More on Mirror Reversal
The following is an email exchange I had with someone who was dissatisfied with my mirror reversal explanation. Out of courtesy to him I have removed his email address.
Subject: Mirror question
Date: Thu, 15 Apr 1999 22:39:58 -0400
From: John C.
To: lystad@iglobal.net
Your answer to the mirror puzzle is quite insufficient. What is written
reveals an acute misunderstanding of the question.
http://www.iglobal.net/lystad/curiosity-shop/mirrors.html
fyi
Subject: Re: Mirror question
Date: Sun, 18 Apr 1999 06:42:55 +0000
From: Garr Lystad
To: John C
In that case I'll try to do better. Do you have any suggestions?
Garr
Subject: RE: Mirror question
Date: Sun, 18 Apr 1999 15:13:57 -0400
From: John C
To: "'Garr Lystad'"
I believe in the Socratic method...
When you say that there is a left-right reversal what you mean is that the
image is reversed with respect to something else...you are making comparison
are you not? What are you comparing?
isn't this fun
-j
Subject: Re: Mirror question
Date: Mon, 19 Apr 1999 06:44:05 +0000
From: Garr Lystad
To: John C
John C wrote:
>
> I believe in the Socratic method...
That's fine.
>
> When you say that there is a left-right reversal what you mean is that the
> image is reversed with respect to something else...you are making comparison
> are you not? What are you comparing?
I only say that right and left are reversed because that is how people
commonly express what they see. If you face a mirror holding a pencil
that is pointing to your left, the pencil in the mirror points to your
left also. So admittedly saying that right and left are reversed is a
poor way to put it. Nevertheless, that's what you commonly hear.
A much more appropriate way to describe what one sees in a mirror is
to say that the "handedness" of a coordinate system is changed. A
right-handed coordinate system, when viewed in a mirror appears as a
left-handed coordinate system. I guess it would be equivalent to say
that the sense of rotations is reversed. What is clockwise in reality
appears counter-clockwise in a mirror.
>
> isn't this fun
Actually it's not only fun, but quite amazing, in a sense, that
strangers separated by large distances can engage in the friendly
exchange of ideas so easily. It is an interesting time in which we live.
Garr
Subject: RE: Mirror question
Date: Mon, 19 Apr 1999 12:52:10 -0400
From: John C
To: "'Garr Lystad'"
Garr wrote:
> I only say that right and left are reversed because that is how people
> commonly express what they see. If you face a mirror holding a pencil
> that is pointing to your left, the pencil in the mirror points to your
> left also. So admittedly saying that right and left are reversed is a
> poor way to put it. Nevertheless, that's what you commonly hear.
I am not dissatisfied with the term "left-right reversed" as I'm sure we
could agree on it's meaning. I guess my question was one of semantics. To
deduce that an object (say the image in a mirror) is "left-right reversed",
we must mean that it is reversed with respect to some other object. Just as
if I said "The British drive on the wrong side of the road.", what I mean is
that I made a comparison between British driving and my own, and found that
we drive on opposite sides of the road.
So...what is the mirror image reversed with respect too???
Garr wrote:
> Actually it's not only fun, but quite amazing, in a sense, that
> strangers separated by large distances can engage in the friendly
> exchange of ideas so easily. It is an interesting time in which we live.
It is exciting to realize the vigor with which future historians will
clamor for insights into our lives during these exciting times. The
Internet may be reborn several times in the next hundred years, we alone
have witnessed it's creation.
-john
Subject: Re: Mirror question
Date: Mon, 19 Apr 1999 20:20:50 +0000
From: Garr Lystad
To: John C
I understand your question better now. I mean the reversal is with
respect to what you see for an object if you just look at it normally,
without using a mirror to look at it.
As an example, take a clock. Put it infront of you and look at its
face. Curl the fingers of your right hand around in the direction that
the hands travel. Your thumb, if extended, will point TOWARD the clock.
Now look at the image of the face of a clock, the same clock if you
like, in a mirror. Curl the fingers of your right hand around in the
direction that the images of the hands travel. Your extended thumb will
point AWAY from the image of the clock. This, I would say, represents
the essence of the change in what you see. What you see without the
mirror and with the mirror are different in this sense.
To relate this back to what people mean when they look in a mirror and
say that right and left are reversed, think about the hands passing the
numbers on the upper half of the clock. If you look directly at the
clock with the 12 at the top the second hand, for instance, will go from
10 to 11, 12, 1, etc. as it moves to your RIGHT. Holding the clock so
you see its face in the mirror with the 12 at the top, the second hand
passes the same sequence of numbers as it moves to your LEFT. What you
see in the mirror is different from the same objects viewed directly.
I must agree with you that semantics is a major problem in discussions
of this nature. What do you think? Did I do better? Maybe some or all
of our emails should be added to my mirror page. Your parts would be
only with your permission, of course. Anyway, I'm interested in your
thoughts on all of the above, both my attempt at explanation and of
improving the page.
John C wrote:
> ...
> It is exciting to realize the vigor with which future historians will
> clamor for insights into our lives during these exciting times. The
> Internet may be reborn several times in the next hundred years, we alone
> have witnessed it's creation.
>
> -john
Yes, I think this is like the invention of writing, or roads and cement
(Rome) or the printing press, or optics. Things will be different from
here on out. I hope the difference will be as good as it now appears.
Garr
Subject: RE: Mirror question
Date: Tue, 20 Apr 1999 08:46:46 -0400
From: John C
To: "'Garr Lystad'"
Your clock example illustrates (and perhaps goes a step towards
defining/proving) that the image in the mirror is reversed left-to-right.
In fact this type of reversal is rather intuitive which can make definitions
and especially proofs very difficult.
Let us accept that the image in the mirror is necessarily reversed
left-to-right. Now we may ask the question which I posed: "What is the
image reversed with respect too?" Your answer is "...the reversal is with
respect to what you see for an object if you just look at it normally,
without using a mirror to look at it."
Your answer, as expected, is missing a crucial step. A translation at
which you are so adept that you fail to mention. It is purely geometric.
You may perform this operation on either of the images. Let us decide to
operate on the original. Can you tell me what translation it is that you
are using to line up these images which we both now know are left-to-right
opposites? Consider your clock, facing a mirror. The hands of the original
turning clockwise, the hands of the mirror image turning counter clock wise.
In your head, before you conclude the reversal, what do you do to the
images?
> I must agree with you that semantics is a major problem in discussions
> of this nature. What do you think? Did I do better? Maybe some or all
> of our emails should be added to my mirror page. Your parts would be
> only with your permission, of course. Anyway, I'm interested in your
> thoughts on all of the above, both my attempt at explanation and of
> improving the page.
You may surely use our conversation anywhere you like. A warning however,
you may well drive away the whole of your viewership with such rational
discussion and logical arguments. I fear logic garners little attention...
> Yes, I think this is like the invention of writing, or roads and cement
> (Rome) or the printing press, or optics. Things will be different from
> here on out. I hope the difference will be as good as it now appears.
You make a good point here. Clearly there are similarities between the
information age and the stone/bronze/industrial ages. What are the
differences? What separates our era from the others? When a new era is
rushed in on top of ours, what will they say about the computer/transistor?
Does each era depend on the preceding? In other words, in a civilization
wholly different from our own (purely conjecture) would we expect to see
these ages in the same order? I would be interested to know how an internet
audience would answer these questions...
Subject: Re: Mirror question
Date: Tue, 20 Apr 1999 21:33:05 +0000
From: Garr Lystad
To: John C
John C wrote:
>...
> The hands of the original
> turning clockwise, the hands of the mirror image turning counter clock wise.
> In your head, before you conclude the reversal, what do you do to the
> images?
I imagine that I'm looking at them. Is that what you mean? Is the
point that it is all relative to an observer?
> ...
> You may surely use our conversation anywhere you like. A warning however,
> you may well drive away the whole of your viewership with such rational
> discussion and logical arguments. I fear logic garners little attention...
Thanks. You are right about people and logic, of course. Still, if
it is labled as supplemental people can easily choose to take it or
leave it. Maybe I'll make them follow a link to get there. People pass
up links all the time and think nothing of it, since it is only one of
many directions to travel in the Web.
> ...
> You make a good point here. Clearly there are similarities between the
> information age and the stone/bronze/industrial ages. What are the
> differences? What separates our era from the others? When a new era is
> rushed in on top of ours, what will they say about the computer/transistor?
> Does each era depend on the preceding? In other words, in a civilization
> wholly different from our own (purely conjecture) would we expect to see
> these ages in the same order? I would be interested to know how an internet
> audience would answer these questions...
That would be interesting, and I bet most people would guess wrong, not
that I know what right is. On the preceding questions, I don't know. I
am reminded of James Burke's books "Connections" and "The Day the
Universe Changed". I think many times in history people were not aware
how major were the major events they may have witnessed. Maybe we're
one up on them knowing that this is a major change, even if we don't
really have much perspective on its implications for the future. Of
course, some people may have a good deal more insight on the
implications than I do.
Which brings me back to your last point. Maybe one of us should put the
question out there and see what sort of responses come back.
Garr
Subject: RE: Mirror question
Date: Wed, 21 Apr 1999 10:01:12 -0400
From: John C
To: "'Garr Lystad'"
> I imagine that I'm looking at them. Is that what you mean? Is the
> point that it is all relative to an observer?
Maybe I'm reaching too much here... I will attach a short description I
wrote for myself. What I was trying to elicit is that in order to compare
the two images, you must rotate the original 180 degress arounnd a verticel
axis. You must "spin it like a coin". Now the faces match up and you may
conclude that there is a left right reversal.
The "punch-line to the puzzle (for myself anyway) is that you must
perform a different translation to evaluate up-down reversal than you do to
evaluate left-right reversal. You must "flip" the original image
"end-over-end". In doing so you reveal that the images are up-down reversed
just as the images are left-right reversed after "spinning".
Unfortunately, our brains are so adept at performing the "spin"
maneuver, that we simply make this translation twice and the result is an
erroneous impression about the nature of a mirror.
I enjoy this puzzle because it reveals something of the intuitions we
use everyday without a deep consideration of their effects. When this question
was first put to me, I was very much consumed with it, as I had intuitions
but no answer. When I finally came upon the aforementioned conclusion, it
seemed perfectly mathematical, logical, and above all, symetric. It is
pleasing to me, as it really seems to me to be the crux of the issue. I
hope it is somewhat satisfactory, and I do hope we may continue to exchange
ideas. I have found however, that this puzzle is not as intriging to others
as it is to myself. I suppose this is the nature of things, though it
frustrates me to no end.
> That would be interesting, and I bet most people would guess wrong, not
> that I know what right is. On the preceding questions, I don't know. I
> am reminded of James Burke's books "Connections" and "The Day the
> Universe Changed". I think many times in history people were not aware
> how major were the major events they may have witnessed. Maybe we're
> one up on them knowing that this is a major change, even if we don't
> really have much perspective on its implications for the future. Of
> course, some people may have a good deal more insight on the
> implications than I do.
> Which brings me back to your last point. Maybe one of us should put the
> question out there and see what sort of responses come back.
It might be very hard to make a case for wrong or right on my previous
questions, though it would be interesting to hear what your audience has to
say. I have given some thought to the last question about order of ages.
It is most unfortunate that we have mearly one sample of the course of human
history. I would very much like to see how things would play out if we
shuffled the cards and redealt...
-john
[Unfortunately I was never able to read John's attachment.]
mirror.vim
Name: mirror.vim
Type: unspecified type (application/octet-stream)
Encoding: quoted-printable
Subject: Re: Mirror question
Date: Thu, 22 Apr 1999 21:09:19 +0000
From: Garr Lystad
To: John C
John,
You are right about the fact that we don't even think about rotating
the clock. Unfortunately I was unable to read your attachment. My Mac seems to
have taken it for a StuffIt document, tried to decompress it, failed, and given
up.
After sending my last email it occured to me that you might be
thinking about turning the clock as the answer to your question. However, it is
not necessary to turn the clock. The simplest example of this is seen by
putting your clock next to the mirror, angled toward it slightly if you like,
and looking at both the clock and its reflection at the same time. No rotation
is performed in that case, but the direct and mirror images are of clocks
turning in different directions.
Another example of no-rotation of the object but a reversal none the
less, is done with a set of coordinate axes. Imagine 3 sharpened pencils whose
axes are perpendicular, each to the other two, with all their erasers stuck in a
ball of clay and all erasers aimed at a single point in the clay. You can label
them X, Y, and Z in such a way that if you point your straight fingers of one
hand in the direction of the X pencil and rotate your arm so that your fingers
can bend to point in the direction of the Y pencil then your thumb will point in
the direction of the Z pencil. If the pencils are labeled so that this is all
true when the actions are performed with your right hand you have a
"right-handed" coordinate system. If instead it works for your left hand it's a
"left-handed" coordinate system.
If you look at a right-handed coordinate system in a mirror, the image
in the mirror is a left-handed coordinate system, and vice versa. You don't
have to move it at all, but the image is clearly not the same as the original
object. The pencils matche up with one hand, the mirror image with the other
hand.
As a final example, consider a clock with a transparent back. Look at
the clock and curl the fingers of your right hand in the direction the hands
turn. Your thumb points in the direction "front of clock to back of clock".
Now look at the mirror image of the clock. Curl the fingers of your right hand
in the direction that the mirror hands turn (i.e. the same direction as before).
Your thumb points in the direction "back of mirror image to front of mirror
image". Of course, this is the coordinate system exampled revisited with a
different prop. The forward/backward reversal, which is what I claim the mirror
does, is most apparent in these last 2 examples, but, I would claim, is also the
explanation for the first example above.
John C wrote:
>
...
> It might be very hard to make a case for wrong or right on my previous
> questions, though it would be interesting to hear what your audience has to
> say. I have given some thought to the last question about order of ages.
> It is most unfortunate that we have mearly one sample of the course of human
> history. I would very much like to see how things would play out if we
> shuffled the cards and redealt...
My audience is rather small, but I can put the question out and see what we
get.
It would be very interesting to be able to reshuffle the cards, as you
put it, very interesting indeed. I wonder if we'd recognize the place if we
could. Actually, I wonder if we'd even still be here if we could.
Garr
Subject: RE: Mirror question
Date: Fri, 23 Apr 1999 08:32:00 -0400
From: John C
To: "'Garr Lystad'"
ok,
I guess we need to remember that when we speek about the geometry of the
mirror's translations (right-to-left, front-to-back, top-to-bottom, etc...)
they are expressed with respect to the mirror itself. This means that
rotating the clock or angling it somehow really doesn't change the shape of
our problem. Considering your suggestion:
GARR:
> The simplest example of this is seen by putting your clock next to the
> mirror, angled toward it slightly if you like, and looking at both the clock
> and its reflection at the same time.
> No rotation is performed in that case, but the direct and mirror images
> are of clocks turning in different directions.
\\
\\
Mirror-> ---------
//
//
I've angled the clock so that I may look at both the faces which seem to be
reversed. But this isn't really changing the problem. The mirror has no
knowledge of the clock's face or hands etc... If you rotate the bottom
clock 180 degrees around a verticle axis, you still get two overlaping
images that are left-right reversed...
\\ \\ //
\\ \\ //
Mirror-> --------- -> ----------
//
//
You've hit upon a good point though. The fact is, the only translation
a mirror truely makes is front-to-back.
If you take 2 images reversed front-to-back and rotate one of these
images 180 degrees, the images will be left-right reversed.
If you take 2 images reversed front-to-back and flip one of these images
180 degrees, the images will be top-bottom reversed.
It is true that a mirror reverses front-to-back. The interesting aspect
of the problem to me is in considering why it seems so very intuitive that a
mirror reverses left-to-right. In fact that reversal is an illusion created
in our minds when we do a comparitive rotation without thinking.
It should also be noted that if you do angle an object against a mirror
and look at it sideways, a left-to-right reversal from your perspective is now a
front-to-back reversal in the mirror's coord sys. I think this is what you
were doing in the first example.
thanks again
-john
ps. I realize these are some serious 3 dimensional translations without
decent pictures to illustrate the concepts. Perhaps someday far far in the
future there will be a better way to express these structures than flat
text...
Subject: Re: Mirror question
Date: Fri, 23 Apr 1999 22:04:20 +0000
From: Garr Lystad
To: John C
John,
It sounds to me like we are in agreement. I suppose now I should make
this all available to the interested reader. As a first pass I think
I'll simply put our correspondence on a page linked to
my mirror page as we discussed. Of course, you are right that it will
be the uncommon reader who will wade through all of our examples and
explanations. But I think when he arrives at the end of your last
message, if he makes it that far, he may just feel that it was all
worthwhile. Maybe some day I'll be able to distill all this down to
something shorter, but maybe not.
It may take me a while to get this done though. I need to put in
extra time at work, prepare for a small legal battle with Toyota, and
I'm planning a trip to Arkansas to see the Gurdon Light, if I'm lucky.
So it may take a little while for all this to appear on line, but it
should get there before too long.
Thanks for the interesting discussion.
Garr
Subject: RE: Mirror question
Date: Sat, 24 Apr 1999 14:24:12 -0400
From: John C
To: "'Garr Lystad'"
Indeed, thank you Garr. Best of luck with Toyota and the Ghost Light...
-john
Subject: Re: Mirror question
Date: Sun, 25 Apr 1999 21:15:35 +0000
From: Garr Lystad
To: John C
You're welcome. Thank you John.
Garr
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