The other mirror test you will probably fail

Why do mirrors appear to swap our left hands for our right hands but not our heads for our toes?

You probably have a preferred answer. If not I recommend you spend some time thinking about it before scrolling down.

Lewis Carroll (the author of Alice in Wonderland, and Through the Looking Glass) is reported to have enjoyed asking people this riddle. The challenge is to come up with a fully satisfying explanation.

There are many famous approaches, including by serious philosophers and writers I admire a lot. All of the ones I've read still leave me unsatisfied. Part of the reason is that the explanations offered didn't feel like they got me any closer to answering the following follow up questions:

• Why is "left/right" swapping the natural direction we imagine our reflections being swapped in, even though they're actually only being flipped front to back?
• Is it possible to construct a scenario where an ordinary mirror does appear to be swapping in the up/down direction and not the left/right direction?

Note - putting a mirror on the ceiling is not an answer to the second question! Sure a mirror on the ceiling appears to swap your head for your toes, but it still also appears to swap your left hand for your right hand. If you don't believe me this is another point at which its worth spending a few minutes thinking it through.

I dislike blog posts on topics like this that tease their answers for too long so, before continuing, I will just sketch my answer below now before going on to a full explanation. Meet "Phlipper", my imaginary friend.

It's ambiguous whether he perceives mirrors to swap in the up/down or the left right directions. That is, until I tell you that this is how he walks.

I am not a particularly talented artist, but maybe these sketches are enough for you to get the idea of how I think you should answer the above questions, and why your default response may not have been sufficient. Otherwise, I've written up a full explanation below.

Full explanation

First, its worth going through a few other explanations and why they aren't sufficient.

One answer I've discussed already is that mirrors swap left for right but not up for down because of where they are placed relative to us. As I mentioned above, this is not satisfying when you notice that wherever you place a mirror, or place yourself relative to a mirror (e.g. lie on your side, stand shoulder on, stand on your head), the person reflected will still always appear to be a version of you but with your right and left hands swapped. If you wiggle your right hand, from their perspective, they will be wiggling their left hand.

Another approach is to claim that mirrors don't swap left and right or up and down at all, they only swap front and back. They effectively perform the same function as one of these toys.

They actually make full-size versions of these. If you imagine pressing yourself into one of these full size toys and stepping back, the imprint you see is the result of inverting yourself through the plane of the "mirror". So neither up and down nor left and right have been swapped, only front and back. Additionally, if you walk around the toy you'll see that from the other side it looks like nothing has been swapped at all?

The reason why left and right hands appear swapped after performing this "imprinting" function is because left and right hands are chiral. That is, any inversion of a left hand on any axis appears like a right hand.

There is nothing factually inaccurate about this answer, but there is some sense that the true explanation is just being kicked down the line. The question remains: when looking at our imprints or mirror images - why can't we shake the feeling that the inversion has taken place along one particular axis?

What this explanation does help illustrate is that whatever is going on is some kind of confusion that is taking place in our minds, and has nothing to do with the physics or optics of mirrors. However what it doesn't answer is why our minds have this particular confusion and not any other.

A full explanation, in my opinion, should provide us with a way to fully rid ourselves of this lingering confusion, or at the very least to imagine a universe in which we had the confusion in the opposite direction... where we had some strong unshakeable intuition that mirrors flipped us head for toe.

One answer that comes pretty close to providing such an answer, and which was sent my way by Ben Pace while I was drafting this blog post, appears in the book Good & Real by Gary Drescher ¹.

Drescher's explanation goes like this:

The mirror treats only the perpendicular dimension differently from the others—the surface facing the mirror faces north, but its reflection faces south.

If we insist on construing the reflection of the front as still the front, and the reflection of the top as still the top, then it follows that the reflection’s right and left are swapped.

But if instead we identify front with front and left with left, then it fol- lows that the reflection’s top and bottom are swapped instead. Thus, it is not the mirror itself, but rather our choice of which of those identifications to emphasize, that determines whether there seems to be horizontal or vertical swapping.

Presumably, people’s approximate left–right symmetry—in contrast with our conspicuous top–bottom and front–back asymmetry—induces a preference for preserving the identity of the top and front rather than the identity of the left or right. After all, it is much easier to construe the reflection of a right hand as a transformed left hand than to construe the reflection of a head as a transformed pair of feet.

I think is account is correct, and that our vertical axis of symmetry as humans is part of the answer. But note how Drescher is hesitant to make an unqualified statement that this is the whole explanation. I think that's because there is still missing step in this account, which is to explain why a human's vertical axis of symmetry leads them to have this particular confusion.

To explain why such an explanation is required, lets go back to the first half of the sketch I introduced at the start of this post:

Notice that the way I have drawn this creature, it actually has two axes of symmetry, both on the horizontal, and on the vertical axis.

Now imagine it looks in a mirror and wiggles it's "left" hand. Which hand does its reflection wiggle?

There are now two possible answers, and they both depend on how the creature translates the front-back mirror inversion into natural terms. If the creature imagines itself "hopping" around but keeping the same foot pointing up when it started, it will perceive its left hand as being swapped for its right hand and so its reflection is wiggling its "right" hand. On the other hand, if its preferred way of moving around consists of backflips:

It will perceive he same original "right" hand wiggling, but its top end will have been swapped for its bottom end.

I used some LLM deep research tools to see if anyone had written about this and it found a few papers I hadn't seen which give a similar explanation but still don't say what it would take for a creature to have the opposite intuition.

Personally in order to arrive at this answer myself, I first had to ask that question, which led me to doodling a version of the images above. So I figured they might be useful for others if I share them.

Of all the papers I did find, the nicest write up of this approach, along with a review of work by other writers appears in this paper. It calls the approach above the "locomotion hypothesis", namely that the canonical way we imagine ourselves walking around a mirror is core to how we perceive what kind of swap has taken place.

This hypothesis makes sense to me, because in my drawing above changing the way the creature moves changes what it perceives the mirror to be swapping.

I ran some polls to test people's intuitions about this, these were the results.

However, when I spot checked people who answered that they did have a satisfying answer and an example of where an ordinary mirror does appear to swap in the head/toe direction, but not in the left/right direction, they were still imagining a mirror placed either on the ceiling or the floor.

Now, there isn't necessarily a "canonical" locomotion for rotating yourself to look at someone hanging from the ceiling... so I wonder if it might be possible there is some way that you can shift your gestalt so that your reflections right hand stays the same. Some people I have spoken to seem fairly convinced this is the case, though I'm not so sure, partially because the case of humans is overdetermined: our interpretation is forced both by the symmetry of the human body (specifically the chirality of left and right) and by our preferred means of locomotion.

Some of the people who participated in these polls also did my shape rotation study, but unfortunately it is far too few to draw any particularly interesting correlation. I wonder if there is some relation between people's visual imaginations and their intuitions about this.

Side quests / follow ups:

Clockland?

Deep research threw up one other paper that seemed to have found a similar approach, though to answer a subtly different question. There is actually a whole in depth correspondence on the question which I've yet to read. Unfortunately all the images linked to in these writeups are hosted on an old and non-responsive FTP server. I'm now on a mission to track down the original images.

When talking to other inkhaven residents about this puzzle, Signore Galilei suggested that a "swirly starfish" creature looking in the mirror would not perceive anything as swapped with anything. In fact, it would be forced to see its mirror image as a "true reflection" rather than any kind of rotation, because it has no axes of symmetry:

This gave me a guess for what the inhabitants of clockland might look like - they'd be creatures whose faces look like clocks.

Similarly because the numberings around clock faces mean they have no axes of symmetry this would also mean that they don't have any kind of "canonical" way to perceive themselves as being rotated on to their reflections.

A mirror bestiary

Another mission I have is to draw more creatures with properties like these, and/or encourage some people who are better at art to have a go. I'd personally love to see some more examples along these lines.

Perception and cognitive science

There are a few philosophers and cognitive scientists who make a big deal out of these kinds of effects. I also think these effects are kind of a big deal, because they relate to the ways in which certain elements of our perception are so baked in to our conceptual scheme that they can go unnoticed even under very deep scrutiny. I'd love to understand some more of the literature about this!

Challenges to the locomotion hypothesis

According to this paper some writers reject the locomotion hypothesis. I'm curious what their arguments are, and whether they can make sense of what is going on for a creature like this.

The case of writing

Finally, a question I'm still a bit puzzled by, is it possible to design a script, so that text written in that script appears to be inverted in the up/down direction but not the the left/right direction?