The Ups and Downs of a Mirror's Magic Reflection

By Burkard Polster and Marty Ross

The Age, 25 February 2008

Why do mirrors interchange left and right, but not up and down? Why does a pimple on your left cheek appear in the mirror to be on your right cheek, whilst your hair remains on the top of your head?

A mirror is flat, its left and right sides not differing from its top and bottom. So why should this happen? Most popular explanations either fail to address the question seriously or are simply not convincing. We hope to do better.

There is a nice trick which demystifies the mirror. To set it up, replace yourself by a photo, and consider instead the photo’s image in the mirror – the mirror’s puzzling behaviour is exactly the same in this scenario.

But now there is an unexpected method of seeing the photo’s mirror image. Imagine the photo has been printed on a transparency. Then, if you look at the transparency from the back, you’ll exactly see the mirror image.

So, comparing a photo to its mirror image is exactly the same as comparing the photo to its flipside. Then, to compare them side-by-side, we need two copies of the photo, the second on a transparency which we flip.

But this flipping is the key to the mirror puzzle. Flipping amounts to rotating the transparency around some chosen axis. The axis is fixed, but other points, such as pimples, swap places to the opposite side of the axis. The edges parallel to the axis are switched, while the other two edges stay fixed. And critically, the choice of which axis to use is not mathematics but psychology.

When you look at any picture with an obvious “up”, like a photo of a person, you will naturally orient the picture and yourself so that both your up and the picture’s up are in the same direction. Therefore, if you have a photo and its mirror image, you will almost certainly place both of them with their ups and your up pointing in the same direction. This corresponds to choosing a vertical turning axis, with the left and right sides swapping. Holding the picture up against a vertical mirror ensures the same alignment of all the ups.

However, sometimes a different juxtaposition of original and mirror image is forced upon us, resulting in a different turning axis. For example, when you compare a mountain and its mirror image in a lake (a horizontal mirror), the turning axis is in fact horizontal. Then, the left and right sides of the mountain stay fixed, and the up and down are switched. And, you get the same effect if you lay a mirror on the floor, and view the image of a friend standing on the other side of the mirror.