[math-fun] Re: mirrors (was The Axiom of Choice for roots of z^2 + 1)
On 11/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
A mirror neither exchanges left with right nor up with down. It
exchanges front with back. This is precisely the definition of "reflection in a plane".
From: "Fred lunnon" <fred.lunnon@gmail.com>
[Not seeing that is] 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
From: Gareth McCaughan <gareth.mccaughan@pobox.com>
[...] 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. [...]
Or front-back.
(There's a *reason* why people commonly think that mirrors swap left and right, which is logically prior to that trope.)
I would say we code for the smallest edit distance, where "smallest" means most likely. So: the image in the mirror is rotated around the vertical axis and flipped left to right. It seems to me that the rotation happens unconsciously (pre-linguistically), and once it has, untangling the rest consciously is tough. Mirrored pairs are called "right-handed" and "left-handed"-- is that the "badly-formulated linguistic trope" you mean, Fred? Or is it the whole treatment of "swap" as a verb, or...?
From: Bill Thurston <wpt4@cornell.edu>
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.
Maybe we can't *access* the question without words(*), but I think I was actually confused when I first came across this one. The mirror obviously flips one axis and not the other, and just as obviously can't. There's a real question of how to make sense of it. Motor/perceptual problem: sequences of transformations and outcomes. Math problem: equivalent expressions. Language problem: in practical (as opposed to tact & felicity) situations, once you have one expression to identify a thing, that's usually good enough: you've identified the thing and the thing is what matters. I guess this shows that a practical problem can have the same structure as a felicity problem. (It's a matter of "perspicuity." Check out Craig Swanson's http://content.perspicuity.com/?q=taxonomy/term/487 .) --Steve * "...just ask the axis." --Jimi Hendrix
Here's a way to exlain the left-right swap to nontechnical folks: Mirrors don't swap left and right at all. Up in a mirror is up for you, down is down for you, left is to the left of you, right is to the right of you. You're just used to seeing people stand facing you, which involves them turning 180 degrees about a vertical axis. Don't believe me? Wear a shirt with words on it: the first letter will be next to your right hand, just as you see it in the mirror. Now let your friend wear it, standing next to you, then turning around to face you. As they turn around, they swap left and right (so that now your left hand is closest to their right hand), and you can read the words as you're used to.
When we contemplate our image in a mirror, biology is definitely involved. - Our Euclidean-space-adapted vision does not deal as well with visually distorted objects. For this reason, we tend to enounter planar mirrors. - Our gravity-adapted vision does not deal as well with objects that are not subject to the gravity we are experiencing. For this reason, we tend to encounter mirrors that preserve the up-down orientation of reflected objects. - When viewing a mirror, we have to face it fairly head-on in order to see our own image. I was going to make a point here, which escapes me now. - Our object-oriented brains perceive reflected images as real objects, subject to same spacial manipulations perceived to apply to natural physical objects. - When comparing two human beings, our face-oriented brains mentally move their images, primarily their faces, to the same location and upright orientation via virtual physical motions. When we do this with a person's natural and mirror images, the plane of reflection becomes the saggital plane of the identified faces, hence we perceive left-right reflection. If we did not perform this mental relocation, we would presumably be more cognizent of the actual front-back reflection in the plane of the mirror.
On Tue, 27 Nov 2007, David Wilson wrote:
When we contemplate our image in a mirror, biology is definitely involved.
- Our Euclidean-space-adapted vision does not deal as well with visually distorted objects. For this reason, we tend to enounter planar mirrors.
- Our gravity-adapted vision does not deal as well with objects that are not subject to the gravity we are experiencing. For this reason, we tend to encounter mirrors that preserve the up-down orientation of reflected objects.
Are you suggesting that there are planar mirrors which would appear to flip images top to bottom instead of appearing to flip them left to right?
On Nov 27, 2007 10:41 AM, Jason <jason@lunkwill.org> wrote:
Are you suggesting that there are planar mirrors which would appear to flip images top to bottom instead of appearing to flip them left to right?
Sure! A mirror on the floor or ceiling, for instance. -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike
participants (4)
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David Wilson -
Jason -
Mike Stay -
Steve Witham