We raised this question for discussons among many small groups of people in our "Geometry and the Imagination" class. At one time I thought the 'correct' answer is that a mirror doesn't actually swap left-to-right, but rather, front-to-back. This is certainly true if one is only thinking about the geometric-optics transformation. Then for another while I thought the even more correct answer is that we're bilaterally symmetric. We have special-purpose brain circuits wired to identify people, and put ourselves in their place. Up to this identification, it's certainly true that mirrors reverse the left and right (of a person). We have the brains we have: it's a fallacy to think that we can think abstracted from all the special- purpose non-mathematical-seeming tricks our brains use, of which this is one. Ultimately, I mellowed. I realized that people were unconvinced by each other's explanations because they weren't actually confused about what a mirror does. There's not an actual question here without words. It's just confusing to find a good way to **talk about it**, and there is no best way---it depends on what you're trying to communicate to whom. Bill On Nov 25, 2007, at 8:28 PM, Gareth McCaughan wrote:

On Sunday 25 November 2007, Fred lunnon wrote:

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Would you believe that I must have asked that dozens of times over the years --- and as far as I can recall, that's the first time it's ever received a correct answer, from people with a technical background included?

How depressing. I'd assumed your reason for describing it as "a claim I fear I may have made previously" was that you took it as obvious that everyone here would have no trouble with it. (My answer would differ from Gene's only in that I'd have added that we think of what a mirror does as exchanging left with right because the things we look at in mirrors -- most notably ourselves, but plenty else besides -- are much nearer to being left-right symmetrical than to being top-bottom symmetrical.)

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It's an excellent example of how a badly-formulated linguistic trope may confuse its users so badly that they become incapable of disentangling its distinct meanings, even after those are explicitly pointed out. WFL

Yes. Though it's not only the linguistic trope that's at fault. (There's a *reason* why people commonly think that mirrors swap left and right, which is logically prior to that trope.)

Wow, I think Andy hit this one out of the ballpark. Very nice! Next question please! On Nov 26, 2007 7:26 AM, Andy Latto <andy.latto@gmail.com> wrote:

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Then for another while I thought the even more correct answer is that we're bilaterally symmetric. We have special-purpose brain circuits wired to identify people, and put ourselves in their place. Up to this identification, it's certainly true that mirrors reverse the left and right (of a person). We have the brains we have: it's a fallacy to think that we can think abstracted from all the special- purpose non-mathematical-seeming tricks our brains use, of which this is one.

I don't think that the special-purpose people-identification hardware is relevant here. Take any almost-bilaterally-symmetric object, and look at it in a mirror. It will seem to be reflected in its axis of symmetry.

I think that it's a different piece of special-purpose hardware that's relevant. We are programmed to "know" that objects translate and rotate, but they don't reflect. If we see two similar objects, and want to compare them, we mentally transform one into the other by translations and rotations, not by reflections, because we "know" that objects get routinely rotated and translated, but not reflected. So in trying to compare an object and its reflection, we look for an orientation-preserving isometry that makes the object look like its reflection. There isn't one, since the actual isometry is an orientation-reversing one. But there's one that's close, if the object is bilaterally symmetrical or nearly so. It's the one which, when composed with the reflection through the object's axis of (approximate) symmetry, produces the actual isometry. So that's the transformation we mentally perform, and we reach the conclusion that the object is reflected through its axis of symmetry or near-symmetry.

One way to see that this is what's going on is that no-one ever mentions the more salient differences between my view of you and my view of your image in the mirror. You are here, on this side of the mirror, and your mirror image is there, on the other side of the mirror. You are facing south, and your image is facing north. But objects translate and rotate all the time, so we are hard-wired to assume that what has happened is a translation or rotation, and to find that transformation. Because of our symmetry, we find a transformation that looks like it works, and the result is off by a reflection in the axis of symmetry. So we describe the effect as a "reflection in the axis of symmetry", by which we mean "some stuff too unimportant to mention, followed by a reflection in the axis of symmetry".

Take an object with no clear approximate axis of symmetry---say a tetrahedron with all 4 vertices different colors, and there will be no clear answer to whether the mirror switched the red vertex with the blue one or the green one, because there are several rotations you could apply first, and none seem better than the others.

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There's not an actual question here without words. It's just confusing to find a good way to **talk about it**, and there is no best way---it depends on what you're trying to communicate to whom.

That's why it's much easier to discuss this in a group like this one, where I can use terms like "orientation-preserving isometry" and expect to be understood.

-- Andy.Latto@pobox.com

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