# The Ames Room Illusion

By Grant Ocean

## The 2-D Pictures and the 3-D World

Hold up one of your hand and keep it close to your eye; extend another arm and hold up that hand. If you focus on both hands alternatively and quickly, you perceive both hands as being the same size. This well-known size constancy is achieved by moving the converging point back and forth, as shown in the diagram below. The Focus1 is placed where it is when we converge our eyes on the near hand, which results in a small sub-visual field volume around the near hand; and the Focus2 is placed near the far hand when we converge our eyes on the far hand, leading to a similarly small sub-visual field volume around the far hand. As a consequence, the similar sub-visual field volume will make both the near hand and the far hand appear the same size. On the other hand, if you focus only on the near hand, the far hand will appear much smaller than the near hand. This outcome is illustrated by the diagram below as well. The visual field that is spreading out from the Focus1 is small for the near hand and large for the far hand. These phenomena of size constancy and micropsia have been already discussed earlier. However, putting these two phenomena together has some important perceptual functions.

If you focus on both hands, you know they are the same size.
And if you focus on the near hand, you know the far hand is smaller. However, the combination of these two understandings is very significant for size perception. When we focus on both objects, we know the actual sizes of these objects. Then, when we focus on the near object, we know the far object appears smaller than before. If we don't know which object is nearer than the other, we simply focus on one object and perceive the other, and later switch them. The object that does not change size is the nearer one. As a result, we know which object's apparent size is different from its actual size. The object that changes size when we switch our focus is the farther one. By how much this farther object has changed its apparent size we can tell approximately how far away this object is from the nearer object. In sum, switching our focus back and forth between objects can help us to know the distance between objects. Take the Ames Room as an example, as shown below. That we are confused about the perceived sizes of these two twins is because we only have information about micropsia, either through the viewing hole or based on the photo taken with a fixed aperture or focal length. As discussed earlier, the fixed converging point will make the farther objects appear smaller than the nearer ones. To overcome the Ames Room illusion, all we need to do is to take down the front wall. In so doing, we can focus on both twins as well as focus on the nearer one and perceive the other. The perceptions of both size constancy and micropsia will help us to know that these two twins have the same actual size and the one who becomes smaller when the nearer one is focused on is at a longer distance from us. And we also know the true shape of the room by focusing on both ends of the room; the higher end of the back wall that appears to be the same size as the shorter end of the wall when the nearer twin is focused on must be farther away from the viewer. As long as we can converge our eyes on all the objects in the Ames Room and perceive them through one fixed converging point, we will not have the illusions about the shape of the room and the sizes of the people or objects in the room.

Let's discuss the photo below, which was taken by Gregory, one more time. The two women in the photo look about the same distance from viewers. Even though the large woman appears to be slightly closer to the viewer than the small woman, their sizes are still tremendously out of proportion considering the distance. The general feeling for the viewers is that they look like being actually quite different in size. This photo attempts to illustrate for us that we are unable to perceive the size of objects accurately without the proper cues to distance. But, we are misguided by this 2-D picture, as well as the above picture of the Ames Room illusion, taken by a camera with a fixed converging point. In the real 3-D world, we can easily focus on both women and figure out effortlessly that they are about the same size and one of them appearing smalleris because she is farther away from us. In addition, judging by the apparent size of the farther woman which is half the height of the nearer woman, we also know that the woman at the backis about twice as faraway from us as the woman in the front. Therefore, the perceived sizes in 2-D pictures are fundamentally different from the real 3-D world; so the pictures like the one below and the Ames Room illusion cannot show us how we perceive in the real world.

The drawing on the left below is based on the picture that was taken in a hallway. Thus we can regard this drawing as a photo picture. This picture has been used to demonstrate that linear perspective plays a vital role in our perception of objects' size at different distances. The fact that the back sphere looks bigger than the front sphere of the same retinal size is attributed to the location of the back sphere in the narrower looking hallway which is a smaller surrounding context or visual field. However, in the real world, perspective is the effect of the visual field volume and the fact that farther objects look smaller is because their portion in a larger visual field is getting smaller. The narrowness of the hallway is not a contributing factor in the size perception of the back sphere. As shown in the picture on the right below, the same large appearing back sphere in the picture on the left now looks smaller in an expanded visual field with the narrowness of the hallway intact. In the real world, the visual field volume for the back sphere is much larger than that for the front sphere. In the picture on the right below a large portion of the visual field volume for the back sphere has been cut out. As such, our perception of the two spheres in a hallway is fundamentally different from our perception of them in the real world.

For a picture to simulate the real world, it has to show a larger visual field volume for the back sphere and a smaller one for the front sphere, which is impossible for a static picture. In the real world, if we focus on the front sphere or on any fixed point ahead of the front sphere as the camera did, the back sphere will appear to be the same size in diameter as the front sphere, assuming that the back sphere is five times bigger in diameter and five times farther away from the viewer than the front sphere. The perceived sizes of the front and back spheres will not change as long as the converging point is in front of the front sphere, which is the case for the picture above. This result is determined by the equation Ps = S / d. You can also confirm this outcome by focusing on your near hand and then perceiving your far hand to check whether it changes size from moment to moment. But, according to Murray et al. the back sphere is perceived by observers to be 17% larger than the front sphere on average. This is impossible in the real world. The reason why the back sphere is perceived slightly larger is that the back sphere happens to be located in a smaller sub-visual field in the picture and our brain manages to extend the picture further to generate a larger sub-visual field for the front sphere. In reality, when we converge our eyes on the back sphere, it will appear to be five times bigger in diameter than the front sphere instead of a tiny 17%. This picture further proves that a 2-D picture cannot accurately represent real 3-D world at all.
One of the distinguishing features of Renaissance art was considered to be its development of highly realistic linear perspective. Giotto di Bondone (1267–1337) is credited with first treating a painting as a window into space, which started the tradition of perspective drawing. Looking at objects through a window has the converging point fixed at the window; as a result, one cannot focus on any object beyond the window. We are actually perceiving everything outside the window with micropsia, that is, all the objects appear smaller than their actual sizes and even smaller when they are further away, which is the essence of linear perspective. Hold up your right hand close to your right eye and look through your pinched thumb and index finger with your left eye closed. In so doing, everything seems to be smaller and you feel that you can squeeze those objects with your fingers. Now view a house across the street between your thumb and index finger. The house is perceived as small as the space between your pinched fingers. Then move your fingers aside and focus on the house and the two pinched fingers with the same space between them. All of a sudden, the house becomes thousands times bigger than before. Similarly, if you can move the window through which you are looking at a house across the street aside, the house will appear to be much bigger than the window. All in all, the artists before Renaissance were not totally wrong when they painted the human figures the same size at different distances because they had emphasized the aspect of size constancy. The artists since Renaissance are not totally right when they are painting the objects based on perspective because they put emphasis merely on the aspect of micropsia. In the real 3-D world we have both aspects for our size perception.
The picture on the right is an example given by Don McCready to show the difference between the linear size and the angular size, which we have already encountered in the second section of this article. In this example, we are looking at a house across the street through a nearby window. McCready claims that we experience both the linear size and the angular size comparisons at the same time, along with the distance comparison. We can say that the house is bigger than the window, referring to the linear sizes of them; and we can also say that the house looks smaller and farther away than the window, referring to their angular sizes. However, this is a 2-D picture; thus there is no way that we can tell the house is actually bigger than the window, except that we can guess the actual size of the house based on our knowledge that a regular house is usually much bigger than a regular window, but this guessing is not based on our perception. This situation is similar to the Ames Room illusion. Even though you know that the twins should have the same size, you still perceive the far corner twin as being smaller in size. It is because you look at the house and the twin through a fixed converging point, i.e., the window and the viewing hole. As I have stated earlier, if you move the window aside or poke your head outside the window, the house will look a lot larger than the window because you can converge your eyes on the house when the converging point is not fixed on the window. This is a fundamental difference between the 2-D picture and the real 3-D world.
Also McCready claims that in everyday conversations "looks larger" often refers to an angular size comparison rather than a linear size comparison. In this picture and also in the real world we cannot compare the sizes of the house and the window as long as we are at the viewing position as shown in the picture. We are focusing on the window, so the size of the window we perceive is approximately its actual size, which is similar but not equivalent to the linear size as discussed before. On the other hand, we are perceiving the house through the window, i.e., perceiving it by means of a fixed converging point at a distance, which is similar but not equivalent to the angular size. It is of course hard or even impossible to compare the actual size of an object with the perceived size of another object at a distance. If we move the window aside and focus on both the house and the window, then we can compare the actual size, or the perceived size at a close distance, of both objects. And if we can focus on a fixed point, such as a viewing hole or a camera, which has equal distance to both the house and the window, then we can compare the perceived size of both objects at an equal distance from our focus. In comparison, it would be easier to compare the actual sizes by focusing on both objects.
Finally, McCready claims that when we say one object "looks larger" than another, we most often are using the verb "looks" to describe not the perceived linear sizes, but the perceived angular sizes for them. Actually, I think that when we say objects "look larger" or "look smaller", we simply mean the way objects look from a fixed converging point. For instance, when we say that the house across the street looks small through the window, we mean the way the house looks as viewed through the window; when we say that one twin looks smaller than another twin in the Ames Room, we mean the way they look through the viewing hole. And when we say that everything looks small as viewed between our pinched thumb and index finger, we mean the way they look between our fingers. The window, the viewing hole, and the space between thumb and index finger are all fixed converging points. Without the fixed converging point, we can just focus on objects at a close range and determine approximately the actual sizes of the objects. Of course, we cannot say that an object looks larger or smaller when we have already found out that it actually is large or small.

Don McCready and other perception researchers have proposed various ways to solve the problem posed by the Gregory's photo, which is similar to the two men drawing at the top right corner of the diagram above. They claim that the picture easily leads to a percept of two men at the same perceived distance. It is because our perceptual system performs a process known as the equal distance scaling resulted from an "equidistance assumption" or an "equidistance tendency". And the perceived linear height of the man on the left is twice that of the man on the right because his perceived angular subtense is twice as large. To make the men's linear sizes look equal, the S' values could be scaled equal by the (cognitive) familiar size cue to linear size or identity constancy if the observer knows their heights; and if the men are unfamiliar, that scaling could occur due to an "equal linear size assumption". In addition, many complex equations and computing processes have been proposed to maintain the linear size constancy (for details, see McCready's web page). In comparison, my solution to the problem is much simpler and perhaps more effective. The two men of the same height will look the way as shown at the top right corner of the diagram above when the viewer focuses on the near man or a converging point in front of the near man such as a viewing hole or a camera lens. But the two men will look the same height as soon as the viewer focuses on both of them. By doing both, the viewer would know the fact that both men have the same height and one man is twice as faraway as the other, as discussed earlier. To repeat my solution to size constancy at the end of this article is to reiterate that there are numerous efforts to solve the problem, but the simplest approach could be the appropriate one if you believe in parsimony.
In the previous article we discussed the visual field volume and in this article I have introduced another important concept, the converging point. Equipped with these two ideas, we can explain adequately a well-known phenomenon called Emmert's law and can also solve the most important size illusion of all and most complicated one, the Moon illusion.

# References

Kaufman, L. & Rock, I. (1962). The moon illusion I. Science, 136, 1023-1031.
Broerse J. et al. (1992). The apparent shape of afterimages in the Ames room. Perception, 21(2): 261-8.

## Related Information on the Web:

http://en.wikipedia.org/wiki/Ames_room
http://www.psychologie.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/ames_room.html