Inconsistent Images

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Take a quick glance at the image to the right, and you see a seated man gazing at a framework cube. This image is detail from Escher's 1958 lithograph 'Belvedere' on the left.

If we look more closely at the cube in this image we will see that it is inconsistent. The opposite corners are joined together and this makes parts of the cube we would expect to be at the back occur at the front. However, still it does contain some of the properties of a cube. If you look at the top half alone it looks like a normal cube. If you just look at the bottom half (below his hands) it still looks like a normal cube. However the two halves are twisted front to back with respect to each other so the whole image fails to be a cube. Numerous other artists have produced inconsistent images. In our work on inconsistent images we will be addressing a number of questions. How does the inconsistency arise? What are the different ways in which inconsistencies may occur? Are there certain conditions in which this would not be an inconsistency? If Escher's cube existed in a space where the space itself was warped then this may well be how a normal cube would appear. We can construct a cube by placing half a cube against a mirror. The symmetrical nature of a cube means this would generate the image of a complete cube. Consider Escher's cube: we take the same half cube but replace a normal mirror with a mirror which also rotates the image by 180 deg. in the plane of the mirror. This is one explanation of how the inconsistency arises. Our world does not contain this type of symmetry . Other types of inconsistency may not arise in this way.
In contrast to the inconsistency of Escher's cube consider what happens when we cannot distinguish between the front and back of these cubes. This type of image is called a Necker cube. The Necker Cube is named after the Swiss crystallographer Louis Albert Necker (1786-1861), who in 1832 first published details of cubic shapes spontaneously reversing in perspective.
This cube is not inconsistent but ambiguous. We cannot tell how it is oriented, i.e. if you focus on the interior 'Y' it could be the interior of a box pointing away from you, or it could be the outside peak of a box ( or pyramid) pointing towards you. We have produced animations of the cube rotating.To view the animations click on the image. (if the animation doesn't work you will need to download Macromedia Flash Player, available here).

When you watch the animation of this cube rotating we see it as a rigid object, rotating with one specific orientation. However if you glance away and back, or change your point of focus, then the orientation can reverse. This flips the front and back faces and the rotation appears to reverse direction. If you focus on the vertical side lines you can see it is still rotating in only one direction. Suppose we recolour the side lines green. If you focus on where the green lines intersect the red, the lowest part of the green lines are in front of the red, whereas the top end green is behind the red. Whilst this does give us a particular orientation it is not strong enough to counter the ambiguous Necker effect. The cube can still flip orientations. The red 'faces dominate our visual processing.

One way to dis-ambiguate the Necker cube is to introduce inner and outer face elements.

This cube's outside face elements are red and inside ones are magenta. One of the bars is coloured blue and green. Reconsider how the ambiguity of the Necker cube arose. Try to focus on the three magenta bars and ignore the rest of the figure. The three magenta bars can be seen as a pointing away from us or towards us. However, the red bars overlap the magenta bars, and the complete cubic structure forces us to see it as pointing away from us. It is the inside back of a cube. The visual ambiguity like this is resolved by the extra information provided by the overlaps.

The cube on the right is, like Escher's, inconsistent. Based on the position of the outer red bars, and the inner magenta bars, we expect the blue bar to be in front of the magenta bar. It is joined to red corners which are at the front of the cube, so its whole length should be at the front, but it is not. This also effects our perception of the magenta bars- we can see them as behind (consistent with the red bars), and also as in front (consistent with the blue overlap).

In the right animation we see a normal cube for most of the rotation. However, the blue/green bar is not always behaving consistently. Follow the progress of this bar and be surprised by an inconsistency. Detecting and processing motion is a very primitive cognitive ability. We expect the cube to continue rotating consistently so the appearance of an inconsistency can lead to a peculiar experience of cognitive dissonance. This is a different type of inconsistency to that of the Escher cube and cannot be explained in the same way. For part of its rotation through space the specific blue bar is suddenly transported from a normal position to the back of the space. We can explain this by thinking of the space as being bounded at its front right edge ( i.e. the y-axis is of finite length). When the blue bar (and only the blue bar!) strikes the edge of the space it re-enters the space at the back wall. It wraps around the space. When it later drops back across the boundary it reappears near the front where we would expect it if the space were normal.

In this next animation, it is not just the blue bar which behaves inconsistently. When the blue bar behaves inconsistently all of the other back side (magenta) bars also move to the front. If you glance away and back at the animation the rotation seems to reverse direction. Rather than a red cube rotating counterclockwise, it temporarily becomes a magenta cube rotating in the opposite direction. Alternatively you can see it as the red cube continuing to rotate in the same direction. The motion itself is inconsistent. The experience of disorientation as it changes direction (twice), but also goes in the same direction, is quite striking. This inconsistency is tied up with the motion of the object and is different again (in the single image at left ask yourself whether the purple bars are in front or behind.) In making all of these bars behave inconsistently we have partially reintroduced the ambiguity of the Necker. The orientation of the red surfaces, near the corners suggest red is at the front, but the overlapping of the purple bars make the purple bars, appear to be at the front, thus an inconsistency.

Another way to produce a definite front and back orientation is to insert different coloured faces. The occlusion of the green side lines by the faces clearly establishes a front and back. This remains consistent when we rotate the cube. As with the earlier frame cube we can introduce inconsistent occlusions.

According to the position of the green corner bars, the blue face should be at the front and the red face in back, but their area of overlap has been made inconsistent. In the animation the surface may appear to bend and dive under rather than continuing its expected course.

When we look at the red face in front of the blue we observe that the faces intersect each other along two edges. The continuity of the 2-dimensional square surfaces means that both edges must be in front or behind. However if we replace the surfaces with wire frames, then we can make individual edges consistent or inconsistent independently of each other. Thus we move on to consider a multi-coloured wire-frame cube below. The cube on the left is consistent, the cube in the middle is inconsistent, whereas the one on the right is consistent, but inconsistent in some parts of its rotation..

We can make any of one, two, or even three lines behave inconsistently. These images and animations include the main variations. Due to symmetries between a redline acting inconsistently and the blue lines it intersects behaving inconsistently, some combinations of occluding lines look the same. We also have cases where the differences only involve a single frame difference in the animation of a parallel red and blue lines acting inconsistently.

So far we have looked at the inconsistent occlusions effecting a single cube. As the single cube rotates the topmost and lowest lines may not intersect any other lines. In theory they could be inconsistently forward or backward, but we could not tell. However we can remedy this by considering more complex wire frame images and animations.

With two cubes in line, as above, the central purple square can behave inconsistently or just the blue line at left-back.

In a plane of cubes, as above, the inconsistency can involve the coloured centre squares, or the lines connecting the outer and inner squares.

In a 3-D cube of cubes, as above, the purple centre cross, the middle red bar, or the top left purple square can behave inconsistently. When we watch these rotate parts of the rigid figure appear to bend and float in surprising ways.


So far we have looked at inconsistent cubes, but inconsistencies can arise in other figures. We will next consider an inconsistent triangle. Look at this image. At first sight it is a picture of a normal triangle. But think closely about the pattern of colours on each face in a triangle. Normally we would see 3 inner faces surrounded by the three outer faces. At other angles we can only see 2 outer faces and 2 inner faces with the other faces occluded. But look at the red face, it is inside at the left but outside at the base: this is impossible. But this is a real 3-D construction. This is called a Penrose triangle(1958) and numerous artists have constructed this physical object. In the animation at left we can see this object rotating.

In 1934 Oscar Reutersvard(1915-2001) produced the following image (below right) of an inconsistent triangle constructed out of individual cubes. In this construction the cubes appear like steps. The second animation (at left) of these cubes contains an extra twist- the lines of cubes behave inconsistently when they pass by each other. In the motion of the rigid body we expect one set bf cubes to pass behind the other but the opposite occurs. This type of inconsistency only occurs in a moving animation.

 

Part of the inconsistency we see in the next animation is dependent on the motion and our expectations of what will happen.Consider the construction to the right. It is part of a sequence of images building up from a single rectangular bar which is being rotated. But if we were simply rotating a bar we would expect the same face colour to be facing us throughout the construction.Thus our first simple hypothesis is inconsistent with the facts and we need to generate another hypothesis. Perhaps the bar itself is also rotating. This would change the face colours. If this were true then we would expect each face to appear for the same amount of time (and cover equal areas). But the red covers about 1/4 of the circle, yellow 1/8th and green about 1/2 . So again there is an inconsistency between our expectation and the image data. A possible answer is that the bar is rotating at different speeds. But this means we need an ontological commitment to 2 extra (possibly arbitrary) accelerating forces. As the animation continues past this point we expect a completed circle to form. But again an inconsistency between our theory and the image produced occurs. View the sequence to find out. A new theory about the motion involves the 3rd dimension. It is simpler than our 2-d theory and does not involve arbitrary accelerations. Interestingly, our initial, default theory was 2-dimensional in nature. Two final 'inconsistencies' occur in this sequence, but we may not immediately recognize them as particularly inconsistent. The sequence grows form a single bar into a compete figure and then it shrinks back down to the single bar. The motion of the boundary is steady (clockwise) but it is constructive for the first half and then destructive. The constancy of the direction tricks us into ignoring the inconsistency present. The cycle of growth and the shrinking has a reflective symmetry. So we are perhaps adopting an additional theory that a reflection of sorts has occurred and not an inconsistency. Finally the symmetry of the cycle is broken, during the growth phase green dominates but to complete the path of the bar red is the main colour. It appears inconsistent but this is a consequence of the difference in location and perspective.

Look at the image to the left . As before it is constructed from a 3-d object. In the animation the distance rotated is not constant, but the acceleration is. We rotate the object 1 deg., then 2 then 3 then... up to 144deg. at a single step. The interesting thing here is that we see most of the transitions as rotations. But this is not justified by the data. At the mid stages each jump is over 50   deg., and should be seen as a jump. The later motions are very jerky but we still interpret them as rotations. Interestingly at the very end when each step is of the order of 100deg. we still see them as rotations, but we also see the arm of the object as precessing (i.e. rotating in the opposite direction). This demonstrates how we impose 'constancy' upon the world. We impose the initial vision of rotation even when the later jerky motion does not justify it. In each of these cases we are seeing different types of inconsistency. The motion of the objects and the expectations this engenders in us is a key part of the inconsistent experience.

Other Optical Impossibilities

In the case of the impossible wire-frame cubes the weirdness arises because incorrect occlusions occur. Due to the shape of the cube we expect one bar to be in front of another at the point of overlap, but the image has been adjusted so that the overlap is reversed. The Penrose triangle relies on a particular assumption that our perception makes. If we see two edges meet at a point then we assume that they are in the same plane and are joined together. We do not perceive them as separated in 3-dimensions and simply overlapping which is what we saw in the Penrose triangle. We will now consider some other optical impossibilities that have different underlying causes.The first ofthese is called Schuster's fork. Similar images are "The Devil's Pitchfork", the "twin-pronged trident" and the "three-stick cleavis". A version of it first appeared in the March 23, 1964 issue of Aviation Week and Space Technology and it was subsequently described by Donald H.Schuster in the December issue of the American Journal of Psychology. If you focus on the top half of the image you see three cylindrical tubes but if you look at the bottom half of the image you see two square cross-section columns. The top and bottom halves of the image are not consistent, thus this is impossible. When you look at the leftmost column you can see that it changes from a flat plane (one side of a square column) at the bottom to half of a curved tube at the top. A second effect can be seen if you look at the second column. At the bottom it is the right hand side of a square column but it fades away to nothing as it rises. A solid object dissapearing to nothing is impossible. This effect can be seen more easily in the image below where we have increased the number of columns.

Some critics of impossible drawings have claimed that the effect is due to the way we process line drawings on pages and the way we interpret them as 3-dimensional. The two images above refute this notion as the effects are produced without lines and are due to shading. This makes the animation of the blue fork on blue background appear surreal.

The image at right

is "Impossible Meander #191" by Oscar Reutersvard(1994). This image has also been called "Double impossibility". As well as the impossibility due to the fading away of the columns the figure is strangely twisted. The top sections slope forward as it goes right, however the base goes in the opposite direction. The impossibility arises because the joining columns are all straight and parallel so that there cannot be a twist.

 

A different cause of impossibility can be seen in the next two images. The image at left is by Bruno Ernst and is called "The wearisome and the easy way to the top"(1984). When you watch the animation you can see that the green ball has a much easier trip to make than the red and blue balls. The balls all travel the same height but each traverses a different number of stairs. This impossibility arises because the flat horizontals twist to become verticals. In the line drawing we do not notice this as the lines are all parallel. In the coloured version, especially in the animation, you can follow the green horizontal surface and just detect the twist.

Perhaps one of the most famous impossible images is the picture at left. It is Escher's "Ascending and Descending"(1960). If you follow the progreess of the monks on the stairs it appears that the stairs always rise, but this is impossible.When you look at the animation below you can see that it is based on the same perceptual assumption that we saw in the Penrose triangle, a 3-d object is seen in a single plane.

 

 

Peter Quigley :: 2003, revised 2005.

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