If you hold this page up to a mirror, the mirrored image will display text that reads from right to left instead of left to right. Why? You may well consider this to be a naïve question scarcely worth the trouble of discussion. And yet, to many people, this is a mysterious and baffling issue.
The puzzlement is nothing new—after all, polished surfaces have been in use for thousands of years. The problem has left many philosophers scratching their heads. According to Plato, “all such appearances are necessary consequences of the combination of the internal and external fire, which forms a unity at the reflecting surfaces.” Lucretius believed that the image “turns inside out.” And Kant reasoned that these objects were merely “sensuous intuitions, that is, appearances whose possibility rests upon the relation of certain things unknown in themselves to something else, namely to our sensibility.” (You probably have questions about these explanations. Can I get back to you?)
In modern times, the problem has not gone away. In the opinion of Martin Gardner in The Ambidextrous Universe (1964), it is all due to a “mental rotation,” and “the mirror has not reversed left and right at all, it has reversed front and back!”
Then in June 1987, there was a renewed flurry of interest when the subject cropped up in the letters section of New Scientist. The result of all this was that I happened to become involved in a three-way correspondence on the subject with Lewis Wolpert (currently emeritus professor of biology as applied to medicine at University College, London) and Richard Gregory (currently emeritus professor of neuropsychology at Bristol University).
By now a subsidiary problem had developed: why do mirrors switch left for right but not top for bottom? Was there some way a mirror could express a preference?
Lewis Wolpert insisted that left and right hands were really inverted. “I still think the real problem with mirrors,” he told me, “is about rotations, and the key thing one needs to explain is why clockwise looks anticlockwise.” What’s more, he maintained that if you rotate a book around its horizontal axis, its mirror image will be not only upside-down but also reversed left-right. To see that this is untrue, try the experiment yourself with this page.
Richard Gregory commented, “[Wolpert] is used to observing embryological, etc., structures, and evidently images are not in his or many other people’s cognitive maps of how things should be.”
Arthur C. Clarke opined that this all had something to do with gravity. Gregory told me, “Whether Arthur Clarke put this thing forward as a joke I am not sure: it was complete nonsense.” He wrote to Clarke, enclosing a typescript of the essay on mirror reversals in his book “Odd Perceptions” but received no reply.
So when you hold a book up to a mirror, what is it that causes the left-right transposition? The answer is . . . you do. What do you do when you want to show the mirror the page? You rotate the book around a vertical axis, switching left for right. And the mirror obligingly displays the result of your rotation. The text in the image reads from right to left.
Why can’t the mirror show an upside-down image? But it can. And once again you are the cause. Beginning with the page facing you, rotate the book around a horizontal axis, switching top and bottom. Once again, the mirror faithfully shows you an upside-down image. And you will notice that the text is not additionally switched left for right. Why should it be?
In case you are confused by the fact that a book is opaque and doesn’t allow you to see the printed page and its mirror image at the same time, imagine that you have written a single lower-case letter b on a sheet of glass. While the letter is still facing you, look at its mirror image: you will see that the letter and its mirror image are exactly the same—b. The mirror has altered nothing. Rotate the sheet of glass around a vertical axis, switching left and right. The letter on the glass has reversed and now looks like a lower-case letter d. And so does its mirror image—again, the mirror has not changed it in any way.
Go back to your starting position showing b, and this time, rotate the sheet of glass around a horizontal axis, switching top and bottom. The letter on the glass now looks like a capital P—and so does its image.
There is a final possibility. Begin with your starting position showing b and rotate the glass twice—once vertically and once horizontally. The glass will now show you what looks like a backwards capital letter P—and so does the mirror, so the letter and its image are still identical. In other words, it is your physical movement of the object that causes any reversals of the image.
Doubters have one further shot in their locker. “Stand in front of a mirror,” they say, “and wave your left hand. The image will wave back with its right hand. So the mirror does cause reversals after all.”
But this is mere word play. Get rid of the words “left” and “right,” and the problem disappears. Wear a wrist watch on your left hand and call this hand the wrist-watch hand. Wave your wrist-watch hand, and the image waves back at you with its wrist-watch hand. Nothing has crossed over. Richard Gregory wrote, “It is amusing that many extremely bright people—Kant, Plato, Martin Gardner, and more immediately some of my cleverest colleagues and students—have got this wrong in various ways.” And the final sentence in his letter to me was the one word: “Amazing.”