I am blogging on this topic rather unwillingly. My thoughts upon this problem are nearly two years old. I was reluctant to blog this one for two reasons: one, this is an old and well known question; hence I expected a handful of articles addressing this one on the net, providing answers close to mine. Surprisingly, I found no answer close enough to mine. And two, this topic is technically way too specific to appear in my blog. However, I have tried my best to use the problem just to illustrate what I want to say.
Well, let’s begin with the question. “why is a mirror image laterally inverted and not vertically inverted ?” I have observed that quite a few people, after a second’s reflection, don’t even realise that there actually is some trouble with the image. The question as such is not clear and hence needs to be defined properly. Here, I have described the ‘trouble’ with the mirror image in a slightly different language.
The mirror has a plane. And it has an axis, perpendicular to the plane. The human body has three directions intrinsically defined along three axes. Feet to head defines the directionality of the vertical axis. Back to front defines the directionality of one of the horizontal axes. Left to right defines the directionality of the other horizontal axis. What the mirror does is, it reverses the directionality of the axis pointing towards the mirror, i.e, the axis perpendicular to the mirror. It does not change the directionality of the axes parallel to the surface of the mirror(at least this is what one would expect). Let’s look at what the mirror does to the human body. The front-back of the image is opposite to that of the object, as one would expect. The top-down of the image is same as that of the object, again as one would expect. However, the left-right of the image is not same as that of the object. This is the trouble that we are referring to in the problem. It’s a serious violation of symmetry between the two directions parallel to the mirror.
One must reflect for a while and convince themselves that above problem is same as the one we have in our mind when we say ‘why is a mirror image laterally inverted and not vertically inverted?’ for the case when the object is a human being (Let us take up the cases of the other objects later). Once formulated this way, it is immediately solved. The left-right is fundamentally different from the other two directions. The top-down and the front-back are defined through asymmetries in the appearance of the human body. While, it is impossible to define left-right just by appearance! They look perfectly alike. But still, we manage to unambiguously define left-right. How? We live in a 3 dimensional space-this is the answer. Any object with two directionalities defined, can be given a third directionality, arbitrarily, as a convention. Once this convention is set, it can be followed unambiguously, since we live in 3 dimensional space. This definition of the third directionality uses the two already defined directionalities. Mathematically bent people can, with a little reflection, convince themselves that this is indeed, ‘defining of the cross product‘. In fact, the ‘left-right = back-front X bottom-top, once the cross product is defined the way it is defined now. Now, why is the left-right reversed in the image? The image preserves the bottom top, but the back-front gets reversed, hence, the left-right, which is defined based on these two directionalities also gets reversed.
About the case of non-human objects, even if the object is perfectly assymmetrical, we attach our left-right to everything (we read and write from the ‘left’ etc) All those troubles are closely related to this. I don’t want to get in to those, since all of them involve just one thing: defining the problem properly.
Now the crux; as said earlier, the purpose of this blog was not to solve the mirror problem(:D). It was to illustrate that mathematics is simply thinking. Putting the problem in a clear language is what we call ‘formalism’.