Saturday, November 26, 2011

Why are left and right reversed when you look in a mirror, but up and down are not?

I encountered this brain-teaser on another blog quite some time ago -- wish I could say which one, but I no longer recall -- and it turns out to be harder to articulate the answer than one might think. I puzzled through it at the time, and then I remembered it again last night and had to re-figure it. Thought I might record my thoughts on it this time around.

One reason I don't remember the original blogger is that he or she simply left it at, "Nobody can really quite say!", which is simply not true: it's just very difficult to say it, but one can indeed figure it out. Try it yourself before you read on, if only to realize that it's not as trivial as one might expect!

Perhaps unsurprisingly, 95% of the answer stems directly from how left and right are defined, and if you want to skip to that, click here. Before that, however, I think it's useful to consider some directions which are not reversed in counter-intuitive ways.

Imagine a woman facing north and looking dead-on into a mirror at eye-level. She is wearing a ring on her "left" finger -- but this is the last time I will use the words "left" and "right" until we come to the answer, so let's say instead that she is wearing a ring on her west finger. Her eyes are facing north, as previously stated, and her head is pointing up.

For the woman in the mirror, all three axes -- north/south, east/west, up/down -- behave exactly as expected for a reflection in the plane of the mirror. Mirror woman's ring is still on her west finger, her head is still pointing up, but now her eyes are facing south. No surprises here, right?

This is because north, south, east, west, up, and down all have definitions which can be expressed independent of both a) any other direction, and b) the content of what you see. (There's actually a couple different ways of thinking about this for up/down, but for an earthbound mirror at eye-level the ambiguities don't come into play, so let's ignore them for now.) The compass points are defined by the Earth's magnetic field -- or by geographical convention if you prefer, but the result is the same -- and up and down are defined in relation to Earth's gravitational field. (Probably.)

In contrast, left and right are not well-behaved in this way; they are defined in relation to two other directions, namely top and front. Uh oh, and there's two new directions, "front" and "top". These are not well-behaved either; they are defined in relation to the content of the scene. For now, however, I am going to treat "top" as synonymous with "up", for simplicity, and then re-examine that assumption later. So when you add it all up:

"Left" is defined as the cross-product of "up/top" and "front", while "front" is defined by visual cues within the scene, such as the direction a person's eyes are pointing. (I think I got my signs right there, but don't beat me up if the correct cross-product is the other way; you get my point!) In our example of a woman facing north straight-on into a mirror, "up" stays constant, because it is (probably) defined in relation to gravity, while "front" reverses compass directions. The real woman's front is north, while the mirror woman's front is south -- we know this because "front" is defined in this case by the side of her body where her eyes are.

Interestingly, if one could make one's brain do the mental contortions, it ought to be possible to visualize it so that the woman's front and back have been reversed, i.e. that her eyes are now on the "back" of her head, her toes point "backwards", her back is now on her "front", etc., and if you could do that, then I believe one would also see that her left/right have not been reversed. The ass-backwards mirror woman still has her ring on her left hand. However, I find this impossible to fully conceptualize; I get about halfway and my mental image ends up looking like the teleporter scene from Spaceballs. It's worth trying this exercise anyway, if only because it helps illuminate why it is so difficult to articulate the answer to the titular question of this blog post. Our assumptions about the definition of left/right and front/back are so ingrained that even when we name the assumptions and express a conscious intention to discard them, we are still stuck with them. We can't help it.

Now imagine the woman turns 90-degrees to her left, so that she is facing west (but the mirror is still north of her). This is actually an easier case. Now we see that front and back are still the same -- west and east respectively -- but left and right have been reversed because they are directly perpendicular to the plane of reflection. It's no surprise that left and right are reversed in this case; how could they not be!

The math gets slightly more complicated if the woman is facing into the mirror at an angle (but still at eye-level, for now), but it ends up working out the same. In these scenarios, rather than one relevant axis staying the same and the other being reversed, both axes are partially transformed. It all works out though: Left/right will be transformed directly by reflection over the plane of the mirror, and indirectly via the same transform of front/back, and together it winds up with left/right being completely reversed.

I've been hand-waving so far on up/down and top/bottom, because they are the same as long as the mirror is at eye level. But what if the mirror is directly above the woman's head, and she is looking straight up at it? This case works out as well, although now it is important the we distinguish "top" as being something defined by the content of the scene, and we will also see that the definition of "up" is a bit slippery and people might have different ways of looking at it -- especially once you moved away from the Earth into space!

I also want to look at some cases where maybe left and right are not reversed in a mirror, because there are no cues to give you front/back or top/bottom. Alas, my sons really need some attention right now, so this will all have to wait for a follow-up post.

3 comments:

  1. Shorter: A reflexive surface reverses axes perpendicular to the surface.

    In fact, mirrors do reverse "up/down". When the above is true for "up/down". Light plays Pong.

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  2. I don't think that answers the question at all!

    Think of it this way: If I look straight on into a mirror, forwards and backwards are reversed because of what you said, but left and right are also reversed, even though that axis is parallel to the surface. If I lay on my side so that now left and right are up and down, up and down are the same in the mirror but left and right are not.

    There's no mystery there, of course, but articulating why that is the case is not simple, and I think your "shorter" fails to do so.

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  3. You think in body parts, no? Light doesn't. Each speck of light from your left hand separately goes forward, is affected by the mirror, and bounces back. Same thing happens at your right side.

    What would be the "right hand" of another person standing opposite you - the hand that is to your own left side - is not a reflection of your own "right hand" when it's a mirror.
    The left side of the mirror actually reflects your left hand, which appears like a right hand because it's reversed.

    Certain concave mirrors may reflect your left hand on the right side, making the reflection more like another person would be. By tha same mechanism, certain mirrors may display your feet above your head.

    You are right that the language is a hurdle.

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