Quibbles and Bits

Of Mirrors, and Combs

Issue 43

Mirror, mirror, on the wall...

Here's a quick challenge for you.  When you look at something in a mirror, why is the image always reversed left-to-right, and never top-to-bottom?  Think about that for a minute.  Always left-to-right.  But never top-to-bottom.

Now lie down on the floor in front of the mirror, with the mirror at your side.  Now the image is reversed top-to-bottom, but not left-to right.

Go read some of the other columns now, and come back to this when you think you have a watertight explanation.  And then read the answer below ….

… Who is The Fairest of Them All?

Suppose you and your identical twin stand face to face.  If I ask both of you to point 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.

Consider, as someone once said to Bill Clinton, 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 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.  Which is very convenient when it comes to combing your 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.  To understand the image of the Virtual World, we need 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 getting them the right way up is easy – but getting them the right way round is a pain in the [somewhere inconvenient].  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 photographic 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 instead of flipping it horizontally we flip the photographic slide vertically, what we will have is a top-to-bottom mirror-image of the original.  Left is on the left, and right is on the right, but top and bottom are inverted.  Now, by the simple expedient of rotating the image 180 degrees clockwise – not flipping it – thus bringing the bottom of the image to the top, we find that the resultant image is now the original left-to-right mirror-image.  You really need to try this for yourself sometime.  And that is the key observation here … viewed from the wrong side the same image can be a top-to-bottom OR a left-to-right mirror image of the original, depending only on how you look at it.  But not both.  And not neither.  We call this an image with “inverted parity”.

Looking at the Virtual World in the mirror is the exact same thing.  The Virtual You in the mirror lives in its 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 – perhaps 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 more often 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 looked slightly odd whenever I saw her in the mirror!

 

 

 

Here are some photographs of a selection of well-known faces.  Three of them are mirror images.  Can you guess which they are?  It hints at 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 it 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?

Once again, we will 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.  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 will often encounter this effect in the elevators of high-end olde-worlde European hotels.

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

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