirrors do what they do?
I think I know the solution, or at least a part of it.
Aisha and Kobir heard a distant bang followed by echoing footsteps. After a minute or so, Auntie Anaximander’s damp features appear round the doorway.
25.ILLUSTRATE:. ANAXIMANDER IN RAINCOAT, LOOKING ROUND DOOR OF ROOM.
ANAXIMANDER: Ah. There you are! Power should be back on soon. What have you been up to?
Auntie Anaximander takes off her raincaot and shakes out the raindrops, spraying them across the room. A few droplets land on the pages of Aisha’s open notebook.
26. ILLUSTRATE: RAINDROPS ACROSS THIS PAGE.
Aisha and Kobir explain that they had been discussing why mirrors reverse left and right, but not top and bottom. Auntie Anaximander, who loves puzzles such as this, sits down on the carpet beside them. She thinks hard for a moment or two.
ANAXIMANDER: You know, in a sense, a mirror doesn’t reverse anything at all.
AISHA: In what sense?
ANAXIMANDER: Well suppose we hold a clock up in front of a mirror, like this.
Anaximander draws this picture on a page of Aisha’s notebook.
27.ILLUSTRATE: CLOCK AND ITS REFLECTION IN MIRROR
ANAXIMANDER: Now let’s draw arrows linking each number on the clock face with the same number reflected in the mirror.
Anaximander adds arrows like this:
28.ILLUSTRATE: AS ABOVE BUT WITH ARROW DRAWN FROM EACH NUMBER ON CLOCK FACE TO IT’S IMAGE IN THE MIRROR (I WILL PROVIDE ROUGH)
ANAXIMANDER: These arrows show that the way the mirror reflects is entirely symmetrical in every direction. The arrows don’t cross over top to bottom. But neither do they cross over left to right.
KOBIR: You know, I think you’re right. In a sense, nothing gets reversed!
Auntie Anaximander is right. The way in which the mirror reflects what’s in front of it is perfectly symmetrical. It is not as if a mirror reflects rays of light differently depending on whether they are coming from your left and right sides rather than your top and bottom. The light is reflected in the same way no matter where it happens to land on the mirror.
So the puzzle has absolutely nothing to do with how light behaves.
KOBIR: But hang on. We said, didn’t we, that the left and right are swapped round? And that top and bottom are unchanged?
ANAXIMANDER: Yes. That is how we would normally describe what we see.
ANAXIMANDER: If we were to replace this mirror with a sheet of glass, and you were to walk round the glass and stand on the other side, looking towards me, in the exact spot you currently see your mirror-self standing, then what would I, standing over here, see?
KOBIR: You would see my right hand to your left, and my left hand to your right.
29.ILLUSTRATE: WE ARE LOOKING AT KOBIR BEHIND BIG SHEET OF GLASS. AUNTIE ANAXIMANDER AND AISHA ARE THIS SIDE, WATCHING.
ANAXIMANDER: True. And of course, when we look at your mirror-image, your left hand appears where your right hand would be if you were standing there, and your right hand appears where your left would be. That’s why we said that the mirror-version of you is reversed left to right. Correct?
ANAXIMANDER: But notice that we have just taken something for granted: the axis about which we rotate you when we imagine you over there.
AISHA: The what?
ANAXIMANDER: When we turn something round, we rotate it about an axis, don’t we? A spinning top, for example, rotates around a vertical axis, doesn’t it?
30. ILLUSTRATE: SPINNING TOP WITH VERTICAL DOTTED LINE LABELLED “VERTICAL AXIS”
And a car wheel rotates around a horizontal axis.
31.ILLUSTRATE CAR WHEEL WITH DOTTED LINE THROUGH MIDDLE LABELLED HORIZONTAL AXIS.
Now, when we imagine you over there in the position your mirror-self seems to be in, we mentally put you there by rotating you about a vertical axis, like this.
Anaximander draws this diagram in the notebook:
32.ILLUSTRATE: SEMI CIRCLE DOTTED ARROW SHOWING HOW KOBIR IS ROTATED BY AN INDICATED VERTICAL AXIS. SEE ROUGH.
KOBIR: That’s right. That’s how I would get over there, by walking round the mirror like that.
ANAXIMANDER: Of course. But what if we were to get you over there not by rotating you around a vertical axis, but a round a horizontal axis? What would happen then?
KOBIR: I’m not sure. Let’s draw it to see.
Kobir now draws this picture:
33.ILLUSTRATE: AS ABOVE, BUT KOBIR IS NOW ROTATED AROUND HOROIZONTAL AXIS, LEAVING HIM STANDING ON HIS HEAD BEHIND THE MIRROR (I WILL PROVIDE ROUGHS)
ANAXIMANDER: Now see what happens! You are standing on your head. And, compared to your mirror image, your left and right sides aren’t switched round.
34.SPLIT ILLUSTRATION: CAPTION 1 “MIRROR IMAGE” RING ON KOBIR’S LEFT, WATCH ON HIS RIGHT, PARTING TO HIS RIGHT. CAPTION 2: “KOBIR STANDING ON HIS HEAD” KOBIR UPSIDE, BUT NOT REVERSED LEFT TO RIGHT.
Your right hand stays over to the right, just as it appears in the mirror. But top and bottom have been switched round!
AISHA: Wow! You’re right.
ANAXIMANDER: So you see, the reason we say mirrors reverse left and right but not top and bottom is due to the fact that we take for granted a particular axis of rotation. But we could just as easily choose a horizontal axis. Then it would be true to say that a mirror reverses top to bottom but not left to right.
I think Auntie Anaximander is correct. She really has hit on the explanation of why mirrors reverse left to right but not top-to bottom. Yes, it’s true to say mirrors reverse left to right, but only if we choose a vertical axis of rotation. Choose a horizontal axis and they then reverse top to bottom.
Kobir furrows his brow.
KOBIR: But why do we take the vertical axis for granted?
ANAXIMANDER: Because people aren’t in the habit of flying through the air like pigeons and settling on their heads. When people normally rotate, it’s almost always about a vertical axis. So we just took for granted a vertical axis of rotation in this case too.
Suddenly the electricity comes on. The chandeliers above their heads flood the hall with light. Auntie Anaximander jumps up and blows out the candle. She suggests they all go downstairs for some hot chocolate. So they do.
35: ILLUSTRATE: AUNTIE, KOBIR AND AISHA LEAVING THE NOW VERY BRIGHTLY-LIT HALL.
I believe Auntie Anaximander is right: this puzzle about why mirrors do what they do is created by our not noticing what we have taken for granted. To solve the puzzle, we need to take a step back and start questioning what we taken for granted.
Actually, I have noticed a similar puzzle about doors. Walk though a door that opens on your left and turn round to come back through it, and the door now opens on your right. But pass through a door that opens at the top (like a cat-flap) and turn to come back through it and the door still opens at the top.
36.ILLUSTRATE: DOOR OPENS AT TOP LIKE A CAT FLAP.
Why does passing through a door reverse the way it opens from left to right, but not from top to bottom? What explains the difference?
The solution is much the same as Auntie Anaximander’s solution to the mirror puzzle. When you pass through a left-opening door and turn around to come back through it, you would normally rotate about a vertical axis, like this:
37.ILLUSTRATE: AS FOR PREVIOUS ILLUSTRATION BUT ONE: EXCEPT WITH AXIS OF ROTATION ADDED: SEE ROUGH.
But what if you were to rotate about a horizontal axis, like this?
38.ILLUSTRATE: AS ABOVE, BUT PERSON FLIPPED HORIZONTALLY, LEAVING THEM UPSIDE DOWN.
Then the door that opened on the left would still open the left on the way back though it, but a door that opened at the bottom would now open at the top. We say that left and right are reversed but not top and bottom only because we take for granted a particular axis of rotation.
In space, where we are weightless, the axis of rotation about which we choose to rotate when turning to come back through a door is less likely to be the vertical axis. You could just as easily spin about a horizontal axis instead.
39.ILLUSTRATE: SPACE MAN LEAVING SHIP THROUGH DOOR, THEN ENTERING THROUGH DOOR UPSIDE DOWN.
So, after years in space, it might start to seem as natural to you to say that a door that opens at the top opens at the bottom when you come back through it as it does to say that a door that opens on the left opens on the right when you return though it.
For creatures that live in a weightless environment, where it’s as easy to rotate about one axis as it is to rotate about the other, perhaps neither the mirror puzzle nor the door puzzle would even be puzzles!
Perhaps you are still not convinced by Auntie Anaximander’s solution. If so, here’s another little experiment you can perform. We said that in a mirror the word “ambulance” is reversed left to right and not top to bottom. So now write out, on a separate piece of paper next to the word “ambulance”, the left-right reversed version of the word. It should look like this:
40. ILLUSTRATE: SEE ROUGH.
You have kept what’s at the top at the top and what’s at the bottom at the bottom, but the right hand end of the word has become its left hand end.
But now suppose that mirrors reversed, not left to right, but top to bottom. What would that look like? What would we see then?
Try to draw it. This time, write on a separate piece of paper underneath the word “ambulance” how the word would look if what’s at the top becomes what’s at the bottom, but the left and right ends remain unchanged. It should look like this:
41.ILLUSTRATE: SEE ROUGH
The word is flipped top to bottom but not left right.
But now look very closely at the two reversed words.
42.ILLUSTRATE: THE TWO REVERSED WORDS (SEE ROUGH), LABELLED “REVERSED TOP TO BOTTOM” AND “REVERSED LEFT TO RIGHT”.
Notice anything peculiar?
The two reversed versions of the word that you have written are actually exactly the same! Turn the second one round to check, like this:
43.ILLUSTRATE: SEE ROUGH (LABEL LEFT RIGHT REVERSED AND TOP BOTTOM REVERSED)
The left-right reversed version of the word just is the top-bottom reversed version.
So why did we call one left-right reversed and the other the top-bottom reversed? Because we just assumed a different axis of rotation in each case. Whether we describe an image as left/right reversed or top/bottom reversed all depends on what axis of rotation we choose.
So I think we have solved the puzzle (or at least we solved this version of it).
Here’s an interesting fact about the mirror puzzle: it can’t be solved by doing science. It can only be solved by thinking philosophically. We didn’t have to any scientific research into how light and mirrors behave. We didn’t have to investigate how our brains work.
44.ILLUSTRATE SCIENTISTS IN LAB EXPERIMENTING WITH MIRRORS, LAZERS, BRAINS ETC.
Even if we had done that sort of scientific research, it wouldn’t have solved the puzzle. In order to solve the puzzle, we need to stop doing science and start doing philosophy. It’s a puzzle that can be solved just by thinking.
45. ILLUSTRATE: ME SLUMPED IN CHAIR, LABELLED “PHILOSOPHER AT WORK”
People sometimes assume that all our questions can be answered by science. They would just assume that the mirror puzzle must have a scientific solution. But it turns out that the mirror puzzle is a puzzle that science can’t solve.
Sometimes, only philosophy will do.