In the Oxford Companion to the Mind (1987), Richard L. Gregory (Professor of Neuropsychology at University of Bristol, UK) in my view, talked all around the solution, without actually delivering it. He certainly understood that there is no rotation in the mirror (see below). However, he seemed to think (like Stephen Law and Umberto Eco) that there is no reversal at all, though he once obliquely referred to an 'inversion', so maybe he knew without knowing that he knew.
In 2002, whilst working in America
In one of his chapters, he gives a lengthy discourse on the mirror paradox, and that was one of my points of contention. He argued that a mirror ‘reverses nothing’, but a second reflection did reverse left to right, which restores the image to what we normally see. I pointed out to him that this was illogical, nevertheless there is a specific case where he is correct. I will return to this specific example at the end of my discourse. I need to say that I have great respect for both Stephen Law and Umberto Eco, as both of these men are far more knowledgeable than me in their respective fields.
Most people explain the mirror reflection in terms of rotation, as it appears that the mirror rotates the image around, and this is particularly compelling for mirror-reflected writing, as I explicate below. But this merely raises another question, effectively transcribing the paradox, not solving it. Why does it rotate the image about the vertical axis, not the horizontal axis? Stephen Law gives an analogy: if you walk through a door that opens on the right side, why does it open on the left side when you come back the other way? The answer is because you turn yourself around. Law argues that if we lived in zero gravity, whereby you could turn yourself upside down to open the door, it would still open the same way, so the implication is that it’s gravity that creates the emphasis on the vertical axis. In fact, Stephen Law speculates that if we lived in a weightless environment then perhaps the ‘mirror puzzle would not even be a puzzle’. But I believe he'd be wrong: it's the left-right symmetry of the human body that creates the emphasis on horizontal over vertical reflection.
All these explanations and descriptions seem to overlook the fact that there is no rotation in the mirror at all – in fact, it’s the lack of rotation that gives us mirror reflection. I believe that most of these explanations actually appreciate this fact; they just fail to explain it. But I have been keeping you in suspense – the answer to this puzzle is deceptively simple: the mirror doesn’t reverse left to right, or top to bottom, it reverses back to front. We live in 3 dimensions, not 2, and a mirror reverses everything in the dimension perpendicular to its plane. So the rotation is a genuine illusion (it doesn’t happen), but the reversal is a true optical phenomenon.
Below is an edited version of my exposition that I sent to Umberto Eco.
Normally, if we want to see something back to front we have to turn it around. Generally we do this by turning the object through its vertical axis but we can also turn it through its horizontal axis. If we turn it through its vertical axis, as happens when someone turns to face us, their left side appears on our right and their right side appears on our left. This is unavoidable. But they could also turn to face us by standing on their hands, in which case they would appear upside down but their left side would still be on our left and their right side on our right. Then if they stood by doing a half cart wheel they would resume their normal stance but left to right would be reversed. The mirror quite literally reverses the image back to front without rotating it through any axis at all. So we don’t see the image upside down but likewise we don’t see the left side on the right or the right side on the left. This is the illusion pure and simple. The illusion, when we face a mirror, is that it appears to rotate us around a vertical axis, when in fact it doesn’t, it turns us back to front. If we look at something between us and the mirror, we see the front of it facing us, and the back of it facing us in the mirror. This is the key to the illusion. When we look at ourselves in a mirror we expect to see ourselves as others see us, but we can only do this when we have 2 mirrors, which appears to really rotate everything about the vertical axis (as Eco contended), but, in actuality, restores front to back to front again. But I’m fast-forwarding - I will elaborate on double reflections later.
In other respects we are not fooled by the mirror’s conservation of left and right. If we see in the mirror someone standing behind us and to our left, we automatically look over our left shoulder, not our right. Where we are fooled is when we reach for something on a table between us and a mirror, as in the case of an object on a dressing table or a bathroom bench top, while watching the object in the mirror. If we reach for an object at the back of the table, we appear in the mirror to be reaching forward towards us, rather than away. Likewise, if we drag an object on the table towards us, we appear in the mirror to be pushing it backwards not forwards. If you doubt this, try shaving or combing your hair with your left hand instead of your right (or your right hand if you're left-handed). We’ve trained our preferred hand through years of practice.
When we look in the rear view mirror at a car parked behind us while standing at a traffic light, we see that the driver is sitting on the same side of the car as we are and we are not confused. Because we know the car is behind us, the same as in the previous example, when we knew that the person standing behind us in the mirror was on our left or right side just as the mirror dictated. If the car was traveling towards us, we would expect to see the driver on the opposite side to us because the car has been turned around it’s vertical axis. If we turned around to look at the driver behind us at the traffic light, we would still see that he or she is on the same side of the car as we are, because both cars are facing the same direction, even though we have turned around to look backwards. Therefore, when we look at the driver in the rear view mirror we can see that left and right have been conserved. So why is it that when we look at the number plate we have to read it backwards, as if it's been rotated?
Writing not only provides the best illustration of the illusion, it also provides the best means to understand it. If you hold up a page of a book with writing on both sides while facing a mirror, the side facing you is readable, but the side facing the mirror is mirror-reversed. However, if the page was transparent, then the writing on the other side would also appear mirror reversed exactly as it does in the mirror. Take a sheet of plastic or cellophane, or anything clear that can be written on. If you hold up this transparent sheet so that the writing is mirror-reversed to you then it will also appear mirror-reversed in the mirror. Likewise if you hold it up so that the writing is readable to you then it will appear readable in the mirror. So where did the illusion of rotation go?
The illusion has gone but the reversal hasn't. Because when you hold up the sheet so you can read it, you are looking at the front of the sheet whereas the image you see in the mirror is the back of the sheet. Left to right is not reversed but front to back is. The front you see in the mirror is actually the back to you. If you were to place yourself between the sheet and the mirror, without changing its orientation, you would see the writing mirror-reversed. The mirror mirror-reverses the back of the sheet. And, of course, you would have to turn yourself around to read it, which only emphasises the illusion that the mirror rotates the image, but actually it doesn’t. The reason writing always appears reversed left to right, is because we always turn it left to right to face the mirror. We do the rotation, not the mirror.
This brings me to the third image created by a second mirror. If you set a book upright on a dressing table (or a table with a mirror behind it) with the front cover facing you, then the back cover will be mirror-reversed in the mirror. If you then took a small mirror (say a shaving mirror) and place it between yourself and the book, but facing back into the main mirror (or background mirror) you can create a third image of the cover in the main mirror. This is very easy to do by small adjustment of the angle of the foreground mirror. Naturally enough (but only because we know in advance) we can read the front cover in the third image exactly as it appears to us on the table. If we didn’t already know this, I believe it would be a complete surprise. The important point is that the image is not rotated at all, it is simply reversed back to front twice, using an intermediary mirror that is facing away from us.
In fact the foreground mirror behaves in exactly the same way as the transparent sheet I referred to in the previous example. If you could see through the foreground mirror so it’s image could be read from the back (in other words if it was a transparent screen with the book cover projected onto it) we would be able to read it exactly as we can in its reflection in the background mirror. The point is though, that the foreground mirror reverses the image, not from left to right but back to front. The foreground mirror only has the writing in the right order because it is facing away from us. If you were to place yourself between it and the background mirror (and turn yourself around) you would see the writing is mirror-reversed as you would with the transparent sheet. So the background mirror mirror-reverses the foreground mirror.
But, as I alluded to earlier, there is a specific situation, and a common one, where a second mirror does translate the image directly from left to right, which upholds the illusion of rotation. We often find ourselves in a bar, or a bathroom, with 2 vertical mirrors joined at right angles like 2 walled mirrors. In this case the image you would see is a double reflection no matter which mirror you looked in. In fact, if there were 2 extended wall mirrors, then there would be 4 images of you, including the prime image. If you were to press your finger into the corner, you would see 4 symmetrical images of it, one of which would be you. Another unique feature of this third image is that it would always remain in the corner of the room as you moved about, whereas the other 2 images would follow you around. This also means, of course, that everyone in the room would see themselves in the corner (assuming they had a clear line of sight).
The (apparent) non-reversed image results from a secondary reflection coming off a primary reflection that you cannot see, because the two reflections simultaneously 'swap' over on the adjacent mirrors. This, in fact, does resemble a rotation about the vertical axis, simply because the mirrors are joined on a vertical axis. And this is what led Umberto Eco to argue that the first mirror image is not reversed left to right but the second image is. He is correct, in this specific case, but only because we create a virtual vertical axis of rotation by the very careful alignment of the 2 mirrors.
So now I have turned a simple answer into more than 2,000 words, and either have confused you completely or explained a common phenomenon thoroughly. I hope the latter.
Footnote: I had a brief discussion with Stephen Law on this topic. We agree to disagree over my belief that science does solve this puzzle. You can visit his post on this subject (and our dialogue) at the following: http://stephenlaw.blogspot.com/2007/04/mirror-puzzle-solution.html And explore the rest of his excellent site.
Addendum 1: If you think this exhausts the subject of mirrors, you should read Richard Feynman's quantum mechanical explanation of reflection in his truly fantastical book, QED, The Strange Theory of Light and Matter.
Addendum 2: There is something I need to tidy up. There is one situation where a mirror does reverse an image left to right, and it’s literally the exception that proves the rule, because it doesn’t reflect the
image back to itself.
If you set up a mirror at 45 degrees along its vertical axis, it allows you to see around corners and there are lots of examples in car parks and other places. Note that you can’t see what’s behind you as you would normally expect from a mirror. Most of these (seen in real life) are convex so that you can see more using a smaller mirror.
The best way to demonstrate this is with a graphic that shows how the light rays reflected at 45 degrees axiomatically reverse their relative orientation from left to right. Note that if you were to tilt a mirror at 45 degrees along the horizontal axis, it would turn an image on the ground (facing you) upside down.
190 comments:
Interesting, is all I can say after reading 2000 words on mirrors, heh.
I never actually thought about it this much, cool though.
Thanks Nate,
If someone accused me of over-explaining it, I would plead guily.
Regards, Paul.
Not sure why you prefer to think that the image is inverted front to back, rather than top to bottom or left to right?
Hi Glibeaux,
It's not what I prefer to think, it's what actually happens.
The image is not inverted top to bottom or from left to right - it's reversed front to back.
When you stand in front of a mirror, your right hand reflects back to you on the right side and your left hand reflects back on the left side. Likewise your head reflects back as being on top and your toes as being on the bottom.
But your front is reflected back as facing you, the opposite direction to how you're standing. It's obvious when you think about it.
Regards, Paul.
Excellent article.
I wonder how many hours I have spent on this seeming paradox, before I read your article.
Hi Harshad,
Glad you appreciate it, and that I've enlightened at least one person.
Regards, Paul.
I have been puzzled by this paradox for years and one evening drove myself crazy trying to figure it out. I actually got closer to the answer when I spent some time looking at a transparency with writing on it trying to figure out the puzzle. You solved it for me with a single line that sent chills up my spine
"We do the rotation, not the mirror."
Suddenly I got it. We are the once who rotate the writing when we hold it up to the mirror!
Gregson Vaux
Hi Gregson
Glad you appreciate the post.
It's amazing that it manages to confuse us so much in the first place. It's really quite simple once you finally work it out.
Regards, Paul.
Your proposed solution to the mirror puzzle is intriguing, but how seriously are we to take it?
As an artist, I once made a replica of the famous painting by Magritte, "This Is Not A Pipe",(pardon my French), except that my version was reversed from left to right, just as one would expect to see the original if it were reflected in a mirror. Would you seriously contend "Aha, no! What you really interchanged were the back and front!" I don't suppose so.
Could it be that what we nominate as the "front" of an image in the mirror isn't really the front of anything, any more than Magritte's painting is really a pipe?
Soz, Gerrie D.
Hi Gerrie,
Yes you can draw anything left to right, effectively creating a mirror image. You can do the same with a film by inserting it back to front in a projector, so they appear to be exactly the same thing, but they're not. It's just that the film has to be turned left to right to make it back to front; it can't be turned upside down.
What I said was that the reason writing appears left to right when we look at it in a mirror is because we always turn it left to right to face the mirror, and the front of it now looks back at us in the opposite direction to what the real writing is facing.
But you could turn it around the horizontal axis, so that it appeared upside down in the mirror but with the writing still going from left to right, not right to left, yet we don't see that as unusual do we?
But if you have the page, or whatever, with the writing facing you and turn it upside down, the writing goes right to left, not left to right. So the item facing you without reflection is never the same as an item facing you in reflection.
But whether it appears upside down or reversed right to left, just depends on which axis you turn it around, horizontal or vertical. Either way, the reflection appears back at you in the opposite direction you are facing, that is what is consistent, always.
Regards, Paul.
"I wrote a long letter to Umberto, who acknowledged receipt of my letter but never followed up with a response. I learned later that he had a fight with throat cancer, so I take it as no slight that he didn’t respond further."
You, sir, are a pompous ass. You took no slight? You think the whole world must respond to every missive you deign to send us?
Hello Faloo Faloo,
If that's your real name. To be honest I was chuffed that he even sent me an aknowledgement, very politely written, over 6 years ago now, and I still have it.
In his response he said he would send a longer response but he never did. It may be that he thought my 'missive' was too trivial, which is fine as well.
I readily acknowledge that Umberto Eco is a real philosopher and I am the amateur.
Regards, Paul.
Your point that we see the mirror image from the opposite direction to that we would have to look from in order to view its source was a good one, but is this really sufficient to guarantee that the image represents a back - front reversal? How do you support the idea that the front of the source corresponds to a something like a "front" or a "back" of its image in the mirror?
I ask this because the planar surface of the mirror has no depth to confer.
Refer again briefly to the idea of Magritte's pipe. If the painting does not represent a pipe. we cannot suppose that it depicts something with a "front" - especially since the image lies in a two-dimensional surface.
On the other hand, whether or not the picture represents a pipe, it still does have a left and right, up and down, which can be genuinely interchanged - as I indicated in my earlier posting.
Gerrie
Hi Gerrie,
The only thing that a piece of paper and a mirror have in common is that they are both a 2 dimensional plane. On a piece of paper, you can do anything you want, which may imitate a mirror or not, as you did with your own rendition of Magritte’s ‘pipe’. Leonardo de Vinci actually wrote left-handed in mirror-writing, so you needed a mirror to read his writing, but a mirror is an optical device in a way that a piece of paper never can be.
A mirror does have optical depth and you can prove that yourself by holding a mirror close to your face with an object in the far background. You will notice that you have to consciously change the focus of your eyes from close up to background just as you would if you weren’t using a mirror at all. This is because the angle subtended by the object in the mirror is exactly the same as the real object at the same distance, and the light rays have to travel the same distance at the same angle; not just the short distance from your eye to the mirror's plane.
On the other hand, you can paint a picture on a planar surface with objects in the foreground and the background, yet you don’t have to change focus to see them, because, as you point out yourself, they are all on the same ‘optical’ plane. But a mirror is different: going back to the mirror close to your face; close one eye, then the other, and the background moves in the mirror. That will never happen with a 2 dimensional painting. A mirror image is genuinely 3D.
Why? Because it’s an optical device that reflects every light ray that falls on it. In his book, QED: The Strange Theory of Light and Matter, Richard Feynman even gives a quantum mechanical explanation of mirror reflection, where every path possible for every light ray is allowed for. Don’t ask me to explain that here; I’m not that clever; but Feynman is.
Getting back to basics, if you place something between yourself and any mirror, you will see the back of it in front of you and the front of it in the mirror, or vice versa – what other proof do you need that a mirror reverses an image back to front?
Regards, Paul.
What other proof do I need?
I used to have a kitten who used to mistake her reflection for a real cat. She would check by looking behind the mirror.
Geraldine D
Hi Geraldine,
Yes, I once saw a pup do that, and yes, cats. It's not at all uncommon to see magpies fight their own reflection in a window.
Some Psychologists and scientists claim that this is evidence that these animals don't have a sense of self, but I believe that's the wrong criterion. It just means they can't recognise themselves. I'm quite sure they appreciate, in their own way, that they have a separate identity, or why would they be so curious about an 'other'.
Regards, Paul.
Paul,
I have occasionally checked your blog since last summer and you may wish to consider the following idea, appropos of recent correspondence:
a) Note that as a hand-held mirror is rotated or moved about, it provides a fully consistent image of the world - quite as if it were a window on to its chiral counterpart or "enantiomorph".
In fact this must be the case, since mathematically, the inversion of a three dimensional manifold is defined as projection through a single point (http://en.wikipedia.org/wiki/Inversion_in_a_point); so there are only two possible worlds in this respect: "right-handed" or "left-handed". If our own world is defined as of the "right-handed" variety then the one apparently represented in a mirror is "left-handed". Presumably it is this feature which motivates your argument that in some sense a mirror re-presents real "depth".
However, consider the following situation:
b) If two mirrors are hung on adjacent walls of an ordinary box-room so that they lie in planes perpendicular to each other, it follows (from a. above) that their respective images will be mutually consistent with the illusion of a single "mirror world" beyond.
We may place an assymmetrical object between, and therefore within sight of both mirrors. (For our purposes it may as well be Ms. Danken's painting of a pipe). If this picture is placed so that it is facing one of the mirrors (A), then the plane in which it lies will be perpendicular to mirror B.
According to your own argument, the reflected image of the painting is not reversed from left to right in mirror A. On the other hand, you claim that mirror B really does invert the same dimension ,ie horizontally in the picture-plane, because of course, this plane is perpendicular to B.
These observations are inconsistent with the point that A and B must agree about the single "mirror world" they seem to look out on: either the horizontal axis of the painting is "really" reversed or it is not.
I thought Gerrie Danken's last posting was a bit cheeky. Even so, it seems to me that if we accept her argument that a mirror-image does not necessarily include a reversal from back to front, it enables us to accept yours - that neither does it invert from left to right.
I hope that this will help to clear things up. Thanks for a great column.
Glibeaux
ps. Perhaps Ms. Danken will not accept this compromise - you know what women can be like!
Hi Glibeaux,
I’ll start with your second point first: a room with adjacent mirrors.
If you face one of the mirrors, in a simulacrum of Gerrie’s painting, with, say, the adjacent mirror on your left. In the adjacent mirror, you will see your left hand towards you and your right hand away from you, which is the opposite to what you see in the mirror facing you – this is your point. But if you rotate your body to reverse this asymmetry, so you are now facing the mirror on your left, with the adjacent mirror on your right, you will see the opposite effect – what happened? What happened is the rotation in both mirrors is the reverse of your real rotation, like 3 linked gears: if you turn the one in the middle, the other 2 both turn the opposite way.
So if you were to look down on the gears, with the same configuration as the mirrors, and paint dots on them, so they each have a yellow and a red dot directly opposite. In the first case I illustrated above, the middle gear has a red dot on its right (your right hand) and a yellow dot on its left (your left hand). The gear on its left has a yellow dot against it and a red dot opposite, whereas the gear to its front, has a yellow dot opposite its yellow dot and a red dot opposite its red dot, so this gear is the opposite to the gear on the prime gear’s left. I wish I could draw it, but you can.
Now rotate the prime gear to its left, so it faces the gear on its left, and what happens?
Everything becomes reversed. If you don’t believe me, cut out some paper gears, or mark some coins and try it.
In a way, this also relates to your first point, which is chirality. Chirality is often called ‘right-handed’ or ‘left-handed’ but it’s a misnomer; it should be clockwise or anti-clockwise, because chirality refers to rotation, spin or twist. And, as I illustrated above, chirality (rotation) is always reversed in a mirror. If you look at a clock in a mirror it runs anti-clockwise. But if you take anything that’s rotating and look at it from the opposite direction, it always changes from anti-clockwise to clockwise or vice versa, so this is completely consistent with a mirror reversing everything from front to back.
I had a look at the article in Wikipedia and it does seem to describe mirror inversion, but I think your use of the term ‘right-handed’ and ‘left-handed’ in this context refers to chirality which is misleading. Even the dictionary definition of chirality doesn’t help. A DNA molecule is described as having a chirality of left-handedness or right-handedness but its to do with the twist in the molecule, and logically it will reverse if you look at it from the opposite direction, so the term left-handed or right-handed is a scientific convention, which is also used to describe fundamental particle spins.
I hope this answers your conundrum; I don't think I can explain it any better.
I don’t know Gerrie, but I hope, for your sake, she has a sense of humour.
I’m glad you like the blog, thanks for your compliments.
Regards, Paul.
Sorry Glibeaux,
I have to make a correction to my last comment. Chirality concerning molecules, screws etc. always run the same way, no matter which way you look at them (contrary to what I said in the previous comment) but it's still clockwise or anti-clockwise, not right-handed or left-handed, even though that's the nomenclature that people use.
If you take a spiral staircase, you go up one way and come down the other. But if there was a mirror above the staircase that you could look up at, you would see yourself going the other direction, both vertically and rotationally.
So, even though a screw or a molecule or whatever has the same chirality, whichever way you look at it, it reverses when you look at it in a mirror. So that's the exception you are looking for.
And,if you stand behind a screw (imagine a spring pointed at a mirror and sliding something like a bead along it) which is different to the case of the spiral staircase (because it's under your feet) the circular rotation of the bead is the same direction in the mirror but the direction along the axial axis would be reversed.
So chirality is reversed in the sense that a 'left-hand' screw becomes a 'right-hand' screw in a mirror.
I've probably really confused you now. I almost confused myself.
Everything else I said in the previous comment I stand by.
Regards, Paul.
Hi Paul,
Also taking your last point first
re: "chirality" (http://chirality.ouvaton.org/homepage.htm)
Also see Concise Oxford Dictionary : "chiral" comes from Greek "kheir", meaning "hand".
More to follow.
Glibeaux
Hi Glibeaux,
I take your point about the origins of 'chirality' and its reference to 'handedness', but it doesn't change anything I've said.
My experience with the word, chirality, is always in a scientific sense, which involves a spiral or a screw or rotation. And all of these are reversed in a mirror, for the same reason, because they are reversed back to front, not right to left.
A right hand becomes a left hand in a mirror, because a right hand is a left hand back to front; you can see this for yourself when you place your 2 hands together.
When it comes to screws and spirals, right and left hand is a convention of terminology but it's a misnomer, because it involves going clockwise or anti-clockwise.
A right hand screw is called such, because if you curl your fingers and protrude the thumb of your 'right hand' it represents the direction of a right-hand screw. If you do the same with your left hand, it represents the direction of a left-hand screw. I was taught this in science at high school.
Regards, Paul.
You seem to have misread my earlier contribution in which I argued that your contention that mirrors "really" reverse back and front but not left to right is logically inconsistent. This is because whatever appears as "back" and "front" in one of two mutually perpendicular mirrors is represented as left and right in the other.
In support of this point, it can be demonstrated that the two mirrors seem to represent different aspects of a single "mirror-world" rather than two distinct "worlds": an unbroken transformation (a translation and a rotation) will bring the mirror initially at A to location B without losing sight of the objective source (which, following your preferred example, might be a human figure). The visual impression is that the image of this figure merely rotates through ninety degrees as the mirror moves to its new location, as though it occupies a single "environment", much as we do.
According to your argument however, when the mirrors are in their original locations at A and B, (and with the real figure facing one of the mirrors), the left and right hands of his image are "reversed" in one, and yet simultaneously "not reversed" in the other - a contradiction.
Apparently this conflict can be avoided if you are willing to accept Gerrie's argument that a mirror does not really invert from back to front.
Regards, Glibeaux
No Glibeaux,
It is you who misunderstands me. I've explain what happens with 2 mirrors at right angles in my original post in detail, so I won't repeat it here, except in summary: there are in fact 3 images, one in the corner where the mirrors meet that is an unreversed image because it is effectively reversed twice.
But I explain what happens with your scenario in my comment before last, with the analogy of gears.
What you don't understand is that what is left to right in one mirror is the back to front in the other mirror. It is entirely consistent. If you use the gears analogy (mark up 3 coins and try it for yourself) you will see that left and right, front and back are interchangable, because they are marks on a coin rather than writing, or right and left hands, which confuse your thinking.
If you stand facing one mirror, your left hand is on your left side and your right hand is on your right side, but in the mirror that you are side on to (say on your left), your left hand is at the front and your right hand is at the back, which is the reverse orientation to the mirror that you are facing, but each mirror consitently reverses back to front not left to right.
So mirrors do not reverse left to right but back to front under all circumstances. Without being able to talk to you face to face or draw diagrams, I can't explain it any better.
Regards, Paul.
Sorry Glibeaux,
This will probably confuse you more, but in my first para of my last comment, I mistakenly say that the image in the corner is 'unreversed', when in fact it is 'reversed' left to right, and it is the only image that is reversed left to right, which confused Umberto Eco as well, and is why I explain it in detail in my original post.
The vertical axis, where the mirrors meet effectively rotate the image left to right about a vertical axis because the mirrors are aligned along a vertical axis, and for no other reason.
This may clarify things or confuse you further. All I know is that it is correct. Of that, I am very confident.
Regards, Paul.
Hello Boys
It’s true that I do not accept Glibeaux’s suggested “compromise”. However, this is because both of you are wrong - not because I am a woman!
Imagine that a word (eg “EAST”) is written on a piece of card. When the card is rotated about a vertical axis and seen from the back, it’s clear that the location of the “E” will now be to the right, and the “T” to the left relative to the viewer. This indicates that the idea as to whether the E is on the right or the left depends on whether the card is viewed from the back or the front.
Now if the word is viewed in a mirror, the E and T will again appear at the right and left of the image respectively. Although the letters in the image and the source share the same left - right coordinates, it is wrong to infer from this that the mirrored letters are unreversed. This is because the image represents the front of the card, and it follows that the mirror really presents a reversal from right to left. This interpretation depends only on the aspect of the card which is viewed in the mirror (front), and not on any depth inherent in the image or the source.
Paul is wrong to say that a mirror reverses only from back to front, whereas Glibeaux is wrong to suggest that other elements of Paul’s argument can be saved by abandoning this mistaken idea.
Whether the rest of Glibeaux’s discussion makes sense I can’t tell. He seems to be saying that if the mirror rotates through 90˚, so does the image-world; but this is also erroneous. The image appears to rotate through 180˚, in agreement with Paul’s false argument, (although through only 90˚ relative to the mirror itself).
Cheers, Gerrie
Hi Gerrie,
For a start, I didn’t know you were a woman until Glibeaux mentioned it. Yes, I know of blokes called Gerrie – perhaps they spell it differently, like with a ‘y’.
I like a good argument, but, like my explanations to Glibeaux, it’s difficult to explain things any better without talking to you face to face, or by drawing diagrams, but I will try.
In response to your 2nd and 3rd paragraphs, I would compare it with developing a film negative back to front (sometimes happens in a magazine). In that case you also get any writing reversed left to right – in fact the whole image is reversed left to right, as well as back to front, because you rotate it around the vertical axis in order to reverse it back to front. (It could be viewed upside down, of course, but the writing would still be reversed, either way.)
Well, this is what happens when you view an image in a mirror, only it reverses back to front without reversing left to right (big difference).
As I explained in my original post, if you write a word, like ‘EAST’ on see-through paper or cellophane, and hold it up reversed (by turning it through its vertical axis) in front of a mirror, you will see the image reversed left to right both in your hand and in the mirror (as you describe in your comment).
However, the image in your hand is the ‘back’ of the paper, and the image in the mirror is the ‘front’. This illustrates my point exactly. The mirror reverses the image back to front without reversing it left to right. If it did reverse left to right, then the word, EAST, would appear unreversed. And this is what happens when you have 2 mirrors aligned vertically, and you observe the image in the corner, which appears ‘unreversed’, yet, in fact, is reversed left to right as well as back to front.
In effect, the image in the corner is exactly like the picture developed from the back-to-front negative, whereas a normal mirror image is not.
I really can’t explain it any better than this.
Regards, Paul.
Hi Gerrie,
The film developed back to front is the same as a mirror image, because it's the same as looking at the back of the transparent sheet, which is not what I said in my last comment.
Umberto Eco, had the same problem. He compared the left right conversion of the double mirror image with a TV image. So my last paragraph of my last comment is incorrect, and I apologise.
I could just delete it and start again, but I don't mind showing my mistakes, and it may make it easier to understand.
The double reversed image in the corner is the same as a normal photograph and a mirror image is the same as a photo developed back to front, which is logical.
A normal photo developed correctly reverses everything left to right because it's the same as when you look at someone facing you - their left hand is on your right and vice versa. Sorry for confusing you.
When you face someone they have to turn around or you have to turn around, which reverses everything left to right. A mirror doesn't do that because it doesn't rotate anything about any axis. Once you understand that it all makes sense.
Regards, Paul.
Hi Paul,
I don't know how many men you know called Geraldine!
As for your argument, I don't really agree with it, but I guess we'd better leave it at that.
Gerrie
Hi Gerrie,
Yes, you're right - I don't know any blokes called Geraldine. And there are lots of women called 'Jo' or 'Joe' as well.
As for our disagreement, it's easy to get confused - if you look at the history of my comments, I've got myself confused on a few occasions.
The key thing is that to turn something around to face you, it has to be rotated, or you have to rotate to face it. But a mirror turns everything back to front with no rotation at all. It really is that simple.
Regards, Paul.
By the way, Gerrie,
This is physics, not philosophy, so the answer is not a matter of opinion. In philosophy, as Bertrand Russell points out, there is not necessarily a 'right' answer - see the quote at the end of this post.
In physics, like mathematics, there is a right and a wrong answer, even in cases where we are yet to find it. In the case of mirror reflection, the answers are known, not unknown.
I'm not trying to be elitist, just pointing out a fundamental difference. There are such things as scientific facts.
Regards, Paul.
Hi Gerrie and Glibeaux,
As I’ve said earlier, I can’t talk to you face to face or draw diagrams, but there is a very simple experiment you can do that should convince both of you that even multiple mirror reflections do not reverse left to right.
What I suggest is that you put a distinct mark on your face to distinguish left from right, preferably near one of your eyes. A coloured eye patch would do it, but also red lipstick above or below one eye would be just as effective – make it bright.
Take a hand held mirror – a shaving mirror would be perfect – and hold it near your face, facing another mirror, like a bathroom mirror. The hand held mirror faces the other mirror – you hold it in front of you, facing away from you, and you are also facing the wall mirror. Hold it near the eye with the colour mark facing the wall mirror.
Now line the hand held mirror up, so you can see multiple reflections of your face, in particular, the coloured mark on your face. You should be able to see 3 to 4 reflections if you can hold the mirror steady enough.
The point is that the coloured patch will appear on the same side in all reflections, which will be the same side that you have it on your face. So if you have it on the right side of your face it will appear on the right side in all the mirror reflections. If it is on your left side it will appear on the left side in all the mirror reflections.
If a mirror reflection reversed left to right, then it would appear alternatively left and right in each reflection, which it doesn’t. Please try this, it’s irrefutable proof.
The only exception is when you have 2 mirrors meeting in a corner at 90 degrees, vertically, and, in that specific case, the image is reversed left to right, as I’ve explained in my original post.
Regards, Paul.
Hello Paul,
The mirror riddle is a gem. I enjoyed your article and the responses. Please allow me to try my hand at it.
First, three postulates:
1 - When people want to compare two nearly identical objects,
a normal strategy is to line them up, facing the same direction.
2 - A powerful way to deal with subtle qualities is to specify
an Operational Definition for the determination of such a quality.
3 - Another normal strategy, when dealing with a confusing setup,
is to insist on not disturbing the scene, leaving it "as is,"
and then analyzing it piece by piece.
Now the explanation:
It often happens in a person gazing in a mirror, that a flash of mental imagery suggests the performance of a certain operation. In this operation, or procedure, the mirror image is frozen, but a self rotation is imagined about the vertical axis in order to face in the same direction as the image. Upon doing this, a "reversal" of Left and Right is perceived in the image. Let’s label this imagined procedure "operational definition A" or shortened to OpDef-A.
This mentally imagined procedure is fleeting in duration and there are no words involved. Later on, the fact that Up and Down were not reversed comes up, and this seems like an unexplained asymmetry.
When analytical thinking and vocalization are brought into play to explain this asymmetry, the rotation is usually not employed, and the situation is examined “as is” and piece by piece. The result is a perceived reversal of front to back, and the left/right reversal vanishes. Let’s label this analytical procedure "operational definition B" or shortened to OpDef-B.
Thinking back on all this and subconsciously drifting back and forth between OpDef-A and OpDef-B, the mirror paradox is experienced in the confusion. It’s first perceived as an unexplained asymmetry with Up and Down not reversed when OpDef-A is in mind. Then it’s compounded upon drifting to OpDef-B as an unexplained vanishing of the Left and Right reversal.
I have set up a discussion forum to deal with all this at
http://mirror-reversal.proboards.com
Please feel welcome.
Hi MikeO,
I think this post has generated more comments than any other post on this blog, and it's 18 months old now.
"The result is a perceived reversal of front to back, and the left/right reversal vanishes."
Well, the fact is that the reversal of front to back is not just 'perceived', it's real, whereas the 'left/right reversal' is an illusion, because the 'rotation' is an illusion.
Plane mirrors follow a very simple law: they reverse everything along the axis perpendicular to the plane of the mirror.
This is a law of physics, not an illusion or a mere perception.
It's that simple, and no other explanation is required.
Regards, Paul.
Hello Again, Paul,
Plane mirrors do indeed reverse the perpendicular component of photons, but the mirror riddle is not concerned with photon trajectories. The mirror riddle is asked by people who PERCEIVE a Left/Right reversal. It’s a people thing. Solving the riddle means satisfying the curiosity of people. Because of this, their perceptions are important to track.
According to my Operational Definitions for the word “reverse,” when a human being thinks within the confines of OpDef-B, as it seems you are doing, then the perceived reversal is Front/Back, and similar sounding to the laws of Physics.
But when a human thinks within the confines of OpDef-A, then the perceived reversal is Left/Right.
As long as they keep applying this Operational Definition, all the photon trajectory proofs in the world won’t shake their feeling that there is STILL a Left/Right reversal. THAT is why this riddle persists. People persist in applying OpDef-A as I’ve outlined it, and the L/R perception persists.
I understand the Physics, and the simplicity there. But people and the language they speak are not so clean and neat. Most of the words spoken in common language have multiple definitions, and this word “reverse” is particularly slippery.
For brevity, I’ve posted here on the two operational definitions of this word that are, contradictory as they may be, in common use. There’s even a third possible definition, but it is not commonly used. I mention that in my website on this riddle.
The essence of the riddle lies within the world of human perception, not the laws of Physics. Solving it means unraveling how people perceive the situation of mirrors and reflections.
Thanks,
MikeO
Sorry MikeO, I disagree.
Once people understand the riddle, they are no longer confused - that's my experience.
In other words, once they understand the 'reality' as opposed to the 'illusion', the problem or riddle, or whatever you want to call it, disappears. Left to right reversal is an illusion, as I explain in my post, and back to front reversal (of the image) is the reality.
It's one of those phenomena, that once you understand it, you wonder how you ever got it wrong in the first place.
Regards, Paul.
If I can add an addendum to my last comment:
"The essence of the riddle lies within the world of human perception, not the laws of Physics."
You are correct in saying that the 'essence of the riddle lies in the world of human perception' but it's the laws of physics that solve the riddle.
"Solving it means unraveling how people perceive the situation of mirrors and reflections."
And that's the subject of my post, which I expound in some detail (including double reflections). So I think we agree on that too.
But having said that, my concluding sentence in my last comment still stands.
Regards, Paul.
HiPaul,
I’ll half agree with half of the concluding sentence in your second-to-last comment, where you wrote: “It's one of those phenomena, that once you understand it, you wonder how you ever got it wrong in the first place.”
I understand it, but I don’t wonder why I got it wrong in the first place.
Plus, I know why other people persist in getting it wrong even after numerous explanations of the Physics. It’s because their subconscious operational definition I labeled OpDeF-A, the one involving a rotation, is so much a reflex action.
This OpDeF-A happens in a flash and people hardly know they are doing it.
It stems from a principle that is usually sound, as I outlined my Postulate #1, at the beginning of my first post (22 January, 2010 11:40).
If you look around the Internet, this mirror puzzle is strewn about in a huge number of places. People are asking it all the time, over and over, and usually not getting satisfied. It’s been going on for centuries, and it will continue on into the next generation. It’s just a normal human reaction to the mirror setup.
As a result of all this I advocate a more gentle approach.
Sure, it’s dead wrong in the Physics, but in culture and in the normal human experience it should be recognized as a valid feeling. It’s the natural result of the normal desire to line up nearly identical objects, and face them in the same direction before comparing them for a subtle difference.
I guess I’m trying to explain WHY the illusion, and I’m predicting that it will persist as long as people’s DNA is the way it is.
Thanks Again,
MikeO
Hi MikeO,
You make some good points. As I said in the beginning, this post has generated more discussion than any other on my blog, and continues to do so, as you and I are currently demonstrating.
In my most recent post (on Quantum Entanglement) I highlighted a significant epistemological point: we only understand new knowledge when we integrate it into existing knowledge. The corollary to this is that the brain axiomatically attempts to understand anything new, or different, in the context of what it already knows or experiences.
This, of course, means that our initial appraisal of a phenomenon can be wrong and we often update it later when we are more familiar with it. In the case of mirror reflections, a stubborn first impression seems to persist. I think that’s the point you’re making, and you rightly point out the evidence of that on the Web. As I say in my post above, even philosophers of the calibre of Umberto Eco and Stephen Law get it wrong.
The point is that when we see something or someone facing us, the normal context is that they have to rotate around the vertical axis to do so, therefore left and right are literally reversed. But in a mirror left and right is not reversed, so we think it is, because it’s the opposite of what we ‘normally’ expect. But as I say in the post, we do the rotation, not the mirror. You have to rotate something to face the mirror and that includes ourselves. You have to think about it in order to understand it. If you use the normal heuristic (that we are familiar with, without mirror reflection) then it’s the opposite of what we see in a mirror. Because most people never analyse it, the perception persists.
I think this is the point that you are making, and, therefore, on that point, we agree.
Where I disagree, is the inference entailed in your following statement:
“when a human being thinks within the confines of OpDef-B, as it seems you are doing, then the perceived reversal is Front/Back, and similar sounding to the laws of Physics.”
Your use of terms: ‘perceived reversal’ and ‘similar sounding’; infers that they are just perceptions and not real. Front to back reversal is ‘real’, whereas the left right reversal is an ‘illusion’, yet you seem reluctant to acknowledge this. Front to back reversal is real because that is the laws of physics and not just ‘similar sounding’ to. This is an important distinction, because it distinguishes what is real from what is illusion.
Regards, Paul.
Hi Paul,
My references to “perceived reversal” and “similar sounding” were not meant to diminish the reality of the Physics involved, but to emphasize the fact that there are two worlds or realms involved in the riddle. One world is the science of optics and the other is the world of an observer’s perceptions. It’s quite possible I overemphasized this point, so please allow me to back up and try again.
I agree with you and the Physics, that the mirror does not reverse (as “reverse” is defined in that realm) anything laterally. I also agree that it’s with Front to Back, along the line perpendicular to the mirror, that a reversal does occur.
But the riddle usually pops up in minds totally unfamiliar with the Physics, and it’s there I want to focus. For minds familiar with science, the Physics can explain away the perceived (illusory) Left/Right reversal, but it can’t explain where it came from in the first place.
For someone who does not hear or follow the Physics, the perceived L/R reversal (as “reverse” is defined in that realm) is quite real and stubborn. Why is this? The Physics is silent. Even physicists who do see the proof of exclusive F/B reversal, somehow often drift back to the feeling that there is still a L/R reversal in there somewhere. Why is this?
At the risk of overdoing it again, I wax philosophical here by quoting Carl Jung: “We should not pretend to understand the world only by the intellect; we apprehend it just as much by feeling. Therefore the judgment of the intellect is, at best, only half of truth, and must, if it be honest, also come to an understanding of its inadequacy.”
***
It’s the origin of, and then the stubborn persistence of this perceived L/R reversal that I want to explain. A secondary goal would be seeing the asymmetry (why no Up/Down reversal perception occurs) explained.
For these reasons I usually try to pose the riddle thusly:
“Why does a mirror SEEM to reverse L/R, but not U/D?”
***
By far, I’ve found that the simplest key to the riddle is what I wrote before as a Postulate: When people want to compare two nearly identical objects, a common strategy is to line them up, facing them in the same direction.
This is contrary to the normal scientific approach, which is to examine things “as is” and not to disturb the scene before analysis. Seeing that these both of these strategies are common and useful, yet also contrary, is also key to deeply appreciating the riddle’s unraveling.
***
The solution in a nutshell is this:
People often desire to compare their image with themselves facing in the same direction. This they do mentally, but it’s difficult. It’s even harder to vocalize. When it’s successful, a rotation around a vertical axis is imagined, which is a L/R reversal. Rotations around an horizontal axis (provoking an U/D reversal perception) are hardly ever imagined because it’s too difficult to do them in actuality.
I go into all this in greater detail on my website at: http://mirror-reversal.proboards.com
Thanks,
MikeO
Hi Mike,
I agree with you that psychology is involved, which is why it’s one of the labels on the post. However, if you read the comments thread, you will see that more than one person has had a ‘Ha-ha’ experience from reading this. In other words, once they worked it out, it was obvious – just like finding a solution to a puzzle.
And the reason that they see it as a L-R reversal and not a U-D reversal is because we are horizontally symmetrical, as most animals are and we are vertically asymmetric in the extreme, as most animals are. And we turn about our horizontal axis to face something – we don’t stand on our heads to look in the opposite direction. It’s all very obvious when you give it just a modicum of thought.
“Why does a mirror SEEM to reverse L/R, but not U/D?”
Well this is the ‘mirror paradox’ precisely stated and the above post explains it in a lot of detail, that anyone, not just physics students, should be able to understand. In fact, did you notice that I make virtually no mention of optics? I explain it completely in terms of perception, which is exactly what you are talking about.
Talking about lining things up, in one of the above comments I suggest the following experiment. Stand in front of a large (bathroom) mirror, holding a small mirror so that you can see multiple reflections of yourself. Make a very significant mark on one side of your face, preferably near your eye, then hold the hand-held mirror near that eye and line it up so that you see multiple reflections of your face. You will notice that the mark always appears on the same side in all reflections, which happens to be the same side of your face that you made the mark. If you mark the right side of your face it appears on the right side in all the reflections and vice versa if you mark the left side. If mirrors reversed left to right it would appear left and right in alternative reflections. This is the best little experiment, that anyone can perform, to dispel the illusion of left-right reversal. The really important thing to understand, is that there is no left-right reversal, in exactly the same way that there is no vertical reversal.
I think we are absolutely in agreement on this and now all we are doing is trying to determine who between us is the most clever.
Regards, Paul.
Hi Paul,
Thank you for engaging me for so long on this topic. Although I have to plead guilty, along with most, to desiring the distinction of being the most clever, finding the most elegant answer to this age old problem is also on my wish list. Part of an elegant answer is brevity and being free of extraneous information.
Yes, we are in more and more agreement on the answer to this riddle. (1) The L/R reversal perception is due to human interference with the standard setup, bringing in a real or imagined rotation about the vertical axis. (2) The non-perception of an U/D reversal by the same humans is due to no envisioned rotation about a horizontal axis.
My last contribution here (best guess) to this would be to object to your use of the phrase “...because...we are vertically asymmetric in the extreme...” in reference to our preferred vertical axis rotations. I’ve seen this argument put forth before, that our bodily asymmetry causes us to avoid the illusory perception of U/D, but I don’t buy it.
It’s gravity, not our body shape, that causes us to avoid any imagined rotation about the horizontal. From an evolutionary point of view it would also be argued that gravity is the cause of our body shape. It’s gravity and the inconvenience of doing a hand-stand, or the lack of familiarity with doing hand-stands, and not a built in body symmetry bias.
As I’ve posted before, utilitarian convenience is what drives many of us to try and line up nearly identical objects to face the same direction before comparison and detection of subtle differences. This lining up is conveniently accomplished with a vertical axis rotation only.
My opinion is that the absence of a perceived (illusory) U/D reversal is a matter of gravitational inconvenience, and not body asymmetry.
MikeO
Well, Stephen Law agrees with you on that point (about gravity). He even claims that if we lived in a gravity-free environment the mirror paradox wouldn't exist, but I disagree with him on that last point, which is why I think it's the symmetry about the horizontal axis that's the key point. In other words, if we lived in a gravity-free environment I believe the illusion would persist, but I might be wrong
Having said that, it would appear that you are right that it's gravity that has created our vertical asymmetry and our preference for rotating around the vertical axis.
However, fish also have the same L/R symmetry U/D assymetry, and they live in a virtual gravity-free environment, which is interesting.
Regards, Paul.
Now this is getting funny. The more I think about it, the more I can agree with you on the body asymmetry point.
If our bodies were vertically symmetric, then doing hand stands wouldn’t be difficult!
Gravity, body shape, muscle strength, and preferred vertical axis rotations all seem pretty intertwined.
Oh gosh, now you’ve got me going, and I can’t resist this.
I’d say fish live with a perception of gravity similar to ours. The difference is they are supported from falling at all points, instead of just a few like us. Food sits in the bottom of their belly like in ours.
That said, I think the vertical asymmetry of all animals is (in addition to gravity) somewhat due to the need for a one-way flow in the alimentary canal.
I thought some more about the fish thing, and they certainly have a sense of up and down, but I expect it's mainly to do with light and perhaps current, and, to a lesser extent, temperature gradient. But how do really deep diving fish orientate themselves, like sperm whales, for example? Can sperm whales change their buoyancy like submarines? I don't know.
Birds, while flying, would use other means as well. A glider pilot once told me that he saw a bird get disoriented flying into a cloud and literally fell out of it upside down. Think about it: you never see a bird flying upside down.
Regards, Paul.
How do really deep diving fish orientate themselves? How about this? They look in a mirror to see which way is not reversed?
But seriously... just like food sits at the bottom of their bellies, fluids in their semicircular canals (or the fish equivalent) ought to do likewise.
Maybe I ought to stick to mirrors...
Another interesting point is why we are fooled. I content it is because we are bi-laterally symmetric and have an innate sense of 'left' vs 'right'. So, even though the image in the mirror is reversed fron vs back, we see it as a rotation because we identify (incorrectly) the 'left' and 'right' hand side of the image instinctively.
I'm not sure a cylindrically symmetric organism would even understand this 'paradox'.
Hi Thaddeus,
Thanks for your interest and your comment.
Despite being almost 2 years old, this post has generated more comment than any other on this blog.
Yes, I pretty much agree with you.
If you don't mind me correcting you, I think you mean 'contend' not 'content'. It's only one letter difference, but, typical of English, means something completely different.
Regards, Paul.
As a philosopher, Paul, what’s the difference in your mind between what is real and what is merely perceived? (22 Jan)
Like you I think, I want to believe that these Laws of Physics were put there by a Loving God to remind us of His Mercy, and not to be fooled by all those pesky ol’ perceptions.
Hi Jed,
So as not to mislead you, I'm not a professional philosopher and I'm not an academic, even though I've studied philosophy at tertiary level.
I find your question very Kantian, who famously talked about the thing-in-itself as something we may never know; we only know a perception of it.
Nature exists in layers, and, depending what layer you look at, you perceive something different. To give an extreme example, looking at the universe at the quantum mechanical level is quite different to looking at the world from our everyday level, which is different again to looking at the universe at a cosmological level.
If you take something like colour (an example that Kant used), it only exists in our minds, not out there. What exists out there is light being reflected off objects of various wavelengths that only become colour in some creature's mind.
Mathematics has given us more insight into what is 'real' than anything else, and the more I learn about mathematics the more fundamental it seems to be the nature of the universe.
As for God, like colour, 'he', 'she' or 'it' only exists in our minds. Whether there be anything external that creates that manifestation is purely conjecture. As for God being 'loving' and 'merciful', that's completely dependent on the beholder. Everyone's experience of 'God' is unique to that person.
Regards, Paul.
The philosophical discussion in the last two comments echoes the sentiment of the philosophy behind Hinduism.
Check this article out:
http://www.livemint.com/Articles/PrintArticle.aspx?artid=0C229B4E-0E85-11DF-8ABF-000B5DABF636
Cheers Paul
Like you I'm no philosopher, but I know a man who is. He told me that Kant is pretty impenetrable, so I asked him whether this is where we get the saying "Thats a load of Kant!" He thought I was being funny, although I was serious at the time.
I tried to find Hrj's hinduism article, but couldnt get it.
Regards Jed.
@Jed
Here's a better link to the Hinduism article:
http://bit.ly/9t36J6
~HRJ
Hi Paul,
An interesting Blog, but I'm not really convinced by your "reversal" argument.
If I see an object before me, for example, a figurine, which is rotating clockwise about a vertical axis, its motion is of course referred to the coordinates of the room in which I am sitting.
The same is true in the case of a reflected image which I take to be "rotating" (clockwise, say). This idea is only possible if the rotation is referred to a notionally unreversed set of coordinates in the mirror, ie the mirror-space is imagined simply to be an extension of my room.
If I thought one of the coordinates seen in the mirror was reversed, I should have to take the apparent rotation as an anticlockwise one despite appearances. Is this what you intended to convey?
Regards, Stu
Hi Stu,
Thanks for your comment.
The co-ordinates in a plane mirror are the same except the axis perpendicular to the mirror is reversed. In other words, if you extend the ‘z’ axis into the mirror (where x and y are the planar axes) it effectively becomes the negative of the axis in the ‘room’, whilst the x and y axes stay the same.
As for rotation, a watch or clock runs anti-clockwise in a mirror, whichever way you look at it – try it.
On the other hand, if you have a transparent clock so you can view it rotating from the back, then it will appear to rotate in the same direction in the mirror, if the image is the back of it in the mirror and the front is facing you in the room (it will appear to rotate clockwise) or vice versa if you turn it around. So it confirms that the mirror reverses everything back to front.
Angular momentum is the odd one, because it’s not reversed along the z axis but it’s reversed along the x and y axes. On the other hand, chirality (direction of a screw) is reversed in all directions.
Regards, Paul.
Thanks for your comments Paul, although I think I have understood your arguments pretty well and have yet to be charmed. I am already aware that a real-world clockwise motion appears to be anticlockwise if viewed in a mirror.
My intended point is that an interpretion or perception of the sense of any rotation depends on the assumpton of a non-reversed set of reference coordinates, whether reflected in a mirror or not. For example, if I am to judge the sense of rotation of clock hands seen in a mirror, this assumes a non-reversed standard of reference.
Conversely, a "z" axis genuinely reversed and extended into the mirror surely implies a reversed standard, according to which the seeming anticlockwise rotation (of clock hands) must now be taken to be clockwise - emphatically in spite of appearances. I am, of course, assuming that the signs of the coordinates provide a mathematical definition for direction and sense of rotation that is independant of subjective impressions.
Regards
Stu
Hi Stu,
For example, if I am to judge the sense of rotation of clock hands seen in a mirror, this assumes a non-reversed standard of reference.
I’m not sure what you mean by this. The ‘z’ axis is reversed – that’s just stating the obvious. The angular momentum vector is not reversed along the z axis, but the clock face is still reversed. In fact, it’s because the clock faced is reversed that the angular momentum is not. You have to turn the clock around to face the mirror, which also reverses the angular momentum vector (to you). The clock is actually going anti-clockwise to you (in the room) but you can’t see it.
Regards, Paul.
Hi Paul
My point concerns the question of what is meant by "reversal" especially of reflected images.
Do not overlook the fact that the sense of the rotation of an object is defined in relation to a reference system, taken to be non-inverted as standard.
If, as you claim, the space "in" a mirror is best represented by system with an inverted z coordinate, then rotations that are apparently reversed relative to their real-world counterpart, are not "in fact" reversed when judged against this inverted system.
The above comment waqs actually mine!
Stu
If you have a clock facing the mirror, the angular momentum vector is reversed in as much as: in the room it comes out of the back of the clock, but in the mirror it comes out of the front of the clock; using the convention that a clockwise rotation has the vector going away from you and an anti-clockwise rotation has the vector coming towards you.
So, even though the vector is not reversed, it is reversed relative to the object and its image, because the image is reversed but not the vector.
I hope that clarifies it for you.
I assume you do understand that the 'z' axis is reversed and that we are not arguing about that.
Regards, Paul.
Hmm
To be sure, I was uncertain about your introduction of the idea of "angular momentum" to support your arguments (it looks obfuscating), and I am not at all convinced that concepts related to dynamics are applicable to the logical or "kinematical" interpretation of images in mirrors.
Can you justify this? I will try to make/check your argument out, but so far it doesn't seem to make sense.
I assume you have understood my point by now, which is based on a straightforward interpretation of appearances rather than physics, for which I have no special love.
As for the "reversal" of the z-axis, I'm open minded, but am inclined to suppose that your presentation uses an oversimplified idea,(I can argue for this), and would be surprised to be persuaded otherwise.
Stu
Hi Stu,
I didn’t introduce angular momentum to confuse you. (I came across it in one of Richard Feynman’s many books, when, from memory, he was discussing symmetry.)
I introduced it because you keep insisting that rotation is not reversed in a mirror (if I haven’t misconstrued you). I assumed that you were referring to angular momentum, because it’s a direct consequence of rotation, and, in fact, it’s not reversed, but only along the z axis.
To me, the fact that the ‘z’ axis is reversed is self-evident – I’m simply stating the obvious – if that’s an oversimplification, I don’t apologise. A mirror reverses everything along the axis that is perpendicular to the plane of the mirror (as I state very early in my original post).
If you have another argument, as you allude to in your last comment, please deliver.
Regards, Paul.
Thanks Paul
Although I was not referring to angular momentum, this sounds quite interesting. My sister gave me a book by Feynman one Christmas so perhaps it is in there.
The point I have been making is straightforward enough, but may be unexpected, so I will try to explain it more carefully.
Using your example of the clock face seen reflected in a mirror, my argument proposes that in describing the image, we are faced with a choice between two mutually exclusive options:- either the hands are seen running anticlockwise or an axis is reversed. (not both)
Consider the following thought-experiment.
A standard set of orthogonal axes (x,y) is drawn on a tabletop, with signs (+ and -) marked. Place an acetate sheet over the diagram and copy it, but with the addition of a circle, centre the origin, and an arrow on its perimeter to indicate a clockwise sense (eg from the +ve y axis to the +ve x).
Flipping the acetate about the y axis transposes the signs on its x axis. Relative to the tabletop the arrow will now indicate a counter-clockwise (+ve y to -ve x) sense, but relative to the axes drawn on the acetate, the arrow remains non-reversed, (clockwise) by definition.
This argument translates directly to the problem of a clock hanging on the rerar wall (say) of a room, and viewed in a mirror. In order to read the motion of the reflected clock hands as "anticlockwise", we must envision the space and walls in the reflection as non-reversed.
In other words, it is necessary to conceive of the reflected space as though it represented a perfectly ordinary interior, not a "reversed" one; and in this sense it becomes difficult to suppose that a mirror image is really reversed as though by simple "laws" of some kind.
Regards
Stu
Hi Stu,
I think you may misperceive what is meant by clockwise and anti-(or counter)clockwise.
A clockwise motion always becomes anticlockwise when you look at it from the opposite direction, and vice versa. It’s like something going away from you becomes something coming towards you if you turn it around – it’s exactly the same principle.
If you take a screw, it combines both these ideas together. A ‘right-hand’ screw turned clockwise goes away from you, but if you reverse it, it turns anticlockwise and comes towards you. A ‘left-hand’ screw does the opposite: turns anti-clockwise going away from you and clockwise coming towards you. A mirror turns a right-hand screw into a left-hand screw and vice versa.
At the risk of confusing you, the angular momentum vector is exactly the same as a right-hand screw. Any object that is rotating has angular momentum, and, for the sake of consistency, the convention is that the direction of the vector is the same as the direction of a right-hand screw. When a mirror turns a right-hand screw into a left-hand screw, it either reverses the rotation or the direction of the screw. Along the x and y axes it reverses the rotation but not the direction of the screw. Along the z axis, it reverses the direction of the screw but not the rotation. No matter which axis you examine, a right-hand screw becomes a left-hand screw and vice versa. Angular momentum (by convention) always follows the right-hand-screw-rule. So, in a mirror, angular momentum is always the opposite direction to the direction of the screw. Along the x and y axes it is the opposite direction to what is in the room (as well as the direction of the screw in the mirror) and, along the z axis, it’s the same direction as in the room (but opposite the direction of the screw in the mirror).
If I’ve overloaded you with information, then ignore the last paragraph; the answer to the dilemma posed in your last comment is actually in the second paragraph of this comment.
This argument translates directly to the problem of a clock hanging on the rear wall (say) of a room, and viewed in a mirror. In order to read the motion of the reflected clock hands as "anticlockwise", we must envision the space and walls in the reflection as non-reversed.
The clock hanging on the wall behind you is no different to a sign on a wall behind you – they both appear back to front in the mirror. Why is this so? Because to observe them in the room, you need to turn around and face them. This gives the illusion that they are ‘turned around’ in the mirror, which they are not. What’s behind you in the room faces you in the mirror – it ‘inverts’ the room totally, but only along the one axis that is perpendicular to the mirror. And the reason that the clock appears anticlockwise and the sign appears back to front, is because the mirror DOES NOT turn them around but translates them exactly back to front.
I actually don’t expect you to get this, but it’s the best I can do.
Regards, Paul.
Hi Stu,
Just to clarify my opening statement in my last comment, in case you think I'm taking the micky.
A rotating object is only clockwise or anticlockwise relevant to an observer - to say it's one or the other relevant to itself doesn't actually make sense.
On the other hand, the angular momentum vector, which is always represented as being along the axis of rotation, remains the same relative to the object no matter what it's orientation. (I've probably confused you again but I hope not.)
Regards, Paul.
Hi Paul,
Thanks, but your reply is simply a patient reiteration of your earlier position - which I understand.
Your "space behind the mirror" is not inverted. The concept that it is simply does not exist - because it cannot. That is my point.
Laws of physical optics, (derived from exhaustive studies of reflections), do not tell us whether reflected images are ultimately intelligible.
Regards,
stu
Hi Stu,
If you understand my position, then there should be nothing to argue about.
On that basis, your last 2 paragraphs make no sense.
Regards, Paul.
Hi Stu,
Just to elaborate on my previous comment.
My point is that if you understood my position, then you would not have made those statements.
In the first statement you infer that there is no inversion of the image, but if inversion is the same as reversal, then that’s categorically incorrect and is plainly contradictory to my position. There is no ‘physical space’ behind the mirror (entirely irrelevant to the discussion) but there is an inverted image reflected by the mirror (completely relevant to the discussion).
Your second statement, that a mirror image is ultimately unintelligible, is just nonsense. If it’s unintelligible, then everything I’ve explained to you, in elaborate detail, must also be unintelligible.
I can only conclude then, that you don’t understand my position at all. In which case, I really can’t help you further.
Regards, Paul.
Thanks Paul
I dare say you are familiar with work by the artist MC Escher. One of his lithographs shows what at first appears to be a tower with a staircase around the top. Hooded figures seem to trudge "up and down" the steps in a never-ending loop; but the title, "Ascending and Descending" is deliberately misleading, for even though the monks might imagine that they are "really climbing steps", they labour under a paradoxical illusion. (There is an interesting entry about this on Wikipedia.)
Similarly, despite cursory appearances, the seeming inversion of the "z-axis" in a mirror must be an illusion as I have already explained. The key to this understanding lies in the last two paragraphs of my contribution of May 9th, which you may care to review.
Just as an inkjet printer decides where the ink should go when running off a copy of "Ascending and Descending", the laws of optics decide where light will go after it "rebounds" from a reflective surface. It is important to realise however, that neither the printer nor the optical laws take an interest in our "real-world" stories about any images which may result. (The laws of physics are off-guard). With this in mind, we are certainly discussing illusions Paul, and not scientific laws as you have mistakenly claimed.
You are correct on one matter however, in that there is still a point of your argument which confuses me; and that concerns the issue of what exactly you mean by "real" when you claim that we see a real reversal in a reflection. You have admitted that you do not suppose that (reflected) space exists behind the mirror, and yet you consider this to be a trivial point with no relevance to the discussion? Surely if the space depicted in a reflection is not real it cannot really be reversed?
Regards Stu
Hi Stu,
The mirror creates the ‘illusion’ of a space behind it, but the image it reflects is a ‘real’ phenomenon. If you can’t comprehend the difference, then there is no point in pursuing this discussion any further. What’s more, it’s an optical phenomenon that obeys the laws of physics, which entails the reversal or ‘inversion’ of the image – that’s a scientific fact.
If you disagree, then there is nothing I can say or do that will enlighten you. You said in an earlier comment that you have no love of physics. Perhaps that explains your incomprehension of the phenomenon.
I understand the basis of Escher’s ‘optical illusion’. In that image he exploits the mind’s inherent ability to interpret 3D images from a 2D projection using ‘perspective’ techniques. In other images (like Relativity and Convex and Concave) he exploits our mental ability to ‘flip’ a projected 3D image on a 2D surface from an ‘internal’ projection to an ‘external’ projection and vice versa. In all cases, these ‘illusions’ have nothing to do with mirror reflections.
Regards, Paul.
Hi Paul
Once it is acknowledged that there is no real space behind a reflection, it becomes obvious that the image seen in a mirror is just that - a picture subject to interpretation. It is conceivable that more than one reasonable interpretation is available should we wish to consider the possibilities.
On the other hand, to claim that your preferred theory is in itself a Law of Nature is certainly to slam the lid on this opportunity, as we have seen.
You are similarly mistaken to argue that the figures in Escher's picture are (supposed to be) subject to an optical illusion. (wikipedia)
Regards Stu
Hi Stu,
I’m no longer sure there is any point in even talking to you.
To quote Wikipedia on Escher’s picture:
'Ascending and Descending, in which lines of people ascend and descend stairs in an infinite loop, on a construction which is impossible to build and possible to draw only by taking advantage of quirks of perception and perspective.'
I believe I said something similar myself, though you exhibit an extraordinary ability to non-comprehend anything I say.
My so-called ‘preferred theory’ is not a preferred theory, it’s scientific fact. Your inability to appreciate that fact is your problem, not mine.
Images are open to interpretation but whether an image is inverted or not, is not open to interpretation – it's either inverted or it’s not – it’s either a fact or it’s not a fact. You want to deny a fact that’s your call.
If you don’t understand something have the courage to admit it. But please don’t pretend, to yourself or to me, that your incomprehension is my ignorance.
Regards, Paul.
Hi Paul
The figures in Escher's picture are meant to be under an illusion about the nature of their world, not an optical illusion as such. The artist makes a wry comment about this on the Wikipedia page that I have seen.
You do not seem to have followed my argument of May 10 either. Ah well, c'est la vie.
Stu
Hi Stu,
I’ll address both of your issues, even though you seem to create or debate issues that actually don’t exist.
In regard to Escher’s comment on his own image, he’s obviously talking about the characters in the image rather than the illusionary nature of the image.
I call his picture an ‘optical illusion’ because it illustrates a physical impossibility. We know it’s an illusion, because we know that, in reality, it simply can’t happen. And it’s called an ‘optical’ illusion because the illusion is purely visual.
Why you even argue over such self-evident trivia is beyond me. You seem to be trying to demonstrate or prove that the use of the term ‘optical illusion’ is somehow incorrect, simply because the author of the image didn’t use it. I shouldn’t have to explain what I mean by ‘optical illusion’, or why the image represents an optical illusion, when it’s bleeding obvious to anyone who looks at it.
What exactly in your 10 May comment (it’s 9 May on my blog because it follows Oz time) are you alluding to?
As far as I can tell, I’ve addressed all your arguments, but you give no indication that you’ve understood anything I’ve said.
In my last comment to this one, I pointed out why the inversion of the image is a fact and not a matter of interpretation – because it’s either one or the other – either true or false. Something that is either true or false can’t be open to interpretation; the person interpreting it is either right or wrong.
You give me the impression that you have no appreciation of science at all. But mirror reflection, despite your objections to the contrary, obeys natural laws.
Perhaps you don’t believe in laws of nature – you think it’s something I’ve appealed to just to win an argument. Science demonstrates, more than any other endeavour, that nature is full of uncertainties, but it also obeys rules or laws that are not amenable to human interpretation.
As Richard Feynman pointed out, when he was part of the panel investigating the Challenger failure, it’s very dangerous to believe that the laws of nature are somehow manmade or can be dictated to by human hubris.
Regards, Paul.
Just to pre-empt you,
As I said in an earlier comment, the fact that there is no space behind the mirror is irrelevant. We are not arguing about the space behind the mirror, we are arguing about the image that the mirror reflects. You seem to be arguing that, because there’s no physical space behind the mirror, there should be no image, but that’s a non-sequitur.
You say, effectively, because there’s nothing physical behind the mirror, then there is nothing to reverse. You’re suggesting, therefore, that we are arguing about ‘nothing’. But that’s not the case: we are arguing about the image that the mirror reflects, which is not nothing.
Then you say, any image is open to interpretation, which is true. But the inversion of an image is either true or false, so it’s not open to interpretation even if some people fail to see it. (Just turn a photo negative back to front and it’s inverted, it can’t be in-between or interpreted as not inverted.)
Just because you don’t understand something doesn’t make it un-real. No one understands quantum mechanics, yet no living scientist would say it’s not real.
You interpret it differently to me (apparently) which means your interpretation is wrong, because it’s false, meaning not true. When it comes to the laws of nature, there are right and wrong answers – it’s not a matter of opinion. If you don’t appreciate the difference between an opinion and a fact, and you can’t tell the difference, then there’s no point to this discussion at all.
I’ve done my best to explain an everyday common phenomenon, and you’ve done your best to not understand it. I can’t explain something to someone who really doesn’t want it explained.
Regards, Paul.
Hi Stu,
I offer you proof.
If you turn a photo negative back to front, is the image not reversed? It either is or it isn’t. Well, a mirror image is reversed exactly the same way, only it’s not turned around. If you don’t believe me, hold up a photo negative to face a mirror and the mirror reflection is exactly what you see in your hands, only you are looking at the back of the negative and the mirror reflection is, of course, the front, but the images are identical. Proof absolute, that mirrors reverse or ‘invert’ images.
Regards, Paul.
Now steady on Paul!
It is not because Escher didn't use the term "optical illusion" that you are incorrect. It is because in my own initial posting (May 22) I clearly said: "even though the monks might imagine that they are 'really climbing steps' they labour under a paradoxical illusion". It was my argument, not Escher's that you failed to read properly.
I went on to suggest that your "inversion of the z-axis" may be an illusion of a similar sort - the consequence of over familiarity with an "everyday" perception. I offered explanations for this too, but you still do not seem to have understood them, even though I write clearly.
My best guess is that this is also due to a hasty misreading on your part.
kind regards
Stu
It was my argument, not Escher's that you failed to read properly.
Your 22 May comment:
You are similarly mistaken to argue that the figures in Escher's picture are (supposed to be) subject to an optical illusion. (wikipedia)
So I went to Wikipedia. But whether it’s your argument or Escher’s own comment, they’re both irrelevant to the discussion. It’s obvious, when I refer to an optical illusion, I’m talking about what the observer of the picture sees (meaning you and me and anyone else) not the characters in the picture. They are not having an optical illusion, because they are in the picture. Red herrings are your speciality in argument.
I went on to suggest that your "inversion of the z-axis" may be an illusion of a similar sort - the consequence of over familiarity with an "everyday" perception.
Except, the “inversion of the z-axis” (or more generally expressed, the back to front reversal) is not an illusion. What’s an illusion is the perception of a left-right reversal. That’s what my entire post is about and seems to be your cognitive stumbling block.
Regards, Paul.
page
Hi Paul
Sorry for the delay in responding to your last posting, but my brain went to sleep on this topic for awhile.
Inasmuch as your argument differs from that of other thinkers, it can easily be shown to be false because you have misrepresented the concept of an inversion. In fact it seems that your idea is "not even wrong" - it's unintelligible!
The mathematical description of a three-dimensional image does not undergo an inversion if it merely rotates through ninety degrees, as you wrongly imply in your analysis. If the reflected image - of a playing-card, for example - seems to rotate through ninety degrees from the xy plane into a putative zy plane in the mirror, then by definition (of a rotation) the image does not undergo inversion.
From this, I conclude that it is very reasonable to regard images seen in the xy ("picture") plane of a reflection as inverted, and to wonder why one axis must always seem to be reversed preferentially. On the other hand, if we accept your argument that the image in the picture-plane is not "really" inverted, it follows that neither is its associated depth - a fact which corroborates the proposal by Stephen Law and others that in a sense, mirrors do not reverse anything.
The preceding point is not only true by definition, but is clearly supported by subjective experience and intuiton. Note that we never directly perceive depth, the distance between ourselves and what we see. Instead we infer depth in our world by visual cues drawn from the familiar perceptual plane with which we are actually confronted. If we do infer that "depth" is inverted, for example by a mirror, it is only because information in the picture-plane seems inverted - a genuinely obvious fact which, for some reason, you have been determined to resist.
It is not easy to uderstand your belief that depth-reversal contributes something to your solution in any case. Surely you are only saying that if (or since) there is no reversal in the xy plane of a reflection, it is meaningless even to claim that it looks reversed? How is inversion of depth expected to influence this simple idea?
If my assessment of your position has been unfair, I'm sure you will provide a powerful argument to the contrary.
Regards,
Stu
Hi Stu,
I don't think I've ever come across anyone more confused on this issue than you are.
For a start, I've made the point many times (both in the original post and in subsequent comments) that the mirror does not 'rotate' anything (there is a singular exception which I'll explain at the end*). In fact, that's the whole point of my entire dissertation.
To invert or reverse a real image or object, be it a photo or a page of writing or your very self, you must rotate it. What makes a mirror unique is that it inverts an image without rotating it at all.
If the back of an image faces a mirror, you see the back of it facing you. Hence it reverses the image back to front because the back that's facing away from you is facing you in the mirror. It is really dead simple.
As for depth, there is real optical depth in a mirror. Optical depth is not only determined by 'visual cues', but also by focal length. A lens, including the eye, must change focal length to focus on near and distant objects. If you stand up very close to a mirror you'll find that near and distant objects have different focal lengths exactly as they would if you were looking at them without the mirror. A flat 2-dimensional picture, including a movie projected onto a screen, doesn't do this. Everything in the picture or on the movie screen has the same focal length.
*A reflected image can be 'rotated' if you put 2 mirrors at a 90 degree angle. Then the image is rotated about the axis where the mirrors meet. I explain this in detail in my original post.
Regards, Paul.
Hi Paul,
My principal concern with your argument or dissertation is that it misrepresents the concept of an inversion. I endeavoured to explain this, but did not mean to imply that a mirror rotates anything. Either I do not write clearly or you do not read properly, Paul.
According to your dissertation, a facet of a reflected image (such as the face of a playing-card) is not inverted if it is understood to lie in a plane parallel to the reflective surface. On the other hand, when the (image of the) same facet is found to be perpendicular to this surface (at a different time, obviously), your theory states that it is inverted.
This (ie your) requirement contradicts accepted definitions, indicating that your position is false by accepted standards.
I have excluded reference to the term "rotation" in this version of my earlier discussion, because you are apt to suppose that it confuses me. If you spot any further evidence of confusion, I should be grateful if you would indicate where the muddled phrase or passage occurs.
However, in the hope that my point has been made sufficiently clearly in the second paragraph above, does it represent your argument correctly or not?
Regards Stu
Hi Stu,
I admit I struggle to understand what your problem is, because, for me, this is so simple.
I think the word inversion may be the wrong term, because I don't even understand how it confuses you.
I think reversal is a much better term: the image is reversed back to front, which means it is reversed along the axis that is perpendicular to the plane of the mirror.
That's it - nothing else needs to be said.
Regards, Paul.
Hi Stu,
Just to clarify.
I don't think your second paragraph relates to anything I've said.
But the third paragraph of my last comment is my dissertation in a nutshell.
If you don't understand that then you'll never understand how a mirror works.
Regards, Paul.
Hi Paul,
The third paragraph of your last posting is, by your own admission, your "dissertation in a nutshell", and it begins thus:
"To invert or reverse a real image or object ... you must rotate it."
It is you, not I who fails to distinguish between a rotation and an inversion. A three-dimensional object cannot be inverted by rotating it. Fortunately, I am not confused by your arguments, because I can see where your own confusion lies.
Kind regards,
Stu
Hi Stu,
If you look up a dictionary, 'invert' has 2 meanings or, to be more precise, 2 usages.
Its most common usage is to mean 'upside down', but it can also mean to 'reverse direction' or 'put in opposite order'. Now, obviously, a mirror doesn't turn anything upside down but it does reverse direction.
When I used the word invert or inversion I obviously meant reverse or reversal. To reverse an object's direction you must rotate it, which is equally obvious. But a mirror reverses the direction of an object without rotating it.
If you understand all this then neither of us is confused.
Just to clarify: my 'dissertation in a nutshell', that I refer to, is from a different comment to the one you reference, though it's equally correct.
You'll also find that this entire argument, that we've been having over the last few comments, is exactly the same as I addressed in my original post.
Regards, Paul.
Hi Stu again,
I'm sorry but you say: "A three-dimensional object cannot be inverted by rotating it."
Then how else do you 'invert' it?
Regards, Paul.
Hi Paul,
I'd rather you did't have to take my word on the answer to this query, Paul, but there is an excellent website called "Dr Math" where you can submit such queries to professionals.
You are familiar, however with the fact that a left-hand threaded screw cannot be superimposed over a right-hand thread, irresective of how it is rotated. This everyday example illustrates a more formal answer to your question.
Regards, Stu
Apologies. At first I only noticed the second of your responses above.
I admit that when I read your initial argument, I did assume that you were attributing "reversal" in a geometrical sense to the appearance of an image in the mirror. Since your reversal is represented neither by a rotation nor an inversion however, my assumption must be wrong. On the other hand, nor can you be referring only to the reversal of the direction of light as it "rebounds" from the surface of the mirror?
As far as I can see in any case, your position is summarized by the observation that the xy coordinates of the image are shared (identical) with those of the source. There is no (xy) reversal hence there can be no puzzle. Unless I have missed something, the stuff about depth is not strictly relevant - interesting though it may be in its own right.
I understand the paradox to be a question about the consistency of our everyday interpretation of images, which applies as well to reversed photos as to actual reflections(That is why I argue that physics is not significantly involved).
Regards, Stu
Hi Stu,
I'm not sure why you suddenly introduced right and left hand screws, again. We discussed them to death in an earlier correspondence.
You can invert a 3-dimensional object by rotating it about any axis, but nominally x,y or z through its centre. I don’t need to ask a ‘Dr Math’ to know that. And it has absolutely nothing to do with right and left hand screws.
Your assumption is correct: I am talking about the reversal of the image in the mirror. You are right: the x,y axes are not reversed. But the z axis is.
This is the elementary and fundamental point I've been making all along. The question is: do you get it?
Regards, Paul.
Hi Paul
You wanted me to explain the distinction between a rotation and an inversion (in three dimensions). I gave the everyday example of left and right threaded screws because you had used it yourself in an earlier posting and I had reason to believe it would be familiar to you. If it was "done to death" previously, you only have yourself to blame because I made no comment or reference to it myself.
Next you state: "You can invert a 3-dimensional object by rotating it about any axis, but nominally x,y or z through its centre. I don’t need to ask a ‘Dr Math’ to know that".
Wrong on every count. You do need Dr Math. In three dimensions, rotation is not the same thing as a reversal of one axis, (eg "x") through the origin (centre). I don't expect you to take my word for it though - ask an expert!
In truth I have been using the term "inversion" deliberately loosely, when "reflection", for the case of the mirror would have been more accurate. This was in order to avoid confusion with commonplace usage - the fact that mirrors "reflect".
Since you have at least made it clear that your dissertation concerns the (geometrical) properties attributed to mirror-images, please note that you also need to be precise about what kind of reversal is exhibited by (your) z-axis. Is it supposed to be reversed by rotation or by reflection (inversion)? There is no generic "reversal" in this case. If your answer is "inversion", you are misrepresenting the meaning of the term; explanation given above, twice.
Regards
Stu
Hi Stu,
You invert something (turn it upside down) by rotating it around the x or z axis by 180 degrees. By rotating it around the y axis you turn it back to front. You can demonstrate this by picking up any object that is lying on a table in front of you and simply doing it. An object that has a clear front, back, top and bottom is ideal. If you need an expert like Dr. Math to explain this to you then I believe you have a cognitive problem. I asked you how you could invert a 3 dimensional object without rotating it because I believe it’s impossible. If you are talking about a geometrical drawing then that is not a 3 dimensional object.
Reversal through one axis is exactly what happens in the case of a mirror reflection, as I’ve been saying repetitively since this argument began. You can’t reverse a 3D object along one axis without turning it inside out. So a 3D solid object can only be reversed along any axis by rotating it around another axis.
A mirror reflection reverses the image along the axis that is perpendicular to the plane of the mirror. A mirror reflection is unique because it reverses the image without rotating anything. I keep telling you this and you keep ignoring it. It’s the crux of this entire discussion. If you can’t understand it then just say so and leave me alone.
In regard to screws and threads, if you physically reverse a right hand screw, it’s still a right hand screw. Turning it around does not change it from a right hand screw to a left hand screw or vice versa. On the other hand, a mirror does turn a right hand screw into a left hand screw and vice versa, as I explained many posts ago.
Regards, Paul.
To be fair Stu, I think you’ve explained how a mirror reflection works only you’re unaware of it. It is the ‘inversion’ that you describe in your own comments, as opposed to a ‘rotation’. A 3D physical, solid object in the real world can only be reversed in direction (along any axis) by turning it around. A mirror reverses the direction of the object in its reflection (along one axis only) without rotation.
So maybe you understand it after all. I sincerely hope so.
Regards, Paul.
Hi Paul,
Yes, it seems we have been at cross-purposes. By "object" I mean a mathematical object, not a tangible, solid one, but clearly I was wrong to assume that you were already familiar with this concept (ie the concept of a mathematical object). In a similar way, a scientist may refer to a model, without meaning something you can physically touch, or to Kate Moss. By "invert" I did indeed mean "to turn inside out" in the sense you have described - in the sense of turning a mathematical object inside out. I used the term "inversion" to distinguish this (mathematical) transformation from "rotation" and from the commonplace usage "reflection" (in everyday speech everything you see in a mirror is a reflection) - I needed to refer to the mathematical transformation (inversion), specifically.
Furthermore, if you actually read my contribution of November 1, you will see that in your recent contribution, you have simply repeated the point I made about screw-threads. Without realising it, you have repeated the very argument I was trying to make to you. Once more, you didn't read it!
You have continued to presume that I don't know what I am talking about, and I suppose this is why, again and again, you don't bother to read, let alone to make sense of what I have written.
Because of this failure to engage, there seems to be little hope that you will become cognizant of the inadequacies of your own arguments, as I have familiarized myself with them.
Regards Stu
Hi Stu,
I think if we were talking face to face rather than remotely, then there would be less misunderstandings and frustration. I admit I didn’t think you understood what I was saying.
The fact is, that using your own description of inversion (reversal without rotation), you have the explanation of mirror reflection both in essence and in fact. I’m still unsure if you appreciate that or not.
Regards, Paul.
Hi Stu again,
You said earlier:
You wanted me to explain the distinction between a rotation and an inversion (in three dimensions). I gave the everyday example of left and right threaded screws because you had used it yourself in an earlier posting and I had reason to believe it would be familiar to you.
This is all very true: it's a good example because it explains that mirror reflection is 'inversion' and not rotation, exactly as I've been arguing all along.
You can't invert a left or right hand thread by rotating it but it does invert in a mirror reflection. In a mirror a right hand thread becomes a left hand thread and a left hand thread becomes right handed, so it becomes inverted by your own definition.
It seems we now agree that a mirror reflection 'inverts' an image, using the definition of inversion that you describe yourself (refer above quote).
Regards, Paul.
Thanks for your further comments, Paul. It's good to imagine that I may finally be getting to grips with this mirror thing; and I'm sure that if we met face to face it would be easier for you to get your message across. None of us likes to suppose that anything so simple as a mirror could keep us fooled for very long!
I see that I have contributed three argumentst against your one, and the first two look pretty watertight. It is good, in a way, that you did not understand these because it gave me the opportunity to propose a third,(17 Oct) which is the more interesting in that it may be slightly weaker:
If, as you claim, the (reflected) image in the "xy" plane is not reversed - and let's say I agree with you - then you have answered the puzzle outright. What can your supposed "real" reversal back to front contribute to this solution one way or another? It's not a trick question, is it?
Of course, I could put words into your mouth, and try to answer the question for you; but that would be unfair. (You may be even cleverer than I think).
Merry Xmas,
Stu
Hi Stu, and a merry Christmas to you too.
If, as you claim, the (reflected) image in the "xy" plane is not reversed - and let's say I agree with you - then you have answered the puzzle outright. What can your supposed "real" reversal back to front contribute to this solution one way or another? It's not a trick question, is it?
I think the issue is that the mirror is 2 dimensional but the world we live in is 3 dimensional. If you take the x and y axis to represent the plane of the mirror then the z axis represents the third dimension that is perpendicular to the mirror. It's along this third dimension that the image is reversed.
If that's the answer you were expecting then you'd be right.
Regards, Paul.
Hi Paul
No, your reply was certainly better than anything I could have come up with.
However, although you have described what you think mirrors do, you have not really explained how this relates to the original puzzle. In particular, we may justifiably ask, if the reflected image is not seen to be reversed in the plane of the glass, why do so many people believe mistakenly that it is?
You have not said so, but I presume you agree with me, tacitly, that the further question as to whether the z-axis is reversed plays no real part in the solution.
Cheers
Stu
Addendum:
A rubber stamp, for example, meets and matches its printed image point-for-point. However, it is generally agreed that the print is an inverted representation of the face of the stamp.
Similarly a reflection in a mirror is understood to present an image of the source that is inverted in the plane of the glass. (ie. in the "xy" plane)
Stu
Hi Stu,
You're the No100 comment on this post. The only post to reach that figure.
The image of a stamp or any mould for that matter, like the old vinyl records, are exactly the same as if the image produced by the stamp was transparent and you could view it from the other side.
So you are correct: a mould or stamp is an inverted or reversed image, the same as a mirror image; because it's the image viewed back-to-front.
In other words, if you could look at a stamp from the other side it would appear like the image it creates on a surface. It's why your hands are mirror images of each other - because the right hand is your left hand back-to-front - just put them together and they are like the stamp and its image at the time you stamp it.
So the image is still turned inside out, not reversed left to right or up to down in the x,y plane. It may be a 2 dimensional image, but it's reversed by looking at it from the opposite direction along the z axis.
Regards, Paul.
Fair enough, Paul, but does the image look reversed? ("So you are correct: a mould or stamp is an inverted or reversed image, the same as a mirror image; because it's the image viewed back-to-front").
Or not? ("So the image is ... not reversed left to right or up to down in the x,y plane").
Regards, Stu
Regards, Stu
If you hold a stamp up to a mirror you should get the same image as it stamps on a piece of paper (I don't have a stamp, so I can't check this for myself). And if the stamp was transparent you would see the same image (from the back). Therefore it's reversed back to front. It's only when you turn the stamp around to face you that it appears reversed left-to-right or top-to-bottom, depending on which way you turn it.
To view the stamp you have to turn it to face you. So if you turn it left to right (to face you) it will appear reversed left to right, and if you turn it upside down then it will appear reversed top to bottom. It's why writing (always) appears reversed left to right in a mirror because one (always) turns it left to right to face the mirror - I point this out in my original post.
If you think about it, the stamp only works because every point on it is reversed in the direction perpendicular to the plane of the stamp. Every point lines up just like your 2 hands do when you place them together. Your 2 hands are reversed back-to-front. If you put one hand over the other they don't line up unless you have the back of one and the front of the other.
I can't explain it any better than that. If you're still questioning me about it after all this time, I really don't think you'll ever understand it.
Regards, Paul.
Hi,
I believe the mirror paradox is deeper than most might imagine. Saying that a mirror works by reversing the axis perpendicular to its surface is fine. However, I do not see that it identifies the essence of the problem.
Changing the x,y,z axes corresponds to a change in reference frame. Most of the time, that doesn't pose a problem, as we can get from one frame to another by physically rotating ourselves, in a 'continuous' manner. However, the mirror inversion results in a reference frame that cannot be achieved continuously from our reference frame (this can be explained mathematically). Something strange is indeed going on in the reflected world, which is quite difficult to reconcile with our own.
This 'something' is a change of orientation. There is a notion of orientation which is intrinsic to the 3-dimensional space that we reside in. There are two choices of orientation, corresponding to the ambiguity in defining left/right, or indeed clockwise/anticlockwise. The mirror effects a change of orientation, albeit in a 'virtual space', which is disconnected (even in a mathematical sense) from ours, and which we may not access physically. It does offer us a glimpse of the other side though, which is fascinating.
Really, all this is most easily seen in a mathematical treatment, and the paradox to me, is whether the mathematical disconnect is truly a physical one - i.e., whether we are really forbidden from the other side.
Hi agtc,
Of course there is nothing physical on the other side of a mirror, but the optical depth that a mirror provides is just as real as the optical depth we have in reality. In other words, there is a focal depth in a mirror where your eyes change focus the same as they would in open space.
This is what makes a mirror special. It's why you can make a room appear twice its size just by putting a mirrored wall at one end. If you painted a mural it would not have the same effect because the focal length for the mural is the same as the wall it is painted on. But, for a mirror, the focal length changes according to how far something is away from the reflected surface.
Regards, Paul.
Hi Paul,
Of course what you have said is right, although I might argue about the precision of your choice of words. But I probably did not make my point too clear in my previous post.
You don't really need an actual mirror to imagine your reflected-self. The real mystery is why we have difficulty reconciling ourselves with our reflected version. I would argue that reflection is a discrete symmetry, as opposed to rotations, and there are grounds to argue that left/right "handedness" cannot be exchanged in our 'physical' world, necessitating the fascinating device called a mirror to effect such an exchange.
In any case, the issue of mirror reflection is, to me, certainly not something to be trivialized, even if one can identify the mechanism by which a mirror works. As a philosopher, I think you would understand what I am driving at.
Hi agtc,
Your comment didn't publish - it went to spam - possibly because of your moniker, but I don't really know. Your first comment published okay.
I agree with you that mirror reflection shouldn't be trivialised, which is why I wrote 2,000 words on the subject.
The real mystery is why we have difficulty reconciling ourselves with our reflected version.
The reason is that the mirror-reflected self is not how others see you. To see that you need 2 mirrors joined at a 90 degree vertical angle.
Regards, Paul.
Sorry atgc - I got your moniker wrong.
Regards, Paul.
Hi Paul,
It is well known that two reflections bring you back to the same handedness. (In some sense, back to this part of the universe, which you originally belonged to...) You don't necessarily have to place them 90 degrees apart.
You could even imagine an odd shaped mirror, one for which you cannot nail down a particular axis which is flipped. But what you do know is that a reflection took place, which changed the attribute of "handedness". This is the crucial property of a mirror reflection, not the swapping of an axis direction, which is particular to a flat mirror (and effects the change in handedness in that case).
Again, I wish to stress that this may be related to the reason why physicists consider continuous symmetries to be part of physical law, rather than discrete symmetries. The reflected version of the universe is in a highly non-trivial sense, different from ours. The doubly-reflected one, however, is more familiar to us, perhaps more "real" even, like a photograph or a video.
Hi atgc,
I'm not sure if you've read my original post, but I explain the difference (in some detail) between double-reflections and the reflection through a 90 degree axis.
It's only in the latter case that you can look yourself in the eye and see yourself as others see you.
The difference is that a double reflection caused by 2 mirrors is normally a front-to-back reversal twice over.
In the case of 2 mirrors aligned at 90 degree along a vertical axis, the image is literally reversed left to right - rotated about the vertical axis - so the effect appears the same, even though the mechanism is different.
I explain this in the original post. Going by your comment, I'm not sure if you've read it.
Regards, Paul.
Hi Paul,
I'm really not sure what the emphasis on two 90 degree-separated mirrors does for our understanding of the paradox. For one, it suggests a strange conspiracy between plane mirrors to have a different mechanism kick in when they are 90 degrees apart (quote " even though the mechanism is different").
Secondly, it is not clear (to me at least) what you mean by "the image is literally reversed left to right - rotated about the vertical axis". A rotation is a rotation, pure and simple. In this case, it's just a 180 degree rotation about the vertical axis, which is, of course, what you would do in order to see yourself from the front. You don't normally say that you have reversed left/right when you have turned half a complete round. 90 degrees is not crucial. Any combination of two reflections can be shown to be a rotation.
Perhaps you mean a different thing from me when you refer to left/right. When you say left, do you refer to a fixed direction in space, or do you refer to "handedness" as in the case of a screw? For me, I am always speaking about handedness, and indeed, in the context of the mirror paradox, this is the natural definition.
I maintain that a (single) mirror reflection does reverse left/right (handedness), which, contrary to many expositions, is not a direction vector like up/down or front/back. (It is really a choice between the two possible orthogonal directions available once "up" and "front" have been defined.)
Your "inversion about the z-axis" is one popular way of explaining how left/right gets switched in a reflection, but I prefer to think in a more general sense --- that a mirror (curved/ bumpy/concave/convex) reflection turns a left-handed universe into a right-handed one.
Hi atgc,
If you read my post I've explained everything that you raise in your comment.
A mirror does not reverse left to right (it always reverses back to front) but a double reflection where 2 mirrors join at 90 degrees does exactly that. It's because the image of one mirror bounces onto the other mirror and back to you directly. The 2 images swap over each other so that they literally swap left to right. This only happens when the mirrors are joined at 90 degrees, hence the mechanism is different. If you read my post the difference is explained in detail (with examples).
The image is the same as rotating about a vertical axis, because swapping from left to right is the same as a rotation through 180 degrees. When you turn to face someone you rotate through 180 degrees and your left side is on their right and your right side is on their left. If you look into the corner of 2 mirrors that are joined vertically at 90 degrees you will see your left side on your right and your right side on your left, exactly the same as when someone else looks at you.
A single reflection does not change handedness - it takes 2 reflections. A double reflection (as I explained in my last comment and the original post) reverses back to front twice, which is why you need 2 mirrors to see the back of your head. Two mirrors joined at 90 degrees is a special case, because you can look directly at yourself (in the eye) and have your handedness reversed, just like other people see you. Any other combination of 2 mirrors won't give you that, even though other combinations do reverse handedness as you point out.
This is exactly the point I've made in my last 2 comments and my original post.
Regards, Paul.
Just to clarify my last point. Other combinations do reverse handedness but only if the mirrors are at an angle. Two mirrors facing each other (in parallel) won't reverse handedness, as I pointed out in a much earlier comment in a dialogue with someone else (back in June 2009).
Regards, Paul.
Hi Paul,
My earlier comment has disappeared mysteriously...
Well, it is now evident to me why we disagree. When I speak of left/right, I refer precisely and unambiguously to a choice of orientation http://en.wikipedia.org/wiki/Orientation_(mathematics). This is exactly the same notion as handedness, clockwise/anticlockwise, chirality, etc. For us humans, left is often misconceived as a direction in space (a vector versus a pseudovector).
I tried to follow your arguments using different definitions of what left and right might mean. Unfortunately, I really do not know what precise meaning you attach to left/right.
For example, you say "A single reflection does not change handedness". But in June 2009, you said "So, even though a screw or a molecule or whatever has the same chirality, whichever way you look at it, it reverses when you look at it in a mirror". So, does a single reflection change handedness/chirality or not?
"If you look into the corner of 2 mirrors that are joined vertically at 90 degrees you will see your left side on your right and your right side on your left". In this case, what, precisely, makes you certain which side of your double reflection is the "left side"? You are allowing the words left/right to do double-duty -- Firstly, as a direction in space (like east/west); secondly, as a chiral property (left-hand <--> left-handed screw, or alternatively, left = front cross product up). For me, the unambiguous way to tell a "left hand" from a "right hand" is to see which way the thumb points in relation to the fingers -- a chiral property. That is how you "know", in your double reflection, that your "left-hand" is on the "right side". Therefore, in my terminology, a double reflection does NOT effect a left/right reversal. It is simply a rotation. Your real-life left hand waving corresponds to your double-reflection's left hand waving.
The usual explanation that a mirror reverses front-back is just one example of a mirror transformation. I wanted, as far as possible, not to invoke mathematics into the discussion, but it may prove unavoidable. As I have mentioned before, things are much clearer in a mathematical language, when we consider the structure of the group of isometries of 3-dimensional space. http://en.wikipedia.org/wiki/Orthogonal_group
Hi atgc,
Your comments keep going to spam. I tried to recover them both but I deleted them by mistake (it wasn't intentional).
Handedness and chirality are not synonymous because chirality involves 3 dimensions. I shouldn't have to give a definition of handedness because it's bloody obvious to any ordinary person what I mean by right and left. But for your benefit, I will say that right means something on the right side and left means something on the left side. Your right hand is normally on your right side and your left hand is normally on your left side. If handedness is reversed then your right hand is seen on your left side and your left hand is seen on your right side. It's the same when someone else is facing you - they see your left hand on your right side and vice versa.
We call chirality right or left handed by convention and it's synonymous with a right and left hand screw. Chirality involves a third dimension as well as rotation along an axis. If you curl your fingers in a half-fist and point out your thumb, then the fingers give the direction of rotation and the thumb gives the direction along the third axis. Because we have a right and left hand this simple exercise determines whether a screw or chirality is left-handed or right-handed. I think it's equally bloody obvious that when I was talking about right and left handedness in my recent comments I wasn't talking about chirality.
If you don't understand this difference then I really can't help you any further and this conversation is a waste of our time.
Regards, Paul.
I notice that Wikipedia does refer to 2 dimensional chirality, but I've always associated chirality with spirals, screws and helixes (also referenced in Wikipedia). The so-called 'right-hand rule' that Wikipedia describes is different to the one I described for 'screws' and the like, because it specifically refers to electromagnetic radiation, but it doesn't change the convention of right and left hand screw. The description I gave and learnt at school applies specifically to electromagnetism in a solenoid but also to right and left hand threads.
The major difference is that right and left handedness can be reversed by rotation around a vertical axis, whereas chirality cannot be reversed by rotation around any axis. I hope this clarifies any misunderstanding.
Regards, Paul.
Hi Paul,
The difficulty in defining left/right is notorious. I have chosen what I feel to be the least ambiguous notion, which is chirality. (Since you seem to understand the word "handedness" differently from me, I shall cease using that word.) I believe that this is the more precise notion that is used in science and mathematics, but perhaps I am misguided.
On the other hand, you would have me imagine that "left/right" is "... obvious", and that "your right hand is normally on your right side and your left hand is normally on your left side." I presume that by your definition, "left/right" corresponds to the line joining your two shoulders. If you now stand with your side facing the mirror, then by your explanations, it is the left/right direction that is switched. Or is it not? Please clarify.
Perhaps we would agree that the direction perpendicular to the plane mirror surface is flipped. This is one way to effect a chirality change, and indeed, Feynman himself explained the problem in this way. As I have said earlier, I prefer to think of a mirror reflection as causing a left/right chirality change, because this notion is unambiguous, and does not depend on how the mirror and subject are arranged. (Note that there is an ambiguity in choosing which chirality is "left", but no ambiguity in saying that the "chirality has changed".)
Also, I do not see the relevance of mentioning 2-dimensional chirality, or indeed, how electromagnetic radiation came into the discussion. Perhaps you meant electromagnetism or electromagnetic induction, in which case, it is fruitful to think about how Maxwell's Laws fare under a mirror reflection. I have a hard time imagining how electrons could find it 'obvious' which direction in space was left.
I also noticed that you proposed an experiment involving a shaving mirror to prove your point. Let me point out that what you see on the main mirror are only the odd-numbered reflections, since you have to count the reflection from the shaving mirror as well. Successive images in the main mirror differ by two reflections (and so automatically have the same chirality). Then again, you also mentioned that a left/right reversal occurs only when two mirrors are aligned at 90 degrees. So I'm not sure what this experiment proves, since there is no occasion for your hypothesis to fail. What happens when two mirrors are placed 89 degrees apart? 89.9? The original chirality is restored under two reflections, not necessarily at right angles. Can the same be said for your "left/right reversal?"
Quote: "If you don't understand this difference then I really can't help you any further and this conversation is a waste of our time."
This is a pity, since I had hoped to bring up a discussion about the philosophy of physics, and I felt that the mirror paradox, like any good paradox, raises rather deep questions about our understanding of the world.
Your comments are still going to spam which is a nuisance.
Perhaps we would agree that the direction perpendicular to the plane mirror surface is flipped.
Yes, this is exactly what I argue continuously in my post and in my comments, which begs the question: what are we arguing about?
If you look at a mirror reflection of yourself your right shoulder is on the right hand side and your left hand shoulder is still on your left side (to you facing the mirror). So right and left handedness is not reversed in a mirror reflection. I also make this very point in my post.
You've picked up on the experiment I referred to in an earlier comment nearly 2 years ago. You raise a valid point: you are looking at every odd reflection (not the even ones), which would explain why all the reflections keep the same handedness as yourself. So I admit you've enlightened me.
If the mirrors are not aligned at 90 degrees then you will still get a right-left hand switch (in the second reflection). But at exactly 90 degrees you can look yourself directly in the eye as others see you. This is the point I've been making since we started this conversation. If you don't believe me, then just try it. Less than 90 degrees your eyes move off centre and greater than 90 degrees the image disappears because the reflected light travels at an angle that reflects away from the opposite mirror.
I'm happy to leave 2 dimensional chirality and electromagnetism out of the discussion because it confuses things (I only raised it because it's in Wikipedia and I thought it may be confusing you).
My last paragraph of my last comment still stands, where chirality specifically applies to helixes, screws and spirals. My understanding of chirality is in regard to molecular structure, where the structure is 3 dimensional and is helical.
Regards, Paul.
Hi Paul,
I hope this doesn't go to spam this time around.
Well, so clearly we diverge in our opinion of what we think left/right should mean.
Let me first clarify that chirality is not a property restricted to screws/molecules and the like. It is a geometrical feature of the spatial manifold itself, with two discrete options. I imagine that humans have the dexterity to carry out some kind of screw-like motion.
I speak of chirality because I think it relates most closely to what a mirror reflection entails.
Quote: "If you look at a mirror reflection of yourself your right shoulder is on the right hand side ..."
This is of course true if you choose to use the notion of left/right as directions in space which you as the subject facing the mirror pre-defined.
But I would like to return to the case where you face the mirror side-on. What do you mean by left/right now? If you say that left/right is not reversed in the mirror, I presume you mean left/right to refer to directions in space parallel to the mirror surface (i.e. the same as in the earlier case where you faced the mirror straight-on). However, what sense can you then attach to which of your two hands is the left hand and which is the right hand? To avoid double duty of the words "left" and "right", you will now call these the "front" and "back" hands instead. Or do I not understand your definitions? If you could provide a consistent definition of left/right once and for all, I believe I will not be so confused.
There is a very strong reason why we somehow sense that what you call the "right-hand" of the image is really a left-hand -- it has the opposite chirality. You cannot shake hands with your mirror image. In my opinion, this is also the reason why some people like me insist that a left-right reversal must have occured, and why I emphasize that left/right are better regarded as chiralities rather than spatial directions.
Hi atgc,
I think this conversation is in danger of becoming nonsensical. You’re the first person I’ve ever had a dialogue with, who doesn’t understand what I mean by left and right.
As I said in an earlier comment, chirality and handedness are not synonymous – this is a very important point – so there should be absolutely no confusion or confounding one with the other.
I know that chirality applies to other things other than molecules, but that’s what the word is usually associated with – in particular the amino acids, L-alanine and D-alanine, one of which is predominant in DNA. When I think about chirality, I always imagine a helix because it incorporates all the right attributes and handedness is completely irrelevant, even though people refer to left and right-handed chirality (out of convention).
As I said in another comment, you can change handedness (relevant to an observer) by rotating an object through 180 degrees around its vertical axis. On the other hand, chirality won’t change no matter what axis you rotate it about.
When it comes to mirrors, chirality is always reversed and handedness isn’t and that’s because the mirror doesn’t rotate the object about any axis. As you said yourself, it just reverses everything along the axis perpendicular to the plane of the mirror.
If you read the last 4 paragraphs you’ll find they entail an inherent consistency with each other. Mirrors do not reverse handedness but they reverse chirality. You can reverse handedness through rotation but you can’t reverse chirality through rotation. A mirror doesn’t rotate an object about any axis. Handedness and chirality are completely different things.
Regards, Paul.
Hi Paul,
Quote: "You’re the first person I’ve ever had a dialogue with, who doesn’t understand what I mean by left and right."
I will take this as a compliment. I do hope you have seriously thought about the precise definition of left and right. I am still waiting for you to provide one, despite your claims that it is "******* obvious".
As far as I know in my scientific work, "handedness" is synonymous with "chirality". But as I mentioned a few posts ago, you believe otherwise, so I have decided not to use the word "handedness" anymore.
Let me assure you that the terms "left-chirality" and "right-chirality" are completely natural. They arise from the chiral nature of your very own hands, unless of course you have special achiral hands. If you do not believe me, please read the second paragraph of http://en.wikipedia.org/wiki/Chirality_(chemistry), which I suspect, you might have already visited.
My basic point has always been the same. The characteristic feature of a mirror is that it changes chirality -- i.e., a "left-type hand" becomes a "right-type hand". I do not feel that a discussion which neglects this can do the mirror paradox any justice.
Would you at least agree with me that your right-type hand, holding a right-handed screw, turns, in a mirror, into a left-type hand holding a left-handed screw, and that this is independent of how you align yourself with the mirror?
You had a lengthy discussion with Glibeaux regarding two mirrors placed 90 degrees apart, where you replied that "What you don't understand is that what is left to right in one mirror is the back to front in the other mirror. It is entirely consistent."
So I take it that you refer to left/right as the left/right "of a mirror", rather than that of the subject. In particular, the left/right of the mirror in your room and the one in my room are different and have no relationship with one another, or indeed, with me and you. If this is the subjective notion of left/right that you wish to use, whose burden is carried by the mirror but not the subject, then I am glad that there is obviously no paradox for you. For you have been humble enough not to impose your intuitive sense of left/right on the mirror and avoid confusion.
I am not as enlightened. As you probably suspect, I am terribly confused. For my benefit, I really hope that you could state once and for all, what you mean by the left/right. Are they two fixed directions in the background like east/west? Or does it follow the subject around as he turns and moves? Or does it follow the mirror in question? Or some combination of the three?
As I explained to Glibeaux, the reason that the left hand is the mirror image of the right hand is because the left hand is effectively identical to the right hand back to front and vice versa, and mirrors reverse back to front, not left to right. To appreciate this you only have to put your 2 hands together.
Wikipedia says our right and left hands are 'chiral' for this reason - they have a mirror symmetry, which is their definition of chirality. And this is consistent with my previous comment, because I state specifically that chirality is reversed in a mirror (as does Wikipedia's definition). But it doesn't mean that mirrors reverse left to right - quite the contrary - and that's what I mean when I refer to handedness (literally right and left directions wrt an observer).
I can now see your confusion: because left and right hands mirror reverse yet right and left do not reverse. Your right hand in the mirror is still on your right side because if you lift it you will see it's still on your right side as you face the mirror. But it looks like your left hand because your left hand is your right hand back to front.
Regards, Paul.
Hi Paul,
I'm afraid that until you declare once and for all what you mean by the left/right "directions" when you make the statement "left/right are not reversed in a mirror", I will still be utterly confused by your claim.
Quote "...and that's what I mean when I refer to handedness (literally right and left directions wrt an observer)"
So over here, your left/right are directions attached to an observer looking into a mirror. But your response to Glibeaux's example indicates that you refer to left/right as directions associated with a mirror (otherwise one mirror's front/back will become another mirror's left/right and you will have left/right reversal in one but not the other). Now, which is it? Directions defined by a mirror, or by an observer?
Quote: "Your right hand in the mirror is still on your right side because if you lift it you will see it's still on your right side as you face the mirror."
But then you say,
"...your left hand is your right hand back to front."
So is the mirror image of the hand which you lifted the right hand or the left hand? Your first sentence says says it's the right hand, because it's on your right side. Your second sentence says it's a left hand, because the mirror image (effected by swapping back/font) of a right hand is a left hand. An on top of it all, you also say that left/right directions (referred to what, exactly?) are not swapped.
Is it any wonder why you have left me confused?
In contrast, I offered instead to look at left/right as opposite chiralities which you agree is swapped by a mirror, unambiguously.
Hi atgc,
If you hold up your right hand so it faces a mirror you’ll notice that your thumb is on the left and your fingers are on the right. If you look in the mirror you’ll see that your thumb is on the left and your fingers are on the right. And I’m certain you can understand that without me defining what I mean by left and right. On the other hand, you’ll see the back of your hand in the real world and the front of your hand in the mirror. So the mirror reverses your hand back to front, not left to right.
I admit that Wikipedia confounding handedness with chirality does confuse the issue. But I can’t think of a better term than handedness to describe left and right succinctly. I could have said horizontal orientation, but its clumsy. But handedness, in the context that I used it, simple means on the left or on the right. The point is, by this definition, handedness is not reversed in a mirror and chirality is, therefore handedness and chirality are different things (according to this definition).
The only reason that we think a mirror reverses left to right is because the human body is almost completely symmetrical about its central vertical axis face-on, even down to our fingers and toes. I make this point in the original post as well.
Regards, Paul.
Hi Paul,
I think we both already agree on the following two statements:
1. A (plane) mirror reverses the directions along it's axis.
2. Chiralities are reversed in a mirror. Thus a left(chiral) hand turns into a right hand.
Here is where we really diverge in opinion. You said
"...I’m certain you can understand that without me defining what I mean by left and right".
To which I have pointed out repeatedly and with examples, that it is crucial to state exactly what you mean. If you wish to discuss left/right as spatial directions, you must at least specify which reference frame you are in (mirror, or subject). I have already used Glibeaux's example to illustrate that your claim that "right/left directions are not reversed in a mirror" can only be consistent if left/right are notions carried by a mirror, which have no a priori relation to the subject's own choice of left/right directions. Furthermore, this can only be a local notion, not a global one (consider a concave mirror, a spoon for example, or see a common youtube video like http://www.youtube.com/watch?v=ay_M1jZ41xE&feature=related. There, you will see that sometimes left/right as you "defined" it, is not switched if you are near the mirror, but switched for me, thousands of miles away. No double reflection funny business had occured. The more consistent way to identify a left/right hand is to look at the chirality.)
Do you agree on this observation?
I am disappointed that despite spending a good amount of time crafting my posts and phrasing questions, you have refused to address my queries directly. In particular, you still have not explained clearly your notion of left/right, turning instead to the claim that it is "obvious", which as I have pointed out countless times, is crucial for understanding your claims.
According to Wikipedia, mirror reflection is how chirality is defined, so to say that chirality is reversed by a mirror is simply stating a definition. On the other hand, a left and right position of an object can only be reversed by rotation. By left and right position, in reference to a mirror, I mean the opposite ends of an object in a horizontal direction parallel to the mirror, which would be the x axis.
A mirror does not rotate an object, which is consistent with the definition of chirality because chirality is not reversed by rotation. So left and right reversal is not caused by a mirror, therefore left and right reversal is not synonymous with chirality by the very definition of chirality. If I substitute my use of ‘handedness’ with ‘left right reversal’ then all my statements are consistent. This equally applies to the human body and the human hand, which are not left right reversed by a mirror.
Regards, Paul.
I won't address your second paragraph, because it's obvious we are not talking about concave mirrors but plane mirrors.
Regards, Paul.
Hi Paul,
I find it strange that you dismiss many things as "obvious", as I had the impression that you would like to discuss philosophy.
Well, now that you have clarified that you take left/right to be horizontal directions referred to a mirror (let's assume the simplest case, where we are on earth so we both know what horizontal means, and if you wish, a plane mirror), we can actually get some useful discussion going.
So your notion of left/right only has meaning for a particular mirror. This is because for a second mirror placed 90 degrees apart from the first one, the notions of left/right do not coincide. Your view of left/right must be a local one. In fact, a mirror's left/right sides do not necessarily coincide with an observer's left and right sides (as directions). In your language, David Beckham's famous right foot becomes a left foot if he turns 180 degrees, and is neither one nor the other if he turns 90 degrees ---well, because the mirror's impression of left/right is more natural than Beckham's whimsical views...
Jokes aside, let us now address the mirror paradox itself. A plane mirror should not be able to distinguish the left/right from the up/down directions (referred to itself), or indeed, any direction in its plane. This is because the only distinguished direction for a mirror is the direction defined by its axis. YET we perceive a left/right reversal of some kind, for instance, we cannot shake hands with our mirror image. This seems to suggest, paradoxically, that a mirror treats left/right differently from up/down.
If we use your definition of left/right, then there is no paradox to speak of, because a mirror should not be able to distinguish its left/right directions from its up/down directions,... AND it doesn't. There is no paradox in the first place! Your prescription appears to have solved our problem: a mirror dutifully does what we expect it to do...
Except for a strange detail. Why can you not shake hands with your mirror image if they are both "right hands"? You and your image are both advancing their "right hands" (based on your defintion) in a valiant attempt to meet each other in a firm handshake.
Quote "...so to say that chirality is reversed by a mirror is simply stating a definition..." Thank you for clarifying my stand. One of my purposes is to resolve the mirror paradox by stating that left/right are chiralities which are switched by a mirror. They should not be thought of as spatial directions, which (if in the plane of a mirror) are not reversed.
P.S.
Quote: "By left and right position, in reference to a mirror, I mean the opposite ends of an object in a horizontal direction parallel to the mirror, which would be the x axis"
I have other objections to this definition, as can easily be seen by realizing that the mirror has no idea what horizontal is. Every direction is a horizontal direction for a mirror placed on the ground, for instance. Also, I do not know what an "x-axis" is, and neither do I expect the mirror to.
Hi atgc
Now that I've addressed your question, I would like you to please address just one question of mine.
A mirror image doesn't rotate an object about any axis: do you agree or disagree?
Regards, Paul.
Hi Paul,
Your question is ill-defined. I suppose you mean "can a mirror reflection be solely described by a composition of (any number of) rotations?"
To which the answer, in 3 dimensions, is no.
I did not answer your question directly because it can be interpreted in another way. There is more than one way to carry out the transformation effected by a mirror. In general, one can use a combination of rotations and reflections. But the prescription always has to include (an odd number of) reflections.
From your reply to stu, on a related idea:
"A reflected image can be 'rotated' if you put 2 mirrors at a 90 degree angle. Then the image is rotated about the axis where the mirrors meet. I explain this in detail in my original post."
Two (an even number) reflections, can be explained solely by a reflection, as you have pointed out yourself. Of course, this does not answer your question, which concerns one mirror reflection, but it is helpful, at least for me, to keep this example in mind.
I hope I have interpreted and answered your question to your liking. Otherwise you may wish to clarify your question again.
I also have some links where rotations and reflections are discussed in more precise detail and generality. I agree with the definitions and presentations found there, so I shall not copy-and-paste the whole content.
http://www.physicsforums.com/archive/index.php/t-300280.html
http://en.wikipedia.org/wiki/Improper_rotation
http://www.physpharm.fmd.uwo.ca/undergrad/tweedweb/4rotation.htm
The truth, one might suspect, is that I have no idea how to post mathematical equations as a comment.
Hi atgc,
Thank you for your response.
We now agree on the following:
1. A mirror only reverses along the axis perpendicular to the plane of the mirror.
2. Chirality is reversed in a mirror.
3. An object is not rotated about any axis by a mirror. The mirror image (in a sole plane mirror) is not rotated about any axis.
4. We disagree that a mirror does not reverse left to right.
Your position on point 4, that a mirror does reverse left to right, contradicts the previous point 3, because an object can only be reversed left to right by a rotation. Therefore my position on point 4 that a mirror doesn’t reverse left to right is the correct one.
This argument is now over.
In response to a previous comment of yours, the reason you can’t shake hands with your own mirror reflection is because of point 3 above: the mirror doesn’t rotate your image.
Regards, Paul.
Hi Paul,
I do not understand your proof-by-contradiction.
We disagree on point 4, because we use different definitions for left and right. I stated very clearly from the outset, that I referred to left and right chiralities, which are reversed in a mirror. So Points 2 and 4 are the same thing for me. A rotation, unlike a mirror, does not change left and right chirality. You have agreed with me on each of these points in your own previous posts, so I do not know why you are so eager to contradict me.
Let's look from your point of view instead. I took great pains to discover that by left/right, you meant the following:
Quote "...By left and right position, in reference to a mirror, I mean the opposite ends of an object in a horizontal direction parallel to the mirror, which would be the x axis..." Indeed, this was how you determined the "left" and "right" sides of the mirror image. One corresponds to the direction of increasing x-values, while the other corresponds to the direction of decreasing x-values.
However, you go on to say "...because an object can only be reversed left to right by a rotation..."
At this point, you speak of left/right-of-an-object. This is a different notion from the left/right directional axes fixed with respect to the mirror, which just a moment ago, you had used. You have "left" and "right" taking on different meanings in different places.
Perhaps you disagree with me on my emphasis on chirality, but I hope you had considered my view as much as I had considered yours, before issuing a claim of contradiction.
Hi atgc
Rotation is the key to understanding this. Chirality and left-right orientation are not the same thing and I can prove it.
For something to have the attribute of chirality it must be asymmetric, which means it must have a left-right orientation, like one of your hands for example.
You can’t make your right hand look like your left hand, or anyone else’s left hand for that matter, by rotation. On the other hand you can change its left-right orientation, which way it points for example, by rotating it.
A mirror changes its chirality but doesn’t change its left-right orientation. Rotating it changes its left-right orientation but doesn’t change its chirality. Therefore left-right orientation and chirality are different things.
It would be nice for you to admit that I’m right.
Regards, Paul.
Hi Paul,
Once again, you do not seem to be reading what I am posting. I have already told you that I use left/right to mean chiralities, which is both standard and precise. Furthermore, if you have ever sat down to think clearly about how left and right are defined, you would realize that they are intimately linked to chirality. The two concepts are inseparable, and hinges on the orientability of the three-dimensional manifold that we inhabit. It is now clear to me, from your last reply (in particular the second and third paragraphs -- when you point into a mirror, does your image not point in the opposite direction, or do you mean yet another "left/right" this time?), that you do not appreciate the concept of chirality sufficiently to discuss it in a precise manner.
I have also noted that you generally use a different notion(s) of left/right, which is NOT chirality. You do not have to repeat what I have already said, nor do you have to provide a proof which I am unable to comprehend. What I contended was your imprecise use of the words "left" and "right", which as I have shown via examples, appeared to take on different meanings in different contexts. This was confusing and I am not the first person to point this out to you. I am not surprised at your response because you had similarly dismissed well-intentioned and valid objections raised by other commentators, stu and Glibeaux, for instance.
From my perspective, you have used the words "handedness", "orientation", "inversion" and "reversal" so loosely that I struggle to be convinced by your statements. I have provided many links where accepted definitions and clear usage of these terms are given, but it does not seem that you have consulted them.
I readily admit that I am pedantic and often ignorant, and you have the license to take me for a fool. Unfortunately, I cannot agree with your imprecise presentation of your stand, as I will be doing an injustice to the mathematics and physics professors who have taught me to properly address mathematical issues.
Finally, as this is your personal blog space, you should have the last word on this discussion. I hope that our correspondence will be useful for future readers of your blog.
Best Wishes
Hi atgc,
Despite your good wishes at the end, I’m not impressed by your lack of grace.
I could ask you: what part of ‘left and right’ don’t you understand?
Don’t answer, it’s a rhetorical question.
I gave you two opportunities to close this debate with good grace, but you declined both times. Somehow, I’m not surprised.
You are less a philosopher than a sophist, and sophistry never compensates for lack of comprehension or failure to admit you’re wrong.
Good bye. Paul.
I've not read the preceding discussion but it strikes me that the issue has everything to do with our egocentric understanding of space. For a member of the Guugu Yimithirr (who possess a geocentric orientation) the mirror paradox would simply not exist: the hand on the Northern side would correspond to the hand on the Northern side and the hand on the Southern Side would correspond to the hand on the Southern Side. For them there would be no reversal.
Apologies, I should have given a link to my source - a fascinating article in the NY Times: http://www.nytimes.com/2010/08/29/magazine/29language-t.html?_r=1&pagewanted=1&fta=y
Best
Jim
Thanks Jim, that's a fascinating article.
I've known that aboriginal people have a sense of direction unsurpassed in this part of the world, and I've been told that even on the football field they have a better sense of orientation to their European fellows - I'm talking Australian Rules football where Kooris have excelled. It's different to other codes of football in the way the direction of play can switch around, plus there are players of different teams all over the paddock, not one side lined up against the other.
I was once told a story of a white man who left a vehicle with some black men to pursue a kangaroo and after they caught it they were no longer in sight of the vehicle. The white bloke went to return the way they had come, but the blacks pointed out the correct way back: 'as the crow flies' (as we say in Oz).
In reference to the mirror paradox, the Guugu Yimithirr would have no problem. Because not only would the north or east (or whatever) leg and hand have the same orientation in the mirror as themselves, but the image would be facing the opposite direction, so they would understand the problem much easier than we do.
Thanks for that. It's provided one of the best ways to explain it that I've come across.
Regards, Paul.
who came up with this paradox?
Hi Anonymous,
Actually, it's all in the mind, which is why it has a 'Psychology' label attached.
If you don't see the paradox then you're unusual. Basically, people ask: why is a mirror reflection reversed left to right but not top to bottom?
That's the mirror paradox. For the answer, read the post.
Regards, Paul.
I realise in hindsight that I made a mistake in my argument with atgc last month when we discussed the images created by 2 mirrors facing each other. So I wish to correct my mistake, for anyone who reads the thread.
The scenario is to hold a small handheld mirror near one eye and you'll see that that eye (whether left or right) always appears on the same side in every reflection. If the mirror reversed left to right then you'd expect it to appear on opposite sides every second reflection.
atgc argued that we see the same side in every second mirror, which is the handheld mirror, because that's the only mirror we see. At the time I agreed with him, but I was right the first time. atgc was incorrect and I can prove it.
If you hold something up between the 2 mirrors with writing on it (like a tube of toothpaste for example) then you'll only see the writing every second reflection. However, the hand that's holding the tube appears on the same side in every reflection. Proof positive that mirrors do not reflect left to right.
Hi Paul
Please read the following sentences in which I use the word "handed" correctly:
"A left-handed guitar, once owned by sixties legend Jimi Hendrix fetched £280,000 yesterday in a London auction.
Experts agree that it may be easier to bring its original owner back from the dead than to transform the instrument into a right-handed guitar merely by rotating it through 180 degrees. Now Australian philosopher, Paul Mealing, claims there is still hope..."
I challenge you to construct a sentence using the word "handed" in the highly individual sense introduced by you in your posting of March 16. If you can't manage this Paul, I reckon no one can - so good luck!
Regards, Stu
Hi Stu,
I’m not sure if you’re aware but Hendrix always played a right-hand guitar. If you look at any image of him with a guitar you’ll see that his favourite instrument was a Stratocaster that he not only turned left-to-right, but also upside-down, so that it still faced the right way. He simply re-strung the guitar back to front.
So if someone wanted to convert a Hendrix guitar back to being right-handed, all they would have to do is re-string it for right-hand playing.
In regard to my post on 16 March, I’m merely pointing out to atgc that handedness in terms of ‘left’ and ‘right’ orientation is not the same as chirality. If you take a screw with a left-hand-thread, you can point it to the right or the left just by rotating it through 180 degrees, but it still remains a left-hand-thread no matter what direction you turn it to.
On the other hand if you reflect it in a mirror it becomes a right-hand-thread no matter what direction you turn it.
Unfortunately, atgc couldn’t grasp this simple difference.
Regards, Paul.
Hi Paul,
Yep, I must admit, I did think Hendrix always played a right-handed guitar, but hoped you wouldn't pick up on it.
stu
Turn your "back" to the mirror. Look, over on your shoulder to the mirror. Voila! Your right hand is attached to the correct shoulder of right side :)
Excellent article Paul.
Thanks Rhyddwr, glad you appreciate it.
This is my second most popular post.
Regards, Paul.
If you are right-handed, then "voila!" the figure in the mirror is left-handed, Rhyddwr. See?
Hi Stu,
Actually, no, the figure in the mirror is not left-handed, but it would be if it was rotated. Humans are symmetrical about the vertical axis, which is why we have the illusion that the mirror rotates them, when it doesn't, it simply turns them back to front, literally.
Regards, Paul.
Hello again, Paul. Good to see you are still on form. Still quick on the draw!
Your insistence that mirrors reverse back to front becomes increasingly mysterious with time. You believe this is a matter of objective fact; and I say it is all down to interpretation - if that.
You seem to be saying that we should not let ourselves be fooled by mere appearances: although the reflected figure may look left-handed, the truth is that he must be right-handed. This is because the laws of physics say that what we think looks a lot like his back really is really his front. Damn clever eh? But how on earth does the mirror work all this out??
Stu
Hi Stu,
Actually, the mirror works nothing out - it just does what it does - obeys physical laws. And his back doesn't look like his front at all, but his left is pretty well identical with his right; hence the illusion of left-right reversal.
If the subject was to hold his right arm out it would point to the right both in reality and in the mirror. It looks like he's left-handed because he looks like someone facing you holding their left hand out pointing to your right.
To appreciate the difference, imagine that his right hand had a large yellow bow attached to his wrist. If he turned around to face you, the bow would appear on the other side - your left - but in the mirror it appears on the same side as you - your right. That’s why mirrors don’t reverse left to right.
Regards, Paul.
Paul, (re your 30 July posting), surely the reflection of a right-handed figure does look like a left-handed one by definition of reflection? Stu
Hi Stu,
Not by 'definition of reflection'. 'The reflection of a right-handed figure [looks] like a left-handed one' because the figure, assuming it's human, is left-right symmetrical - no other reason.
Regards, Paul.
I don't see what you're driving at Paul. After all, the mirror image of a right-handed guitar looks like a left-handed one - assuming it's still a guitar as you might say.
This is due entirely to its lack of symmetry.
The feature under discussion which enables us to think of the reflection as an image of the original (guitar) is the only relevant example of symmetry in the problem.
Merry Xmas
Stu
Hi Stu,
The mirror image of a right-handed guitarist looks like a left handed guitarist who is facing you because their left hand is on your ‘right’ side.
A left hand guitarist who is facing the same direction as you would have their guitar pointing to your right, but if they turned to face you then their guitar would still point to their right, but your left. In a mirror reflection, however, their guitar would still point to your right. Think about it. So the mirror doesn’t rotate them.
This is the illusion of a mirror reflection: it looks like the subject has been rotated around their vertical axis, when they haven’t. Instead they’ve been turned back to front.
Regards, Paul.
We seem to be talking at cross purposes, Paul, but I don't see how this can be.
In your own post of 18 March, 2011, you stated that "Chirality is reversed in a mirror."
Quite so; for this was exactly my point when I said that the reflection of a right-handed person is a left-handed one. Even so you wanted to contradict me. Why? What's to disagree with?
You have tried to argue that the change of chirality in a reflected figure is due somehow to its symmetry,(31 Oct 2014) but this is entirely wrong.
A right handed glove lacks symmetry; therefore, under reflection, its image is a left-handed glove; and unsurprisingly the same is true for people. (People lack symmetry).
By way of contrast, consider the case of a sphere. Since this is symmetrical it is identical with its reflection and obviously not subject to reversal from back to front by your laws of physics (assuming, as always, that the reflection still represents a sphere).
regards
Stu
Hi Stu,
You are one very confused individual. I don’t believe I’ve ever suggested that chirality has something to do with symmetry. Chirality, by definition, is asymmetric.
The mistake that people make is to confound chirality with handedness, which, I admit, is very easy to do, because our right and left hands have reverse chirality, as you point out. But right and left are directions, which can change without changing chirality. This is the most important point to understand, as it’s the crux of the issue.
If you have an object pointing to the right (like a hand) you can change its direction by rotating it through 180 degrees (either way) to point to the left, but its chirality remains unchanged. The same is true of a screw or a helical spring or any device that expresses chirality.
In a mirror reflection the direction (left or right) remains unchanged but the chirality is reversed. This is why your left hand looks like your right hand in a mirror, because they have reverse chirality.
They are also symmetrical when placed alongside each other (just touch your thumbs). So they are symmetrical about each other but individually asymmetrical. This is also a possible source of confusion.
But they are also the reflection of each other back to front, which is what mirror reflection does. To appreciate this just put your hands together (as in prayer); they are identical back-to-front.
I hope this exposition hasn’t confused you more. To me, it’s all very consistent.
Btw, is Stu short for Stuart? There are a lot Stuart(s) where I work, but being Aussies they tend to get called Stuie.
Regards, Paul.
Hello Paul
Yes, Stu is short for Stuart. I have an Australian friend who has always called me Stui. Now I know why!
I'm not as confused as you suppose. The terms "handedness"and "chirality" are used interchangeably in many scientific disciplines including chemistry, physics and informally in mathematics. This is not a matter of "confounding" anything and I simply assumed you knew about it. It is convenient to describe an asymmetrical object as right-handed (say) for the sake of comparing it with its nominally "left-handed" counterpart. It is commonly used in chemistry for example, to refer to the distinction between molecular enantiomers. The designation of left or right-handedness in such a context is purely conventional, and refers to the (conventionally defined) left or right-handed polarisation of a light ray transmitted through an "optically active" solution.
By the way, you are right to point out that our left and right hands have opposite chirality, but this is hardly surprising, since Lord Kelvin, who first coined the term did so with regard to the opposite "handedness" of our hands. "Chirality" is derived from the Greek word for "hand"!
Regards
Stu
Hi Stu,
Yes, I agree with everything you say - I'm well aware of the 'right hand, left hand' convention used in physics, chemistry and biology to describe chirality and spin.
Please note that it doesn't disagree with or contradict anything I've said about mirror reflection.
And you're right in that a right-handed person looks like a left-handed person in a mirror, and vice versa, but the mirror doesn't reverse left to right - it's an illusion. It's because the mirror looks like it's rotated the person about the vertical axis, when it hasn't - it's reversed them back to front without any rotation at all.
And yes, the illusion is helped by the fact that one's left hand looks identical to one's right hand in the mirror and vice versa, BUT, as I pointed out in my last comment, the right hand is identical to the left hand back-to-front and vice versa. Or, if you like, the right hand is the reverse chirality of the left hand, and chirality is reversed in a mirror; therefore the right hand looks like the left hand in a mirror and vice versa. It doesn't matter which way you look at it, chirality is reversed, but left to right orientation is not, which is the key point.
And humans are symmetrical about the vertical axis, including their hands – the left hand is symmetrical to the right, which is why they have reverse chirality.
Regards, Paul.
Thanks Paul,
Although, with reluctance, I am repeating what I have already said in previous posts, I feel obliged to reiterate that as far as I am concerned, an image in a mirror is nothing but a type of picture – an illusion. For this reason I have argued, unsuccessfully it seems, (for you remain unconvinced), that the apparent reversal in a mirror from back to front is no less of an illusion than any other aspect of the geometry depicted there. For me, it has proved impossible even to interpret your claim that there is a law of physics devoted to the “reality” of reflection from back-to-front contrasting with the “mere illusion” of a reversal from left-to-right. Perhaps I would be more sympathetic to your strange hypothesis if you were able to convince me that there is no illusion of a left-right reversal. However...
However, I waited at a bus stop this morning, and eventually a bus came. Although I cannot deny that physical laws are implicated somehow in the arrival of the bus, I never would try to argue that there are laws of physics devoted to the coming of buses.
Do you really not see my point?
Regards,Stu
Hi Stu,
There is a very simple experiment you can do that proves I am right. This is a case where there are right and wrong answers, not just matters of opinion. So you can’t convince me of an opinion that I know is wrong.
What you need is a wall mirror, a hand-held mirror (like a shaving mirror) and something small with large writing on one side (like a tube of toothpaste or the box it came in). As you can see, all these ‘instruments’ can be found in a bathroom.
In one hand hold the mirror in front of you so it faces the wall mirror and in the other hand hold the tube of toothpaste between the 2 mirrors.
You need steady hands, but you should be able to see at least 4 images of the toothpaste which is enough to prove my point. What you will see is that in every second image you see one side of the toothpaste (with writing) and in the other in-between images you see the other side (without writing), but in all the images you see the same hand on the same side holding the toothpaste (left or right, depending on which hand you used).
This PROVES that mirrors reverse back-to-front but not left-to-right.
If this doesn’t convince you that I am right, then there is nothing I can say.
Regards, Paul.
Well Paul, if there were big prizes for ignoring your opponent's argument, you would surely win them all; so perhaps there is something I can learn from you. In view of that, I admit defeat.
It is depressing, but finally, I give up.
Stu
Hi Stu,
I think that's the pot calling the kettle black, as you've obviously ignored my arguments.
I believe I've addressed all your points and more. You're simply miffed because I can’t agree with you.
The point is that you will always believe you’re right no matter what arguments you hear, or, even if someone can physically prove you wrong as I have done.
You can continue to be ignorant – that’s totally your choice – but don’t blame me simply because you don’t understand what I’m telling you, or you refuse to even do an experiment that proves you wrong.
Regards, Paul.
"the mirror doesn't reverse left to right - it's an illusion. It's because the mirror looks like it's rotated the person about the vertical axis, when it hasn't - it's reversed them back to front without any rotation at all."
I agree Paul. This is an extremely clear and persuasive point. I have followed this post since it started and this the simplest and most convincing explanation so far. There is no rotation. We end to think there is because that's how we would simulate the effect. Excellent, Thanks.
Jim
Hi Paul
In my post of 16 Jan, I gave a critique that specifically addresses particular aspects of your thesis. Contrary to your accusation that I have ignored what you “tell” me, this demonstrates clearly that I have read, understood and taken an interest in what you have had to say.
I have never disagreed with your postings about what we must expect to see when we look into mirrors, (eg 16 Jan), but I am at a loss to see how these examples are supposed to address my objection to your argument, explicitly or otherwise. They fail to demonstrate that the image of depth-reversal in a mirror is not an illusion; for this is what you have claimed.
Elsewhere I have argued in different ways that the seeming depth-reversal must be illusory. If you could have reasoned against this point, I would have been genuinely interested rather than”mifffed” as you say, but I'm insulted by your ploy to continue repeating your old assertions as though you think I am stupid, instead of getting to grips with the significantly more interesting issues I have raised. You are the ”Journeyman Philosopher”, after all
(You know very well that you need to persist with your special pleading for the idea that reversal back-to-front is “real” at any price - including that of your intellectual integrity, otherwise the rest of your argument turns to mush).
Pardon my ignorance
Stu.
Hi Stu,
It is not ‘special pleading’, it’s a known fact.
I am not an oracle or a guru or anything like that; I am a dilettante philosopher, and I’ve never claimed to be anything else. When people, who obviously know more than me, make a point that’s contrary to mine I acknowledge that. So I’ve no need to uphold my ‘intellectual integrity’. It’s just that what I’ve stated, in this particular case, is what actually happens, and I have proved it.
I addressed your 16 Jan post by suggesting you perform an experiment (perhaps you don’t have a handheld mirror) that demonstrates back-to-front reversal explicitly. I couldn’t be more forthright or more explicit in addressing the very issue you raised.
You refuse to even acknowledge this; instead you claim that I refuse to address this very point. We can’t even agree on what we say to each other, so what is the point of even having this conversation.
Regards, Paul.
Hi Paul,
OK, perhaps I can see that you honestly do believe that your “experiments” address the question I have raised, although this still perplexes me.
My point (or question) is, and always has been, that what we see in mirrors is illusory, whereas by contrast you have argued, or at any rate you have asserted, that an aspect of the image is real. That is the interesting difference between us. That is what needs exploration, and that is the question that your examples do not address.
Regardless of whether I agree with your experiments, (and why shouldn't I?), I do not understand how they are supposed to prove that a reflection, or any aspect of it, is not wholly illusory. In your own posting, of 31st October, you referred to the reflection of a figure, and qualified this with the words “assuming it's human”, which intrigued me at the time although I let it go without serious comment. If you are willing to have doubts about the figure being human, what stops you from doubting the reality of your reversal back to front? In what, exactly, does your “reality” or this reversal subsist? Laws of physics? Or laws that deal with things that may not even be human? Pretty good for a Halowe'en posting, I thought. By the way, your other comment in the same post (about the definition of reflection) is mistaken, but at the time I let that go also. There's no need to pick you up on every damn thing, but now it has become relevant.
I am not just picking nits here. If there really is a law of nature which certifies your point of view, (and yes, a point of view is all it is unless you can prove it) then to be sure, no other interpretation is valid. You win! On the other hand, if I am right and a reflection is nothing but a type of picture, its interpretation is not subject to laws of physics. Indeed, I have proposed somewhere above that reflected images do not seem to be consistent with any genuinely “physical“ interpretation. Therefore other readings that happen to be mathematically equivalent are logically admissible. In particular, unless I have made a blunder, the reflected image in a vertical plane mirror can indeed be construed as a reversal of the source from left to right (a mathematical reflection in a putative vertical axis) composed with a rotation (of the reflected image) through 180 degrees about a vertical. Obviously this rotation makes no difference to the appearance of the image.
Even if my calculation is awry, and I am wrong for this particular case, the general argument still holds good.
The “mirror paradox” does not arise just because people are naeve and have been fooled again. It is surely a “fair enough” and quite interesting question about what things actually look like. As far as my own answer is concerned, reflections seem to reverse “preferentially” simply because they do; for that is, by definition, what a reflection is. Since I was a boy, I have found this fact (and it realllyis a fact), truly remarkable.
Regards, Stu
ps. Why are pirates called pirates? Because they AARRR! (still my favourite joke)
Hi Stu,
Looking back at the 31 Oct posts, I think that when you say a ‘left handed figure looks like a right handed figure’ as a ‘definition of reflection’, you’re referring to chirality being reversed. And my reference to 'assuming it's human' is simply a perfectly logical caveat.
There is one specific situation where a reflection will be reversed left-to-right: if you have 2 mirrors meeting at 90 degrees in the vertical plane. A good example is when a bathroom (or any room) has 2 wall mirrors meeting in a corner. This has particular relevance to your 4th paragraph because the 2 mirrors literally rotate the image through 180 degrees - so one's left hand is on the right side and vice versa. This is REAL left-to-right reversal and not an illusion.
Regarding your last paragraph, the so-called mirror paradox is purely a psychological phenomenon, because physically there is no paradox at all, but physically the mirror reverses the image back-to-front for REAL.
Regards, Paul.
Hi Stu,
It has occurred to me that the issue you’re trying to raise is the fact that there is nothing physical behind the mirror. Someone else in this long string of comments made that point, but I haven’t gone looking for it. Also the perception of depth in a mirror is an illusion, as it’s a 2 dimensional plane yet it reflects a 3 dimensional image. Is this the point you’re trying to make?
Be that as it may, the ‘image’ is reflected back to front with respect to the subject facing the mirror, whatever that subject may be, including an empty room. In fact, if you think about the empty room scenario, it has to reflect it back to front out of sheer physical necessity. The back wall is reflected back facing itself – think about it.
As for your last couple of sentences:
As far as my own answer is concerned, reflections seem to reverse “preferentially” simply because they do; for that is, by definition, what a reflection is. Since I was a boy, I have found this fact (and it really is a fact), truly remarkable.
This makes no sense to me, because the term ‘by definition’ in this context is a non sequitur. You use the word ‘fact’ very loosely here. Reversal back-to-front in a mirror is a FACT, not just my opinion, and it will remain a FACT with or without your belief in it.
Regards, Paul.
Hi Paul
In answer to your question:
The very first thing I wrote in the above post was “My point is, and always has been that what we see in mirrors is illusory”.
What is the point I have been trying to make? Think about it. D'uh.
The sun appears to go around the earth, but someone realised eventually that other interpretations consistent with this appearance may be valid.
Similarly, if whatever we see in a mirror is an illusion, any interpretation consistent with what it looks like is as valid as any other.
You suppose that your way of talking about reflections in mirrors is the only correct one, and in much the same way I daresay you think that Wittgenstein's “duckrabbit” is REALLY a duck. “It's bleedin' obvious when you think about it!” Quite; and like so many things in life, it has the terrific advantage that the less you think about it the more bleedin' obvious it becomes.
Regards
Stu
Hi Stu,
I’m sorry that after all this correspondence that you have no further understanding of this issue than when you started. When we start resorting to insults, it’s time to give up. I’ve exhausted all my explanations, descriptions, demonstrations etc, and I can no longer help you.
Regards, Paul.
Hi Paul
I have explained repeatedly that there is at least one interpretation, logically consistent with what we observe of a reflected image, that differs significantly from yours, so until you are able to prove that your viewpoint is upheld by some law, both of these interpretations are of equal (logical) status. (I would certainly contend that the situation is very like the example in Wittgenstein's “duckrabbit”).
Since I have lost all hope that you will ever understand this simple proposal, let alone that you might argue effectively against it, I agree that there is no more to be said, and that there is nothing further you can do to help me.
I am sorry that my last response offended you. I have no real excuse, except to indicate that it reflects my exasperation upon realising at last that despite my best efforts, nothing I have said over the course of our exchange seems to have made the slightest scrap of sense to you. Frankly I thought that my argument was pretty straightforward, and not very subtle; but after finding you still “wondering” what my point might be I lost all patience.
Stu
It’s symptomatic of your misunderstanding of the issue that you compare this with Wittgenstein’s dual image scenario. It’s a completely different type of illusion. In the case of the mirror reflection, the left-right reversal is a mistake because the brain ‘expects’ the reflected subject to be opposite-handed as it would be if it was not a reflection, but it’s not.
The whole point is that, contrary to your assertion, not all interpretations are equally valid – in fact, quite the opposite.
Regards, Paul.
Btw: your interpretation is not ‘consistent with what we observe of a reflected image’. In a mirror the right hand is reflected back on the right side and the left hand is reflected back on the left side so there is no left-right reversal – couldn’t be more simple. So please don’t complain that I haven’t addressed your contention because I’ve addressed it ad nauseam.
Regards, Paul.
http://www.huffingtonpost.com/2015/02/18/why-mirrors-flip-things-sideways-but-not-upside-down_n_6704792.html
Hi Jim,
Yes, she says exactly what I say in my post, even down to talking about standing on your hands.
Regards, Paul.
I know. I told you that you were right. A journeyman no more. Not on this subject anyway.
Thanks Jim.
Jim is also wrong.
The reflection we see in a mirror is just a type of picture, and as with any picture, the image is subject to interpretation, rather than to physical laws. Obviously we construe the reflected image as a sort of altered representation of the world in which we live, but that leaves the question open as to whether there may be differing ways to characterize this “alteration” consistent with our actual experience of the image. I have reasoned that more than one valid interpretaion (of a reflection) is indeed possible. That is to say, more than one is logically consistent with what is actually observed. It is irrelevant that these decriptions seem mutually inconsistent: the reflection is just a picture. In particular, it is valid to argue that a reflection seems to represent reversal of the (notional) left-right axis (with respect to the source) followed by a rotation (about the vertical) through 180 degrees -which conflicts with your view. Can you develop an argument against these suggestions, or not? Do you think I have made a mistake? (Do you understand what I mean, or not?) I am still left with no way of knowing. You seem to read with your fist.
My proposal contrasts with yours which asserts that there is only one true interpretation of the problem consistent with what we actually see, (yours); but instead of making an effort to support this claim by arguing fairly against my critique, your mode of debate depends entirely on perpetual repetition of your simple “Mealing Perspective” - even though you must have realized that it is the very point at issue. If it is correct, then surely you would have had nothing to fear by addressing my analysis properly - and had you managed this, you would certainly have found it necessary to quit repeating yourself.
I think you ventured somewhere, or certainly implied, that since you know you're right, it would be a waste of time to take criticism seriously. This approach evidently wins the admiration of some characters, and is certainly consistent with my own experience of your basic outlook.
In summary however, my evaluation has addressed your simple idea specifically, finding weakness in the point you say you cannot help repeating “ad nauseum” when challenged.
Regards, Stu
Hi Stu,
You are right: there is more than one interpretation of the image and your interpretation that there is a left-right reversal is the most obvious one. However, this is an illusion and my interpretation is the correct one, and that is the point of this entire discourse. What’s more, I spend 2,000 words in the main text of this post explaining why this is the case.
Regards, Paul.
Paul
I found your blog post tonight while googling about the mirror paradox and while I originally was not swayed by your explanation, I now believe from some things you said in the comments that you are probably right on the money. I tried to follow the rambling discussion with Stu but skipped after it seemed to be going nowhere (literally for years). So if I missed a point of enlightenment I apologize for dragging this out further.
By using the words that you chose to use you may have confused the issue further. To say that an image is reversed 'back to front' to me means that I would look in a mirror and see the back of my head. I'd like to suggest a simplification of your explanation.
Using your explanation of x, y, and z axes. The x and y axes are on the surface of the mirror and the z axis is perpendicular to the surface of the mirror and zero on the z axis is at the mirror surface.
Consider that every point in our real world in front of the mirror has an x,y,z coordinate. The corresponding coordinate in the mirror would be x,y,-z. The mirror simply mathematically reflects thru the z axis.
Geo
Hi Geo,
You prefer Geo to George?
Yes, everything you say makes sense. Except, if the mirror didn't reverse the image back to front, then you would see the back of your head (if you think about it). We know that's impossible because we've looked in mirrors all our lives. But if you use 2 mirrors facing each other then you will see the back of something every second reflection. So a mirror behind you will show the back of your head every second reflection, but only if your head is transparent.
Does that make sense? In fact, using a small hand mirror near your face under one eye and holding something small (like a tube of toothpaste) you will see that you'll see the reverse side of the tube every second reflection but always held on the same side and under the same eye. So this proves that (plane) mirrors don't reflect left to right, but back to front.
Best regards, Paul.
Paul
Yes of course it doesn't make sense, which is why I pointed it out. I see what you're trying to say with it but if you ask any normal English speaking person what is meant by an image "reversed front to back" you won't get anything like what you're meaning. The average man-on-the-street will mentally turn the image on the vertical (probably) or horizontal axis.
I'm just pointing out that the phrase doesn't contribute to clarity for most people.
Geo
Hi Geo,
I appreciate your point, but it's the truth and sometimes the truth is not obvious or intuitive. If the statement provokes people into thinking about something differently, then I've achieved my aim. In other words, people have to realise what back-to-front really means, and in the case of mirror reflection it may mean the opposite to what they think. It's one of those issues where people have to work it out for themselves otherwise they simply don't believe it.
Thanks for your feedback.
Regards, Paul.
It's simply a case of parity inversion, but not of left-right or up-down, but forward-backward. The closest parts of us to the mirror are also the closest part of the reflection to us. We only have one inverted form because inverting two dimensions restores the original form. Inverting three dimensions is an odd number and so would just result in that same inverted form. Our minds try to make sense of the inverted form in the mirror by an imagined rotation, etc.
Hi Tony,
Correct.
Quote: a mirror reverses everything in the dimension perpendicular to its plane
This post has generated more comments and more arguments than anything else I've written.
Best regards, Paul.
Why does a mirror reverse left and right but not up and down?
Stand in front of a full size mirror and hold your arms out, the mirror reflecting your entire person. Does your reflected left-arm appear on your right-hand side? Does your reflected right-arm appear on your left-hand side? No.
Make two little signs: LEFT and RIGHT. Hold these signs in the appropriate hand, LEFT in the left-hand, RIGHT in the right-hand. Hold these signs when you have your arms out in front of the mirror. The mirror reflects two blank pieces of paper.
Now flip the signs so that they face the mirror. The reflected LEFT sign is on your left-hand side and the reflected RIGHT sign is on your right-hand-side. Not reversed.
But the writing is reversed. LEFT is TFEL and RIGHT is THGIR. Has the mirror reversed left and right? No. You have, when you flipped the signs so that the signs might face the mirror.
I'm not sure if you've read my post, but we come to the same conclusion:
The mirror reflects back-to-front, not right-to-left or top-to-bottom.
It's a 2 dimensional plane and it reflects along the axis perpendicular to the plane.
I explain this in some detail in the main text.
Flip your calling cards along their bottom edge, not on a margin. Why does the mirror reverse up and down ↑ ↓ but not left and right → ← ?
I answered that in the post.
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