Monday, 27 January 2014

… Who is The Fairest of Them All?

I left you with the conundrum of why a reflection in the mirror is reflected horizontally and not vertically.  This is particularly puzzling when you consider that if you lie down in front of the mirror, the direction which was left-right when standing up becomes up-down when lying down, and for some bizarre reason is not inverted when lying down.  And the up-down, which is not inverted when standing up, becomes the left-right when lying down and is inverted again!  What the..?….

Suppose you and your identical twin stand face to face.  If I ask both of you to point to your left, each will point to a different side.  If I asked you both to point forwards, you would point at each other, again in opposite directions.  On the other hand, if I had asked both of you to point West, or North, you would both point in the same direction.  By stipulating Left/Right and Forward/Backward, I am stipulating a frame of reference which is different for both you and your twin.  Does this help?  No, not really.  But I will come back to it.

Time to consider the optics of the problem.  What we see in the mirror is what we call a “Virtual Image”.  This means that the light rays that reach our eyes appear to have come from a particular place - behind the mirror - whereas in fact they did not.  They reflected off the mirror.  Our eyes (in fact anything in the world that detects light) can only detect the fact that light has impinged upon them.  They cannot tell the direction from which the light came, and they certainly cannot discern the path it took along the way, like we can with a tennis ball for example.  We can only infer these things.  So when we look into a mirror, we see not light rays bouncing off its surface, but an entirely false “Virtual” image of ourselves standing behind it.  This is very convenient when it comes to combing our hair.

In fact what you see in the mirror is an entire “Virtual World”.  A reflection of You and everything else around you.  It is this Virtual World which has the apparent property of being inverted horizontally and not vertically.  In this case, to understand the image of the Virtual World, we need to start with a good old-fashioned photographic slide, the kind you can put in a projector, or hold up to the light and squint at.  Those of you old enough to have loaded a stack of slides into a projector will know that (apart from getting them the right way up) there is a right and a wrong orientation.  If you get it wrong, the slide will come out as a mirror image.  The “Virtual World” in the mirror is just like that old picture slide.  The real image is oriented a certain way, but the virtual image is oriented differently.  In order to see the virtual image, we simply flip the picture slide over and look at it from the other side.

If we flip the picture slide vertically, we can see that what we have is a top-to-bottom mirror-image of the original.  Left is on the left, and right is on the right.  Now, by the simple expedient of rotating the image 180 degrees clockwise to bring the bottom of the image to the top, we find that the resultant image is now a left-to-right mirror-image of the original.  You really need to try this for yourself sometime.  And that is the hidden truth in all this.  We have drawn back the curtain and revealed Professor Oz in all his glory.  The same image is simultaneously a top-to-bottom OR a left-to-right mirror image of the original, depending only on how you look at it.  We call this an image with "inverted parity".

Looking at a Virtual World in the mirror is the exact same thing.  The Virtual You in the mirror lives in a Virtual World, but, like the picture slide, we are in effect seeing it from “behind”, or in "inverted parity", and therefore flipped side-to-side.  Or top-to bottom, if we just lie down.

Apart from being able to comb your hair, this has consequences that you may not have thought of.  How many of you know someone - usually a wife - who complains about how she never looks good in pictures?  I know mine does.  As a result, their appearances in the family album are inevitably few and far between.  Most people have an asymmetric face.  The left half differs from the right half.  Occasionally the differences are subtle, but usually they are quite marked.  As an individual, for the most part, the image you recognize as being that of your face, is the one you see looking back at you from the mirror.  It is, of course, a “mirror image” of your real face.  Since your facial features are asymmetric, it is a different image from how everyone else in the world sees your face.  Everyone else only sees your face as it really is.  You only see the mirror image.  So when you see a photograph of yourself, suddenly your see it as everyone else sees it, but this is not how you are used to seeing yourself.  It is a mirror-image of what you have become accustomed to believing you look like.  And, quite often, it looks plain wrong.  As I write this, I suddenly recall a vivid memory of being a small child, and wondering why my mother always pulled an odd face whenever she looked in the mirror!

Here is a trick to try on the “non-photogenic” wife.  Print out a picture of her which has been flipped horizontally, but don’t tell what you’re doing, and see what she thinks!

I need to put this post to bed with a further observation.  The “mirror-image” nature of the image in the mirror is the consequence of light being reflected off the surface of the mirror.  Mathematically, the reflection off the mirror’s surface flips the “parity” of the virtual image.  This parity has two states, “normal” and “inverted”.  By reflecting off the mirror’s surface, the parity of the virtual image flips to the “inverted” state.  Suppose we then reflect the light off a second surface.  This should flip the parity of the image back to “normal” again.  How might that work?  Here’s how we can test this.  We take two mirrors and join them together at exactly 90 degrees.  Hopefully, where the two mirrors meet will be as smooth a join as possible.  We now stand in front of the pair of mirrors, and gaze directly into the “Vee” of the join.  Light reflecting off the mirror will actually have two reflections, once off each surface of the compound mirror.  What will we see?  Guesses, anyone?…

Once again, we see ourselves in the mirror.  Except this time the image is not a “mirror-image”.  If I stick my arm out and point left, the fellow in the mirror sticks his opposite arm out and points right.  Indeed, he points to HIS left.  If I hold up a newspaper, the fellow in the mirror also holds up a newspaper and all the newsprint on it is normally aligned and perfectly readable.  This is just like where we started off, with me and my identical twin standing face-to-face.  The only annoying thing is that there is a line that insists on interrupting the image and it insists on going right between the eyes of the fellow in the mirror.  It is, of course, the artifact of the less-than-perfect joint between the two mirrors.  What you see in this mirror is your face exactly as it looks to other people (OK, with an annoying line down the middle).  Or how it looks in photographs.  This is an example of a non-inverting mirror.  You can often get this effect in the elevators of high-end olde-worlde European hotels. 

So why are non-inverting mirrors not at all popular?  Well, you might want to try combing your hair in one….