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SCIENCE ARTICLE - The Mirror Paradox
WHY ARE SO MANY CONFUSED BY THIS SIMPLE TRICK QUESTION?
by Robert Rankin

The paradox concerns the often-stated question Why does a plain flat mirror reverse an image left to right but not top to bottom? The confusion derives from the fact that the question is built on an incorrect assumption: that mirrors reverse images left to right . They don't, so the question itself is non-sensical. Let me explain.

The question implies that a reflection is identical to a rotation. This is not the case but when we look into a mirror we are fooled into thinking the mirror has turned our image around. Our image in the mirror looks like we simply have to rotate ourselves left to right in order to superimpose ourselves on it. But this does not work. It looks like it works because of our symmetry. Put a red dot on your right arm and a blue dot on your left arm to make them different. Now, in your mind, rotate yourself to match the mirror image. Sorry, it cannot be done. The blue dot and the red dot are now superimposed, meaning the being you are seeing in the reflection is not really you. If your hair has a part, it will be on the wrong side of your head. This is easier to see if you use a non-symmetrical object. So the left arm always stays on the left and the right arm stays on the right. Likewise, the head always stays at the top and the feet at the bottom. The mirror makes no distinction between up and down and left and right, so contradicting the initial question.

An easy way to show that a mirror does not reverse anything is to stand in front of a mirror, hold a clear sheet of perspex in front of you, between you and the mirror, and, using a felt pen, write a word on the clear sheet. Notice that the image in the mirror is not reversed and easily read. What confuses this demonstration is that often a word is written on an opaque piece of paper which is then held up in front of the mirror. Of course the image of this is reversed because you have turned the paper around. In this case, if you could look through the paper towards the mirror you would see the word written back-to-front on the paper.

This is really the end of the story but words often only go so far in explaining physical concepts and other approaches are often more revealing. The issue is better shown geometrically by drawing a simple ray diagram as shown in the Figure 1. A ray diagram is constructed using two rays from one point on the arrow object to find the position of that point in the image. The usual rays taken are one parallel to the axis of the mirror and the other a reflection from the centre of the mirror, but other rays can be used. The ray parallel to the axis strikes the mirror perpendicularly and so is bounced back on itself. The other ray bounces symmetrically off the mirror surface with the angle of reflection equal to the angle of incidence. As can be seen from the diagram, these rays diverge, so the image is said to be virtual. The image cannot be projected onto a screen but the rays can be converged to a point again by passing through the convex lens of the eye and onto the eye's screen known as the retina. When the virtual rays approach the eye they appear to have come from a point behind the mirror, hence the dotted virtual extensions to the real rays. The two virtual rays meet behind the mirror at a point the same distance behind the mirror as the object is in front so producing a same-size, upright and virtual image. You can confirm the image is upright by drawing rays from the tail fins of the arrow object using the same rules for rays used above for the arrow tip.

So if mirrors do not reverse images laterally, why is the word FIRE written backwards on the front of fire engines? Simple. Look at the ray diagram. Move your eye into the position shown in Figure 2 where you are seated in the driver's seat of a car with the rear view mirror in front of you and the fire engine behind you. Now Imagine you are driving a transparent fire engine and seated behind the FIRE sign attached to the front of your vehicle. From your position in the fire engine cabin, the word FIRE is not reversed. Think back to the situation explained above using a transparent perspex sheet. Back in your car, if you rotate and look back towards the fire engine the word FIRE will be reversed. No mirrors needed for this to happen.
By moving the eye around inside the ray diagram, in front and behind the object, the whole situation will become clearer. The ray diagrams only show the arrow standing vertically but to make this discussion more understandable it is a good idea to turn the ray diagram through 90 degrees anti-clockwise so that the arrow is lying down. Then, the tip of the object arrow is where the F of FIRE is and the tail fins of the arrow are where the E of FIRE would be.

So, back to the original question. The problem with it is that the question is imposing on the mirror a property it does not possess.

If you would like to comment on this explanation and suggest changes or improvements to the explanation, please email me at  info@rankin.com.au

Copyright Robert Rankin, 2014