03 April 2009

Mirror, Mirror

I ran across an interesting optics problem at Cognitive Daily yesterday. The claim is that "your reflection occupies the same proportion of the mirror as you walk away." This is easily demonstrated to be false. The clearest, most obvious demonstration (also posted in a comment at the link) is to use a dry erase marker to trace the image of your hand as it sits directly on a mirror; at this point, the image on the surface of the mirror will be identical in size to the hand itself. Now, walk back several feet and try to line the hand's new reflection up with the outline of it. The image on the surface of the mirror will not be the same size, and, hence, does not occupy the same proportion of the mirror.

I'm not sure, but I suspect the confusion is between two uses of the word "image." In optics, the "image" is not the "image" as it is on the surface of the mirror. It is the image as it appears, and it is a virtual image located behind the mirror. In an ordinary, flat mirror, this virtual image is the same size as the object being reflected, just as if I had an identical twin and was looking at her through a window, and we were the same distance from the mirror. So the size of the virtual image does not change for a fixed object. However, the apparent size of the image as seen on the surface of the mirror will change, just as my twin would appear smaller if we both moved farther away from the hypothetical window. To say that the image always occupies the same proportion of the mirror is as nonsensical as saying that, if my identical twin and I keep moving further and further from the window, I will see my twin's image occupying the exact same proportion of the window. That can only happen if my twin does not move. As soon as she moves back from the window, her image in the window becomes smaller.

Another way to see that this is pure nonsense is to take a mirror that is much smaller than you are, say one the size of a picture frame. Place it close to your body; all you can see in it is the portion of your body closest to the mirror. Now, lean it against something and walk backwards from it. Eventually, you'll be able to see your whole body in it. So the proportion of body:mirror goes from a value larger than 1, where the body's image will not fit in the mirror, to a value less than 1.

I also found a NY Times article discussing this, and it may just be that it was reported badly. There's a link to the primary article at Cognitive Daily, but, alas, I don't have a subscription.

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