Dawkins's Mirror Question

A really good book is one that makes you want to put it down for a while so that you can think.

This is taken from Richard Dawkins's latest book The Oxford Book of Modern Science Writing. See the review in Science (Mail me if you need access).

For years as a college tutor at Oxford, I would try the intelligence and reasoning powers of entrance candidates by asking them at interview to muse aloud on the conundrum of why mirror images appear left-right reversed but not upside down. It is a provocative puzzle, which is hard to situate among academic disciplines. Is it a question in psychology, in physics, in philosophy, in geometry, or just commonsense? I wasn't necessarily expecting my candidates to "know the right answer." I wanted to hear them think aloud, wanted to see if the question piqued their interest and their curiosity. If it did, they would probably be fun to teach.

The book is maybe good bedtime reading, but not the paragraph above. I read the paragraph and I couldn't sleep till had some sort of satisfactory answer. It's rather intriguing and wanted to discuss this below. My friends in Mathematics have a better idea of this, but I am taking a stab at it here.


It is about particular symmetry? That lead me away from thinking about mirrors to drawing mirror images and considering that we reflect images about a particular axis. This is a practical issue when it comes to making stamps, or printing for example.

(Pardon my sloppy drawing. It's much harder to draw accurate mirror images than is popularly believed.)
Upon drawing this simple figure, I realised that mirroring (reflection) is a not really a 2D transform, but is a 3D transform since there is no strictly planar or 2-D operation(rotate, shift) can be performed that convert A to B. Of course if you reverse the X-axis then you automatically reflect. It's like looking at the 2-D figure from the bottom instead of the top.

Thinking about it in another way, if you had simply an outline figure (shown on top), you wouldn't really see any lateral inversion. If viewed from one perspective, there is no lateral inversion. That's because the figure being hollow does not have a correct viewing side.

So, according to me it boils down to a frame of reference. By imposing your frame of reference onto the mirrored image you get the perception of lateral inversion. In that parallel universe of mirror images, the X axis is reversed and hence correctly speaking left is actually right! The absolute values of all the points along X have not changed, only the signs have.

Coming back to the other question of top and bottom inversion, the simple four figure shows that top bottom is achieved by rotation around the Z-axis, or by two repeated mirrorings on the Y and X axis. So, you can ask why rotation only changes top and bottom, but not left and right. They are simply two different transformations.

More generally speaking, there is really left or right, or top and bottom. It's all relative all perceptions are due to the frame of reference. So, if you perform a transformation then you have to choose the appropriate frame of reference or grid to view it correctly. In a more extreme case, even in case of a distortion, the square will be a square in the distorted frame, but appears warped only because you impose your original frame on the 'new' square.

So, I think the problem is one of pure geometry and not of psychology or even physics. Perhaps, humans are predisposed to think of particular frames of references which cause this effect.


Anonymous said...

Interesting post. Another explanation: Imagine you are looking at another person. Assume that +ve x is right and +ve y is up. Then, your +ive x is the -ive x of the other person, but you share the same +ive y. Looking at a reflection is really like looking at another person - hence the lateral inversion but not the up-down inversion. If you were standing on top a mirror, then you would see both types of inversion - this is because if you imagined another person standing upside down in place of your reflection, then her +y is the same as your -y and her +x is the same as your -x. Hence, inversion in both perspectives.

Hirak said...

Great! I like your explanation better. I had the -ve x and +ve x figured out.
But, the top of the mirror was really a clever way of thinking about the problem which reveals the false perception of axes.

Then, you realise that in fact there is bottom-top inversion at all times. Only we don't perceive it as such because the -ve y axis folds up.

To add another example:
A reflection in the water is a 45 degrees in which there is left-right inversion and a top-to-side inversion.

By a consistent interchange from one system of coordinates to another, you can achieve rotation and reflection.

Ganesh Hegde said...

Perhaps I don't understand the problem correctly, but Snell's law of reflection should be the obvious answer.
You 'see' only those rays that reach you on reflection, the ones that don't, need to perceived from elsewhere, much like the quintessential mirror shot in hindi movies, where the audience gets to see the person standing in front of the camera, but the actor himself/herself, doesnt.
The inverse image will still form, but you need to be in another position to view it.

Hirak said...


I don't know how Snell's Law would explain it.

But, you are right it's a matter or perspective, or technically, coordinate systems in use.