Why is it, asked the Author, interrupting a silence during which each of us had tended his own thoughts,
that flat mirrors reverse an image from left to right, but not from top to bottom? The gazes that were turned upon him ranged from the disdainful to the incredulous, and he grew defensive.
Well, mirror writing is back to front, isn't it, not upside down? And what you hold in your right hand, your image holds in its left, doesn't it, but it is not standing on its head? Why isn't the reversal in both dimensions?
The problem may be one of semantics rather than optics, suggested the Philosopher tactfully, looking towards the Physicist. That gentleman was staring abstractedly out of the window, as if he had not heard what had been said, but I could see furrows in his brow that had not been there earlier.
Before asking WHY such-and-such is the case,, said the Mathematician, a trifle brutally,
the sagacious first ask, IS such-and-such the case? The fact of the matter is that plane mirrors do NOT reverse an image, either left to right or top to bottom. Sensing from the looks that were turned upon him a swing of sympathy towards the Author, he modified his tone.
I see that I shall have to be more explicit. Imagine this situation, if you will: you are standing before a mirror, in your hand a sheet of paper with the letter b upon it. What do you see in the mirror? The other side of the paper is what you see. What do you do if you want to see the mirror image of the letter b? You turn the paper round! YOU have reversed the image. If you turn the paper round by rotating it 180° about its y axis (
Flipping it round from left to right I heard the Physicist whisper to the Author)
you will have turned it into the shape of a d. If however you rotate it 180° about its x axis (
Flipping it over top to bottom)
you will have turned it into the shape of a p. The mirror faithfully reflects these rotations that you have made. If you need any more convincing, draw the letter b on a sheet of transparent paper. When you look into the mirror, you will see reflected the same shape that you see by looking directly at the paper.
The Author was still dissatisfied.
If the mirror is not reversing the image, how do you explain the fact that the paper I am holding in my right hand, my image is holding in its left?
The Mathematician's voice rose a tone or two.
My dear chap, your mirror image does not have a right and left hand of its own, any more than a lamp-post does. In movie comedies, of course, when you look in a mirror what you see is quite likely to be somebody pretending to be your image - Harpo in Duck Soup, Lou Costello in The Naughty Nineties, Bob Hope in The Princess and the Pirate, among many others - and in that case, of course, what the victim holds in his right hand, they must hold in their left; but note the change of reference from one person to another. Your image however is not a person. You define what is right and what is left; if you are holding the paper in your right hand, you will see it in the reflected hand to your right. If you believe that a mirror reverses sides, try lying horizontally in front of one; do you see the shoes that are upon your feet on your image's head?
The Physicist applauded quietly.
Bravo, my dear sir, a veritable tour de force! I found that all most convincing. He repeated the last few words, but this time it sounded more like 'almost convincing.'
It would constitute the definitive account of the subject if you would just fill in one small omission. While you were talking, I cut the outline of a letter b from a piece of card. The card happened to be white on one side and black on the other, so that by turning it this way and that I can see a black b, a black q, a white d, or a white p. He demonstrated the point as he spoke.
Now if I repeat your transparent paper experiment with my cut-out, I find, as you so pertinently point out, that when I turn the letter b around, I can see the appearance of a letter d both in the mirror and in the object in my hand. However, the reflection is black, whilst the direct view is white. Now, you have totally convinced me that the mirror has not reversed the shape of the black b; I am forced to conclude therefore that it has reversed the colour of the white d. Would you care to give us the benefit of your knowledge on this hitherto undocumented effect?
The Mathematician opened his mouth as if to speak, but no words came, only the faintest of hoarse sighs like a whispered groan. Then he looked hastily at his wrist, exclaimed
My goodness, is it that time already! and hurried away.
The next day, a few small strips of sticking plaster adorned his chin. When the Author gleefully asked if he had cut himself trying to shave whilst lying on his right side and holding the razor in his left hand, his reply was brusque and, I thought, unnecessarily coarse.