Take another look at this picture of the Rokeby Venus from last week's post on mirrors in art:
Now, imagine you're actually in the room with Venus, as depicted in this painting. You suspend your astonishment long enough to conduct a quick test of the principle of how a flat mirror works. Consider what would happen to Venus' face in the mirror as you approach it. As you walk towards the mirror, would the proportion of the reflection taken up by Venus's face increase or decrease? In the painting, the face takes about 2/3 the width of the mirror. Would that proportion get bigger or smaller as you get closer? Let's make this a poll:
Now imagine the cupid moves the mirror so you see your own reflection in it. As you walk away from the mirror, what happens to your reflection?
We have an unusually savvy and educated readership here, but for most people, those questions are quite difficult, despite the fact that we see mirrors every day. In fact, as Marco Bertamini pointed out in a comment on last week's post, his team of researchers got it wrong in their article about the Venus effect. Even art critics discussing the role of mirrors can make incorrect generalizations about how mirrors work, a point Marco Bertamini's team of researchers made in their article about the Venus effect.
As you get farther away from the Venus, the projection of her face in the mirror approaches the actual size of her face. As you get closer, the proportion of the mirror occupied by her face decreases. So the face in the mirror is only too big if you assume the viewer is located relatively close to the mirror. If the experts can't get this detail right, what chance do non-experts have?
Bertamini has continued to investigate our perception of mirror images. In 2007, a team led by Rebecca Lawson and including Bertamini and Dan Liu, came to a startling conclusion: "There is no percept for the 2-D projection on the surface of a mirror or window." In other words, we don't ever directly perceive the image on the mirror -- we see only the object that's reflected.
We are quite good at judging the size of objects if we know their distance from us. We almost automatically take distance into account when judging an object's size, and despite the fact that objects literally appear smaller when they are farther away (they take up a smaller portion of our retina), we believe that they are the same size. In other words, we don't think cars get smaller as they drive away. But when looking at a mirror, most people are suddenly much less accurate at judging the size of a reflection.
Lawson's team had students stand two meters away from a large mirror. A researcher held a bamboo stick above the student's head, behind a piece of wood so that the student couldn't see the stick directly, only via its reflection. Students were asked to judge the length of the stick itself by extending a tape measure held at their side to match the length of the stick. Then they used the same method to estimate the length of the image of the stick in the mirror (as if they were holding the tape measure directly up to the mirror). Here are the results:
The purple bars show their estimates of length (the sticks were of varying lengths; they've all been normalized to 1 for this illustration). The correct answer is shown with the arrows. As you can see, while responses were very accurate for the estimate of the physical stick, respondents thought the image of the stick in the mirror was much larger than it really was.
As a control, the researchers also held the stick directly against the mirror, and as you'd expect, the students could accurately estimate its length.
In a second experiment, the students viewed the sticks from two different viewpoints -- 1.5 meters away from the mirror or 6 meters away (the sticks were always held directly over the viewers' head). The same error was repeated, no matter how far away the observers stood from the mirror. The students were also asked a question similar to our first poll question: would the proportion of the mirror taken up by the stick increase or decrease as they moved away from the mirror? Most respondents said, incorrectly, that the proportion would decrease.
In two additional experiments, the researchers kept the sticks in the same place while asking the viewers to estimate their size and the size of their reflections. No matter how far away viewers stood from the mirror, their estimate of the size of the reflection didn't change significantly, despite the fact that it changes considerably.
In several of these experiments, students also looked at the objects through a window (with the same relative distance to the physical object and its projection on the window). The results were similar (and the physical principles, in fact, are also essentially identical).
As this diagram shows, we'll instinctively move closer to a window to see more of what's outside:
If someone is the same distance from the window as we are, we can see their whole body even if the window is only half the size of the person we're looking at -- their image in the window is only half their actual size. But if we move away from the window (as in the bottom panel), we can't see their whole body -- their image as a proportion of the window has gotten larger.
Of course, if we're looking at our own reflection, it's impossible to be a different distance from the mirror as our reflection, which means the reflection itself is always half our actual size, and it doesn't change, no matter how far we move from the mirror.
Lawson, R., Bertamini, M., & Liu, D. (2007). Overestimation of the projected size of objects on the surface of mirrors and windows. Journal of Experimental Psychology: Human Perception and Performance, 33 (5), 1027-1044 DOI: 10.1037/0096-1523.33.5.1027
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...this is like an xkcd strip just wiating to happen.
Reminds me of an interesting self-portrait:
http://imagecache.allposters.com/images/pic/130/009_575-010~Norman-Rock…
Interesting results from the two polls. I expected our readers might do a little better than average, but they actually didn't do very well at all. Only 40 percent correctly said that the Venus's face would get smaller as you approach the mirror. And only 12 percent correctly indicated that your reflection occupies the same proportion of the mirror as you walk away.
Nikhil: That's an interesting one, because the angle of Rockwell's face indicates that he is looking at us (the viewers) in the mirror, rather than looking at himself.
Imagine we are standing behind a half silvered mirror looking at the scene of Rockwell making this painting from the perspective shown in the image. Then Rockwell could also be looking (via both mirrors) at the exact scene he is painting. (But of course, in that scenario there would be no constraint on what gets drawn on the in-picture canvas -- it need not be his face at all.)
-kevin
What does this mean for telescopes that use mirrors?
Okay, can you please help me out here? I still don't understand the second question.
I understand the first part - as you get closer to the mirror, less of it is taken up by the face, since you can see more of it in the same area of mirror, akin to getting closer to a window and looking out of it.
But the second part doesn't make sense to me!
Dave, you said:
"...your reflection occupies the same proportion of the mirror as you walk away."
But this is clearly not true!
If I am one mile away from the mirror, clearly, much less of the mirror is taken up by my image than if I am right in front of it, therfore my reflection occupies a lesser proportion of the mirror!
What's wrong with the logic here??
I just tried this with a wall-mirror. I got close enough to it that the image of my toes was at the very bottom edge of the mirror, and the image of the top of my head was, well, level with my head. Then I walked back about 6 feet. The image of the top of my head was at roughly the same height, but the image of my toes was about a foot or so up from the bottom of the mirror. Thus my image occupied a smaller proportion of the visible surface of the mirror as I moved further away.
Flymises:
Have you ever tried that? The same portion of the mirror should be taken up by your image, no matter how far away you are.
Qalmea:
I think the difference is due to the fact that your body is not completely flat. Your toes point out a little in front of the rest of your body. This means that when you back off from the mirror, you end up seeing some of the floor too. But your heels should be on the same spot on the mirror no matter how far away you stand.
Er, no. Just tried it again. Five inches from the mirror, a button was directly in line with the doorknob adjacent to the mirror, and the top of my head was at head-height. Six feet back, that button was nearly a foot up. The image of my torso had clearly shrunk. I could not judge the exact position of the image of my heels, but close to the mirror, they were not far from the bottom edge of the mirror. Further from the mirror, they were much further up.
Try this. Put your hand up against a mirror. The reflection is the same size as the hand. Outline the hand with a dry-erase marker. Now walk back 6 or 8 feet and hold up your hand, trying to line it up with the outline. The new reflection of the hand will be smaller than the hand traced on the mirror. That is not occupying the same proportion of the mirror's face.
To see if there was something special about starting with something directly on the surface of the mirror, I traced the image of my left hand as seen from about a foot away and stepped back to 6 or 8 feet from the mirror. The new reflected image of my hand was still smaller on the glass than the outline.
It is important to keep in mind the difference between looking at our own face (where the eyes are, and therefore the viewpoint is located) and looking at other objects, including our own hand, which does not correspond to the viewpoint. For the whole body the same thing as for to face may apply as an approximation, because the eyes are coplanar with the body for a vertical mirror.
So, with respect to the face, you can use some sticky note papers to mark the top and bottom of the face and then move in and out. You should also close one eye of course because otherwise you have two viewpoints (even though they are near each others they do create a problem for short distances). Very short distances are best avoided anyway, someone may have a very long chin, more importantly you would be working with difficult angles (as distance goes to zero the angle goes to infinity).
You will find that the head will always fit between the two markers. This is true even if you use a telescope. I know this is very counterintuitive. Note that the proportion of the mirror taken up is fixed for the person with the telescope, not for a person standing next to the mirror and looking for the guy with the telescope in it.
For other objects (not located at the viewpoint) the size changes, as in the example of the hand discussed by Qalmea. Here the relationship between the size of the object and the size of the reflection depends on their relative distance and also on the distance of the observer.
As an illustration, if you see a photograph with a person in front of a mirror and the reflection in the mirror is the same size as the person, then the photograph was probably taken with a telephoto lens (so the viewpoint was far from the object relative to the distance between object and mirror).
cheers, MB
Sorry, but it doesn't work even with the face. Put your face directly next to a mirror. The reflection will be about the same size as the actual face. Mark some notable features on the face, like the center of the eyes or the cheekbones (the width isn't good, because the apparent visual width will change if you accidentally turn your head even a little bit). I just marked the centers of my eyes. Now, pull away from the mirror and try to keep both eyes lined up with those marks. It doesn't work. The image as seen on the surface of the mirror is smaller.
The image as defined by optics does stay the same size, but that (virtual) image is behind the mirror, at the same distance from the mirror as its object is. It is not the image as it appears on the surface of the mirror. It's just like looking at an identical twin through a window, but you are always the same distance from the mirror as your twin. As you move away from the mirror, your twin gets further from you, and appears smaller in the "window" of the mirror.
Better yet, start with a mirror smaller than the face. If you put the mirror directly next to your face, the entire image of the face is too big to be visible on the surface of the mirror. Now, pull the mirror in front of you a foot or two, and the entire face is visible.
I just tried your experiment, and I was unable to get the mirror close enough to my face to not see the same portion of my face as when I'm any other distance away. Perhaps the effect would disappear if the eye was actually touching the mirror, but the physics works out as long as there is some nonzero distance between your eye and the mirror, and the thing you're looking at is in the same plane as your eye.
I also tried marking the mirror with post-its and looking through one eye as Marco suggested, and indeed, the size of the reflection does not change, no matter the distance from the mirror.
The physics behind this problem is quite simple. If you're not seeing it it's possible that your mirror isn't perfectly flat or you're making some other error.
Start with the mirror not with face.You will find that the head will always fit between the two markers. This is true even if you use a telescope. I know this is very counterintuitive. Note that the proportion of the mirror taken up is fixed for the person with the telescope, not for a person standing next to the mirror and looking for the guy with the telescope in it.
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