On 11/28/07, David Gale <gale@math.berkeley.edu> wrote:
If you Google "mirror, left, right" you get over 9,500,000 hits.
Terrifying! If you're not already confused when you start, you will be by the time you've ploughed thru' that lot ...
Some of the discussions among the dozen or so I looked at are very good and most of them make the same basic point which I describe as follows.
An experimental finding is called an ARTIFACT if it turns out to have been produced not by nature but by the experimenters themselves and their apparatus. Scientists are of course very much aware and wary of this possibility. Well, the famous mirror phenomenon we have been discussing is an example, par excellence, of an artifact. People say things like, a mirror reverses right and left but not up and down, or words to that effect. Not true. It is we, the observers who do the reversing or up-side-down-ing. Simplest example: You are reading a book and decide to hold the page you are looking at up to a mirror. Will the letters in the image be "backward" or "upside down"? Obviously it depends in whether YOU the experimenter turned the book about a vertical or horizontal axis.
This is the point several people have made earlier --- that the observation has implicitly factored out proper isometries.
The case of, say, a room reflected in a mirror is a bit more subtle. It's certainly true that what you see in the mirror is different in an essential way from what you see when you turn your back on the mirror and look at the room directly. Namely, everything in the mirror is "backward". For example the hands of the clock on the back wall are at two o'clock but in the mirror they seem to be at ten o'clock, etc. Nevertheless, I assert that it's you, not the mirror that's doing the reversing. First you looked at the room directly, (or you imagine you have done so) and then you TURN and look at the image in the mirror. But how did you turn to face the mirror? Probably about a vertical axis, although (assuming you're sufficiently agile) you could have done it by standing in your head, in which case everything would be upside down.
Why should it be any different? It's the same phenomenon again --- the proper isometries have been filtered out as inessential (see below).
Finally, even the mirror is irrelevant. On the store window you see "MERRY CHRISTMAS" in big letters. You go inside and look at the same window and, -what do you know,- the letters run backwards. I've never heard anyone claim that windows reverse right and left. It's you, the experimenter with your apparatus, in this case your body, that accounts for the observed phenomenon. Yes?
This comment considerably enhances the potential for confusion / entertainment by introducing a completely different aspect of isometry: altering the dimension of the space under discussion. An improper isometry in (n-1)-space is expressible as the restriction of a proper isometry in n-space --- typically, a rotation through one half-turn, using the (n-2)-dimensional mirror as axis. The shopwindow is a Euclidean 2-space, the observer's motion in 3-space. Abbott's flatlander perceives a "left-right swap" in a (1-dimensional) mirror; we solid citizens perceive only a rotation [as we are obliged to --- an improper isometry involving ourselves being improbably drastic, both psychologically and physically].
Again, many of these and similar examples are given in the articles I referred to at the beginning.
Fred Lunnon