Youtube presentation on the mirror puzzle

There is a delightful and very popular Youtube video (Physics Girl, presented by Meg Chetwood), demonstrating what seems to be the theory proposed by Richard Gregory, and emphasising the fact that humans have left-right symmetry but not top bottom symmetry.

The video is misleading in one important respect. What a mirror does (as opposed to how we describe what it does) is entirely independent of how the object reflected in the mirror has been presented. The actual transformation produced by a mirror depends only on the orientation of the mirror, though how we think about it may depend on how we describe ourselves, e.g. as having a top and a bottom, a left side and a right side, a front and a back.

One of the commentators referred to this image, which almost makes this point:

If the mirror is tilted at 45 degrees to a person's vertical axis it will reflect a vertical person as a horizontal one, neither parallel to nor perpendicular to the mirror.

Physics Girl's demonstration is excellent at debunking one straightforward source of confusion. Mirrors do not reverse side-to-side any more than they reverse up-and-down. But it doesn't go any way towards addressing a deeper puzzle about mirrors. This puzzle is independent of how the thing to be reflected in the mirror is presented (right way up, upside down, etc.) It concerns instead the actual transformation produced by a mirror. And solving it is not just a matter of debunking it; the solution turns out to require drawing both on facts about the nature of space and on facts about the environment that human beings involved in.

What exactly is the puzzle?

To help get a clearer view of the most interesting puzzle in the vicinity, we need to go back to basics. The most familiar formulation of the mirror puzzle is:

"Why do mirrors reverse left and right but not up and down?"

One possible way of interpreting this question - and this seems to be the way that both Richard Gregory and Meg Chetwood have interpreted it - is as follows:

"Why do mirrors reverse side-to-side but not up-and-down?"

The answer, as a little reflection (or the video from the previous section) clearly shows, is that the question is misconceived. Mirrors do not reverse side-to-side and it's a straightforward mistake to think that they do. If I stand facing a mirror with a banana on one side, the image of the banana in the mirror will be on the same side of the vertical plane down the centre of the mirror as the real banana is.

Thus interpreted, the mirror puzzle is not puzzling. Indeed, a new puzzle presents itself: why did people find the mirror puzzle puzzling? The answer to the new puzzle is not far to find. There are other possible interpretations of the mirror puzzle which lack any such straightforward answer. Here is one of them:

"Why do mirrors reverse left-handedness and right-handedness but not up and down?"

This question is not misconceived and is not resolved by the video. The phenomenon that it highlights is a familiar one. Text reflected in a mirror is hard to read: p becomes q, and vice versa. If I stand in front of the mirror holding a banana in my left hand, I will be presented with an image as of somebody holding a banana in their right hand. If I catch a glimpse of someone writing with their left hand in a mirror, without realizing I am watching them via a mirror, I will incorrectly judge them to be right-handed.

So why is it that mirrors reverse left-handedness and right-handedness but not up and down? To answer that question, we need to think a bit more about what transformation a mirror really performs.

What a mirror really does

Mirrors do not flip things around. They turn things inside out.

Perhaps the most general way to think about what the mirror does is this:

All the surfaces visible from the perspective of the mirror's reflective surface form a large hollow container with a very complex shape. In producing the reflection, the mirror turns that container inside out, something like a sock being pulled inside out, except that a real sock may have different colours and textures on the inside and the outside, whereas the imaginary sock turned inside out by the mirror has exactly the same visible features on the inside as on the outside, which becomes the inside after the reversal.

This explains why the reflection of a left hand is incongruent with the left hand, but congruent with the right hand. A left-hand glove turned inside out fits on a right hand, and vice versa.

The puzzle is resolved; mirrors flip left-handedness and right-handedness because they turn things inside-out. But this resolution immediately invites a new puzzle:

Why do mirrors flip left-handedness and right-handedness but not up-handedness and down-handedness?

There is an difference between the up-down direction and the side-to-side direction. The side-to-side direction is associated with the arbitrary labels 'left' and 'right' which allow us to distinguish what is flipped by a mirror. The up-and-down direction is associated with no such labels. The new puzzle is to explain the source of this difference. Restating it:

Why do we associate what is flipped by a mirror with the side-to-side direction rather than the up-and-down direction?

The solution to this new puzzle is as follows: our sides are so similar that we needed an arbitrary convention to label them; but no arbitrary convention is needed to label our top and bottom ends. Understanding why this is the correct solution will take a few more sections.

The parity transformation; or, the relation between left-handed and right-handed gloves

Before attacking the puzzle head-on, we need to say something a little more general about the transformation that results from turning inside-out, the transformation that relates left-handed and right-handed gloves, the transformation between the objects in the spatial scene presented by the mirror and the objects which cast those reflections. This transformation is known as parity, and the property it reverses is known as chirality. I'm related to my mirror image by the parity transformation; I and the image have opposite chirality. We are chiral opposites of one another, also called enantiomorphs.

Three-dimensional space (or at least regions of it that are accesible for us) constitute what mathematicians call an orientable surface. On an orientable surface, a consistent notion of clockwise rotation is definable in a continuous way over all points. This property of a three-dimensional space is provably equivalent to it being impossible to move a left glove around along any path (without turning it inside out) to bring it into perfect congruence with a right-handed glove. So, in our three-dimensional space the chirality of an object is preserved no matter how much it is moved around. In fact, chirality is preserved no matter how much an object is moved around or stretched or squeezed in any direction.

Some objects - those which have at least one axis of rotational symmetry - can be brought into congruence with their chiral opposites. A sphere of uniform colour, for example, looks just the same in a mirror. And a cylinder could also be brought into congruence with its mirror image. These objects are achiral. Most objects, though, do not have any axes of rotational symmetry. Even cylindrical or spherical objects, if their surface is marked with a design with no non-trivial rotational asymmetries (like a 'L') can be distinguished from their chiral opposites. They are chiral.

Now that we understand these notions, we can apply them to clearly restate and answer our remaining puzzle:

Why do we associate chirality with the side-to-side direction rather than the up-and-down direction?

The answer to this question is not about mirrors, or about the mathematics of chirality, or about anything so abstract. It's about the environment that humans evolved in, and the physiology and form of life that we adopted in response to this environment.

Chirality and human ecology

We did not need the concept of chirality to operate in the environments in which we evolved. However, we did need to be able to distinguish one of our sides from the other. For this, we developed arbitrary labels: 'left' and 'right'. Given the symmetries of a human body, this one arbitrary labelling then allowed us to describe all facts about relative orientation, including distinguishing between chiral opposites.

Hands are the most familiar pairs of chiral opposites; but of course feet, ears, and even eyes also qualify. As Hofstadter pointed out in the quotation above this also applies to sub-microscopic parts, such as a person's DNA. The reason these pairs of body parts are chiral opposites, of course, relates to our approximate plane of symmetry, which runs vertically down through our body and matches one side to the other. At least for most people, their external appearance viewed from different sides has much more in common than their external appearance viewed from the top vs. the bottom, or from the front vs. the back.

This gives us an easy way to distinguish between front and back, and top and bottom, which we lack with respect to side-to-side. Your front has the eyes on; the top has the hair on; the bottom has the feet on. If as an evolving human you want to tell a friend from which side of them the tiger is approaching, you can say 'face-side' or 'hair-side', or the equivalent in your developing language. But you can't say 'hand-side' - your friend has two hands, and will not know to which one of them you mean to refer. So arbitrary labels - left and right - will have been needed early on for practical purposes.

The side-to-side axis is where an arbitrary label is *useful*. If I want to tell someone in a 3d maze to go forwards, I just say 'forwards'. To make them go down a ladder, I just say 'down'. But if I want them to turn one way rather than another, I need to have some (arbitrary) labelling system. So here we need terms like 'left' and 'right'.

Supplementing the directional axes up/down, side/side, forward/backwards, an agreed distinction between left and right allows us to fully characterize the shapes of things including their chirality. Since one arbitrary choice is enough to achieve full expressive adequacy with respect to orientations, there is no need for any other contrastive pairs to be associated with any other axis. This resolves the last part of our puzzle.

How contingent is our understanding of the difference between left and right, and its relation to chirality? The following parable is an attempt to answer this question.

A spatial parable

Imagine some creatures that lack any distinguished sense of up and down, as opposed to side-to-side, and who move around freely in all three dimensions.

For us, horizontal orientation is not easily detectable, unless we have a compass or can see the sun as a reference point, or some well known land-mark that looks different from different directions. But we can easily detect which direction is up and which is down, using gravity (at least at normal locations on earth). Our imagined creatures lacking this ability, we may suppose, would have a clear sense of a distinction between the forwards and backwards directions (via their usual direction of motion?) , but there would be no reason for them to prefer any way of dividing things up between 'vertical' and 'horizontal'.

Our creatures would still need to describe the difference between a left glove and a right glove. They would do it not via an arbitrary choice of orientation in the side-by-side axis, since they have no such axis but via a choice of direction of roll: counter-clockwise as opposed to clockwise. A right-handed glove is then a clockwise glove: one on which the thumb and four fingers (in that order) run anti-clockwise when the hand is held out forwards, facing down.

When you were small, and learning how to tell the time on an analogue clock face, you learned the distinction between these two directions. (Perhaps you learned it earlier still.) In doing so, you probably remembered it via the hand moving from left to right as it goes over the top, then from right to left as it passes below. But the creatures in our less restricted environment have no privileged sense of 'above' or 'below' that distinguishes either term from 'to the side'; so they would have to remember it without reducing it to a combination of a stipulated left-right distinction and a known up-down distinction.

We leave you with the following thought (with apologies to Wittgenstein):

If a clock could talk, it would not say that mirrors flip left and right; it would say that mirrors flip clockwise and anticlockwise.