1145: "Sky Color"

This forum is for the individual discussion thread that goes with each new comic.

Moderators: Moderators General, Prelates, Magistrates

User avatar
ThirdParty
Posts: 351
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

Re: 1145: "Sky Color"

Postby ThirdParty » Fri Jan 25, 2013 4:24 pm UTC

pgimeno wrote:watching a mountain reflected in a lake: the mirrored mountain is upside down, because we compare the images by looking up and down (thus turning the eyes over an horizontal axis).
The mirrored mountain is upside-down because the lake flipped the image vertically; the vertical axis was the one perpendicular to the reflecting surface. Because mountains don't have top-bottom symmetry, there's no way to mistake this for any other kind of flip.

Look, take a book, hold it up to a mirror in the usual way (face toward the mirror, top pointing upward). The cover will appear to have been flipped left-to-right. Then, without changing the orientation of your eyes, rotate the book ninety degrees (face still toward the mirror, top now pointing rightward). Now the cover of the book will appear to have been flipped high-to-low.

Kit.
Posts: 1117
Joined: Thu Jun 16, 2011 5:14 pm UTC

Re: 1145: "Sky Color"

Postby Kit. » Fri Jan 25, 2013 6:19 pm UTC

ThirdParty wrote:Look, take a book, hold it up to a mirror in the usual way (face toward the mirror, top pointing upward). The cover will appear to have been flipped left-to-right. Then, without changing the orientation of your eyes, rotate the book ninety degrees (face still toward the mirror, top now pointing rightward). Now the cover of the book will appear to have been flipped high-to-low.

STOP.
Your explanations are becoming too confusing.
There is a simpler one:

Get a book.
Go to a mirror.
Face the mirror.
Open a book, read some text in it.
Now, don't move the book.
Look into the mirror and try to read the same text there.
Now you can see what the mirror actually flips.

pgimeno
Posts: 7
Joined: Thu Sep 08, 2011 4:36 pm UTC

Re: 1145: "Sky Color"

Postby pgimeno » Sat Jan 26, 2013 6:40 pm UTC

ThirdParty wrote:
pgimeno wrote:watching a mountain reflected in a lake: the mirrored mountain is upside down, because we compare the images by looking up and down (thus turning the eyes over an horizontal axis).
The mirrored mountain is upside-down because the lake flipped the image vertically; the vertical axis was the one perpendicular to the reflecting surface. Because mountains don't have top-bottom symmetry, there's no way to mistake this for any other kind of flip.

Some mountains do have approximate left to right symmetry, yet we don't perceive that the image flips their left and right, which was your implication. The mirror's normal doesn't have to be vertical to give an appearance of vertical inversion; the trick is over what axis you compare and perhaps mentally rotate the images.

ThirdParty wrote:Look, take a book, hold it up to a mirror in the usual way (face toward the mirror, top pointing upward). The cover will appear to have been flipped left-to-right. Then, without changing the orientation of your eyes, rotate the book ninety degrees (face still toward the mirror, top now pointing rightward). Now the cover of the book will appear to have been flipped high-to-low.

I don't hold books that I read that way. I hold them in front of me, almost horizontally. When I look at them, the letters are upright. If I am in front of a mirror and look at the book's reflection, the letters will have left and right in the same place, but will have up and down flipped. I guess Kit agrees, judging by the explanation given.

When you hold the book the way you propose, you are setting up the scenario for mentally rotating the image (or physically rotating the book to face the mirror, or physically rotate your eyes or your neck or your waist etc. to look at both images) over a vertical axis in order to compare both and decide which side is flipped. If you are in front of the mirror and hold the book open vertically with the text facing you, in order to make it face the mirror to look at the inverted image while keeping the letters pointing up, you have to rotate the book over a vertical axis, which is the axis that preserves up and down, and preserving up and down is the invariant you implicitly postulate in your framing.

It's been said in the thread: mirrors invert front and back. But rotating the image over certain axes converts the front-back inversion into something else, which is what gives the illusion that a different axis is inverted. Our tendency to compare the mirror reflection with the original by rotating over a vertical axis is what gives the illusion that left and right are what is inverted.

Yes, symmetry can play a role, in that it's mentally easier to match a figure with left and right inverted than with up and down or front and back inverted, because of the similarity of both sides. But you can get out of that prejudice by asking yourself: what aspect would I have if my left and right were preserved, but my top and bottom were inverted? And what aspect would I have if both were preserved by my front and back were inverted?

After getting rid of that prejudice, the axis over which you rotate to compare the original image with the mirror one is what decides the inversion axis.

User avatar
ThirdParty
Posts: 351
Joined: Wed Sep 19, 2012 3:53 pm UTC
Location: USA

Re: 1145: "Sky Color"

Postby ThirdParty » Sun Jan 27, 2013 12:34 am UTC

Kit. wrote:Now you can see what the mirror actually flips.
The mirror actually flips the image front to back (i.e. the axis perpendicular to the mirror's surface). I stated that clearly in my first post to the thread, and pgimeno didn't object to it. Now we're talking about something else: why people tend to misinterpret front-back flips as horizontal flips rather than misinterpreting them as vertical flips.

pgimeno wrote:It's been said in the thread: mirrors invert front and back. But rotating the image over certain axes converts the front-back inversion into something else, which is what gives the illusion that a different axis is inverted. Our tendency to compare the mirror reflection with the original by rotating over a vertical axis is what gives the illusion that left and right are what is inverted.

Yes, symmetry can play a role, in that it's mentally easier to match a figure with left and right inverted than with up and down or front and back inverted, because of the similarity of both sides.
Okay, I'm more-or-less content with that description.

Consider your own image while standing in front of a wall mirror. The image flipped front-to-back, but it's difficult to think of the image on a 2-D surface as having experienced a 3-D flip, so instead we tend to interpret it as having experienced a rotation plus a 2-D flip. There are two choices: we could think of the image as having been rotated around a vertical axis and then flipped horizontally, or we could think of it as having been rotated around the horizontal axis and then flipped vertically.

The image looks fairly similar to how you would look if rotated around the vertical axis--its head, face, feet, etc. are all in the right place, and the only thing wrong are relatively-minor discrepancies like which arm its wristwatch is on. It doesn't look very similar to how you would look if rotated around the horizontal axis--such a rotation would result in relatively-major discrepancies like the fact that you would be doing a headstand and the image isn't. So interpreting the image as rotated around the vertical axis (with the minor discrepancies caused by a horizontal flip) is more intuitive than interpreting it as rotated around the horizontal axis (with the major discrepancies caused by a vertical flip).

Furthermore, people rotate themselves around vertical axes--e.g. turning around--more often than they rotate themselves around horizontal axes--e.g. doing a headstand. So we're more familiar with that rotation, which again makes us more likely to choose the roated-around-the-vertical-axis-and-horizontally-flipped interpretation.

Kit.
Posts: 1117
Joined: Thu Jun 16, 2011 5:14 pm UTC

Re: 1145: "Sky Color"

Postby Kit. » Tue Jan 29, 2013 5:04 pm UTC

ThirdParty wrote:Now we're talking about something else: why people tend to misinterpret front-back flips as horizontal flips rather than misinterpreting them as vertical flips.

Do they really misinterpret it in their actions?
Or do they do it only in their explanations of the subject, with their actions that involve analyzing mirrored images being generally correct?

And when their actions are incorrect, does it only affect the cases where writing is involved? Or are there more generic cases as well?

User avatar
addams
Posts: 10332
Joined: Sun Sep 12, 2010 4:44 am UTC
Location: Oregon Coast: 97444

Re: 1145: "Sky Color"

Postby addams » Tue Jan 29, 2013 5:15 pm UTC

Kit. wrote:
ThirdParty wrote:Now we're talking about something else: why people tend to misinterpret front-back flips as horizontal flips rather than misinterpreting them as vertical flips.

Do they really misinterpret it in their actions?
Or do they do it only in their explanations of the subject, with their actions that involve analyzing mirrored images being generally correct?

And when their actions are incorrect, does it only affect the cases where writing is involved? Or are there more generic cases as well?

Talk to people that work in dentistry.
Or; Try it.
Very difficult, for me. Yet; Doable, with practice.
Four hands two mirrors one mouth. It gets complicated.
Life is, just, an exchange of electrons; It is up to us to give it meaning.

We are all in The Gutter.
Some of us see The Gutter.
Some of us see The Stars.
by mr. Oscar Wilde.

Those that want to Know; Know.
Those that do not Know; Don't tell them.
They do terrible things to people that Tell Them.

User avatar
orthogon
Posts: 3103
Joined: Thu May 17, 2012 7:52 am UTC
Location: The Airy 1830 ellipsoid

Re: 1145: "Sky Color"

Postby orthogon » Tue Jan 29, 2013 6:22 pm UTC

Kit. wrote:
ThirdParty wrote:Now we're talking about something else: why people tend to misinterpret front-back flips as horizontal flips rather than misinterpreting them as vertical flips.

Do they really misinterpret it in their actions?
Or do they do it only in their explanations of the subject, with their actions that involve analyzing mirrored images being generally correct?

And when their actions are incorrect, does it only affect the cases where writing is involved? Or are there more generic cases as well?

It's not really a misinterpretation, since if

Code: Select all

  (-1  0  0)   (-1  0  0)   ( 1  0  0)
R=( 0  1  0) F=( 0  1  0) M=( 0  1  0)
  ( 0  0 -1)   ( 0  0  1)   ( 0  0 -1)

[I say again, can anyone get the "math" tag to work?]

where R is a 180 degree rotation about the y axis, F is a reflection in the y-z plane, and M is the "true" transformation, a reflection in the x-y plane, then FR=M, i.e. the result is the same: the two models will make exactly the same predictions and imply the same actions.

In terms of the mathematics, the single transformation M is simpler than the two-step FR, and corresponds to a more elegant theory of what the mirror does, but it doesn't seem that way to humans. The likely reasons for this have already been mentioned: rotation in the horizontal plane is a low-cost exercise that we do all the time, and although we can't physically reflect objects horizontally, that doesn't seem such a big deal because of the near-symmetry of human beings (and lots of other objects), and the arbitrariness of left and right. By contrast up and down are distinctly different directions for people living in a gravitational field, and so are forwards and backwards (forwards is the direction we see in, schoolteachers excepted). Reflection in those directions feels more unlikely.
xtifr wrote:... and orthogon merely sounds undecided.

Kit.
Posts: 1117
Joined: Thu Jun 16, 2011 5:14 pm UTC

Re: 1145: "Sky Color"

Postby Kit. » Wed Jan 30, 2013 12:38 pm UTC

orthogon wrote:but it doesn't seem that way to humans.

Actually, it does. They aren't just used to speaking about it properly.

After all, what do people use mirrors for, generally?
To make stuff in back appear in front, don't they?

Although the abilities of a mirror to rotate left-to-right and top-to-bottom have some marginal use by humans too.

User avatar
addams
Posts: 10332
Joined: Sun Sep 12, 2010 4:44 am UTC
Location: Oregon Coast: 97444

Re: 1145: "Sky Color"

Postby addams » Wed Jan 30, 2013 3:23 pm UTC

O.K. Now, explain the upside-down image in a concave mirrored surface.
Life is, just, an exchange of electrons; It is up to us to give it meaning.

We are all in The Gutter.
Some of us see The Gutter.
Some of us see The Stars.
by mr. Oscar Wilde.

Those that want to Know; Know.
Those that do not Know; Don't tell them.
They do terrible things to people that Tell Them.

Kit.
Posts: 1117
Joined: Thu Jun 16, 2011 5:14 pm UTC

Re: 1145: "Sky Color"

Postby Kit. » Wed Jan 30, 2013 3:48 pm UTC

addams wrote:O.K. Now, explain the upside-down image in a concave mirrored surface.

It's not reflection-specific. Refraction can produce it as well (and - in photo lenses - usually does).

User avatar
Red Hal
Magically Delicious
Posts: 1445
Joined: Wed Nov 28, 2007 2:42 pm UTC

Re: 1145: "Sky Color"

Postby Red Hal » Wed Jan 30, 2013 4:28 pm UTC

Plane mirrors do not rotate anything. They reverse along an axis normal to the plane. If the z axis is defined as the axis normal to the plane, then the virtual image of any object will be identical to the real object but with the z-coordinate sign inverted. By way of example, define a set of three orthogonal axes such that the x- and y- axes lie on the reflective surface of a plane mirror, and the z-axis is normal to the mirror. Under that system, a cube formed by the set of points (1,1,1, 2,1,1, 2,2,1, 1,2,1, 1,1,2, 2,1,2, 2,2,2, 1,2,2) would have its virtual image formed at the set of points (1,1,-1, 2,1,-1, 2,2,-1, 1,2,-1, 1,1,-2, 2,1,-2, 2,2,-2, 1,2,-2). It really is as simple as that.

As to why the image in a concave mirror is upside-down, that's because if the object is beyond the focal point it forms a real rather than a virtual image; that is, the image formed by the mirror lies in front of it rather than behind. If you step closer - inside the focal point - you'll see that your reflection is now the right way up.

Edit: Here, take this: http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/mirray.html
Lost Greatest Silent Baby X Y Z. "There is no one who loves pain itself, who seeks after it and wants to have it, simply because it is pain..."

User avatar
addams
Posts: 10332
Joined: Sun Sep 12, 2010 4:44 am UTC
Location: Oregon Coast: 97444

Re: 1145: "Sky Color"

Postby addams » Wed Jan 30, 2013 4:54 pm UTC

That is a good link.
You win.
Life is, just, an exchange of electrons; It is up to us to give it meaning.

We are all in The Gutter.
Some of us see The Gutter.
Some of us see The Stars.
by mr. Oscar Wilde.

Those that want to Know; Know.
Those that do not Know; Don't tell them.
They do terrible things to people that Tell Them.


Return to “Individual XKCD Comic Threads”

Who is online

Users browsing this forum: No registered users and 108 guests