Reflection: A transformation under which each point in a shape appears at an equal distance on the opposite side of a given line—the line of reflection.

It’s not often I have occasion to shout at the television, but a recent episode of the BBC’s long-running television series QI precipitated just such an outburst. The cause of my vexation was their answer to the question that forms the title of this post. The offending episode was the R Series: Reflections, and the explanation was an excellent approximation to gibberish, involving as it did some business about “The mirror doesn’t flip things around; we flip things around,” intoned by Sandi Toksvig as she stood in front of a mirror fiddled with a bit of card with the word BOSS written on it (see above). To be fair to Toksvig, it probably wasn’t her idea, and she did manage to deliver the entire farrago while wearing the sort of anxious expression people wear when they’re not entirely convinced by their own argument.

The answer to the question is really that it is ill-posed. Mirrors actually don’t reverse left and right, for the simple reason that mirrors have no way of telling left from right. They have no left-right asymmetry, in other words. The only asymmetry they do possess is in the plane of reflection—stuff in front of the mirror is the real world; stuff “behind” the mirror is the reflected world.

That’s what’s being described in the definition at the head of this post, which refers to a two-dimensional reflection, like this:

Here we have a “line of reflection”, corresponding to the mirror; a letter “B” in front of the mirror, representing the real world; and a reflected letter “B” behind the mirror. Every point on the reflected “B” is the same distance from the mirror as the corresponding point on the original “B”. So because the spine of the “B” is the farthest part from the mirror, its reflection also lies farthest from the mirror. Conversely, the curved parts of the letter lie closer to the mirror on both sides. And it’s that preservation of “near” and “far” on either side of the mirror plane which causes the reflection to be a reversed image of the original. If we travel from one side of the mirror to the other, we encounter in turn spine-curves-mirror-curves-spine. So, actually (and pace Toksvig), the mirror very much does “flip things around”. Indeed, many introductory geometry texts gloss the word “reflection” as “a flip”.

The same thing happens when a three-dimensional person stands in front of a real mirror:

The reflected image’s left and right (and head and feet) are pointing in the same direction as the real person’s. What has been flipped in the mirror image is the direction in which the nose and toes are pointing. So the mirror has reversed front and back, not left and right. If you lie down with your feet pointing at the mirror, the reflection will also have its feet pointing at the mirror—so on this occasion the mirror leaves your front and back, and left and right, in the same positions, but reverses you top to bottom. Only if you stand sideways on to the mirror does it truly reverse your left and right—but that’s because you’ve chosen to place one side of your body close to the mirror and the other far from it, not because the mirror has some magical ability to tell left from right.

So why do we always think the mirror image has reversed left and right, no matter how we orientate ourselves before the mirror? The answer, I think, lies in the single plane of symmetry in our own bodies. Our fronts are very different from our backs, our heads are very different from our feet, but our left side is very similar to our right side. So it’s very difficult for us to see the mirror reflection as having reversed front and back—instead, we see it as another person who has turned around to face us. In which case, their right hand now moves when we move our left hand, and vice versa. No matter what the orientation of the reflection, we always interpret it as a left-right reversed person, because a head-foot reversed person or a front-back reversed person is harder to conceptualize.

Remove the left-right symmetry, and we stop talking about mirrors reversing left and right.

This barber’s pole lacks a clear left-right distinction. So instead, we find ourselves saying that it “spirals the opposite way”. We’re still, apparently, unable to discern that what has been reversed is the front and back, but now we can’t blame a left-right switch either.

So if anyone ever asks you the title question, permit yourself the slightest of headshakes and the faintest of smiles, and say: “But mirrors don’t reverse left and right; they preserve near and far.”