The Helmhotlz’ Irradiation Illusion

The Helmhotlz’ irradiation illusion, as shown in the two figures below, is created by two figures of the identical size (both inside and outside squares), but the small black square which is inside the larger white square (shown in the figure on the left below) looks smaller than the small white square inside the larger black square (shown in the figure on the right below). This is a very strong illusion and the size difference between the small squares inside the larger squares is obvious and impressive. However, this is not a popular illusion in the textbook and almost none of the websites dedicated to the visual illusions has included this size illusion. Thus, it is hard to find any explanation for the illusion. Helmholtz, the discoverer, himself used the principle of irradiation to explain the observation; the term "irradiation" refers to the spreading of light areas into adjacent dark areas, where an increase in the size of a bright area would occur at the expense of an adjacent dark area. This leads to apparent displacement of a black-white boundary so that the contour appears shifted in the direction of the dark area.

        

Since the publication of Murray et al.´s study in 2006, we now know that two objects that project the same visual image size on the retina can appear to occupy very different properties of the visual field; thus, a object that appears to occupy a larger portion of the visual field can activate a larger area in primary visual cortex than an object of equal image size that is perceived to be smaller in the visual field. We can assume that the small squares in the standard Helmhotlz illusion are the same size on the retina; the perceived difference in size must have happened in the primary visual cortex. The next question is why the white small square is perceived to be larger than the black small square in the primary visual cortex. As is well known, light exerts physical force or energy which acts upon our retina and generates sensory stimulation to produce vision. It is also a fact that white color reflects (or re-emits) more light energy (or photons to be specific) than black color; therefore, the small white square puts forth a higher level of energy than the small black square. Because the two small squares are identical in size, hence they have the same size but re-emit different densities of photons. The small white square has a higher density of photons (generated by a higher intensity of light force); thus it may activate a larger area in the primary visual cortex. The more light energy an object reflects, the larger the object tends to be perceived. But, as shown in the figure below, the circles of the same size and same brightness can be perceived to be different in size. The middle orange circle in the larger black circle on the left below appears to be brighter and also larger than the middle circle on the right below although these two middle circles have the exactly same color, brightness and size. As such, the fact is that the relatively brighter object of the same size due to its darker surroundings generates a larger cortex activity and hence is perceived as larger in size.

So far we only know this fact that a brighter object or relatively brighter object with its dark surroundings produces a larger image in our brain. But we do not know the exact mechanism of the perception. We can merely assume that it could be the light energy or photons that exert a force on our cortex. Nevertheless, I am certain that it is a force, no matter whether it is generated by the photons or something else, that has an illusory effect on our perception of the Helmhotlz illusion. A brighter square within a darker larger square or a relatively brighter circle inside a larger darker circle produces more force than a less bright square or circle. The strength of the force is positively correlated with the perceived size of these objects. As a result, we can conclude that a brighter object or relatively brighter object with its darker surroundings exerts a stronger force on its surroundings and on our cortex which in turn produces a larger image in our brain.We can also call this conclusion a principle. Equipped with this principle, we can explain another much more popular shape illusion, called the Cafè wall illusion, and the bulging checkerboard illusion. The successful application of this principle will further prove my force approach to these illusions.

The Cafè Wall Illusion

The famous Café wall illusion, as shown in the figure on the left below, was discovered by the prominent illusion researcher Richard Gregory in 1973. The illusion is created when the tiles of black and white are in the “half-shifted condition” and the mortar lines (the horizontal lines between the tiles) appear to slope alternately upward and downward. But the lines are actually parallel and all tiles are of the same size; when the tiles align (shown in the figure in the middle below)  or make up a checkerboard (shown in the figure on the right below), the illusion disappears.

                 

          

        When there are so many theories to explain one phenomenon, it usually means that no one theory is convincing enough to stop others from pursuing a better understanding. In this case, I am going to use the principle gained from our understanding of the Helmhotlz illusion discussed earlier to analyze the Cafè wall illusion: A brighter object or relatively brighter object with its darker surroundings exerts a stronger force on its surroundings and on our cortex which in turn produces a larger image in our brain. Let´s look at the standard Cafè wall illusion figure below closely. The black and white tiles are not aligned; otherwise the mortar lines would be parallel to each other. Instead, the black and white tiles are stacked on top of each other only half way. (Note: In the standard Cafè wall illusion figure that I drew the tiles are not tacked half way; merely about a third of  the white tiles is stacked by a black tile.) Now let´s look at the figure on the right which is a cut-out of a small part of the standard illusion figure below. We are going to pay close attention to the white square in the middle of the figure. This white square has black tiles or squares on its sides and both black and white tiles on its top and bottom. About a third of its top and bottom is surrounded by the black tiles. The figure on the right looks similar to the figure on the right above. If we imagine the white tiles on top and bottom of the middle white square are black color, then the cut-out figure on the right would be almost identical to the Helmhotlz illusion figure on the right above. The red rectangle that I have drawn on the middle white square is the only part of the white square that is surrounded by the black tiles on the three sides, i.e., its top, left and bottom sides. Its right side is surrounded by a black tile but with a gap. If we can ignore the gap between the red rectangle and its right side black tile, we can regard this cut-out figure as a Helmhotlz figure with a rectangle rather than a square in the middle. According to the principle, the white rectangle with its black surroundings exerts a stronger force on its surroundings; thus the rectangle will expand like a metal heated up or a balloon blown up. The force of the expansion spreads in all directions. When it expands sideways, the force will not affect the mortar lines. Nevertheless, when the rectangle expands upwards and downwards, its inflating force will push the mortar line on its top upwards and push the mortar line at its bottom downwards. Since only the left third part of the white square is exerting an expanding force on the mortar lines above and underneath, only left parts of the mortar lines, not the whole mortar lines, under and above the white square expand, which is like a pair of parallel bars being pulled apart from one end. The result is that the end of the parallel bars being pulled is getting wider than the other end; as a result, a wedge shape is formed. Similarly, the mortar lines directly above and beneath the middle white square form a wedge with the wider end on its left and the narrower end on its right. So this wedge is pointing toward left. When you look at the standard Cafè wall illusion figure below, you will notice that in the same row as the cut-out all the white squares are placed the same way as the sample cut-out. The left sides of the white squares have consistently more expanding force than the right sides; therefore, the mortar lines on the whole form a wedge toward left. This imbalanced force within a single white square is essential for the illusion. When the expanding force is the same for the entire top part and bottom part of the white squares as in the checkerboard figure and aligned figure (shown at beginning of this section), the illusion disappears. If somebody asks why the mortar lines do not appear jagged due to a succession of wedges formed by each white square, I wouldassume that the mortar lines are rigid objects and they only bend as a whole.

 

        The force explanation I have offered based on the principle can predict many variations of the standard Cafè wall illusion. As shown on the right, the white small square inside the larger lighter gray square appears to be slightly smaller than the white small square inside the larger black square. According to the principle, we would predict that the white tiles surrounded by the lighter gray tiles will exert a weaker force on the mortar lines; hence the wedges will be significantly diminished. The figure on the left is the same figure as used for illustrating the standard Cafè wall illusion; but the color of the black tiles has been changed into the light gray to match the color of the second figure on the right. As you can see, the mortar lines do not appear to tilt one way or other, and the mortar lines seem to be parallel to each other. The illusory effect is by and large gone. As a result, the Helmhotlz illusion figures on the right can correctly predict the outcome of the variations of the Cafè wall illusion. This experiment also tells us that there is a threshold for the force to bend mortar lines. When the force exerting on the mortar lines is under the threshold, the mortar lines will not tilt at all and the illusion will be weakened or eliminated completely. If this is the case, we can predict that the wedges in the standard Cafè wall illusion will not be formed by the mortar lines when the width of the mortar lines is increased to a certain point. The logic is that the mortar lines have to be bent by a certain amount of force; when the force pushing the mortar lines is fixed, the stronger the mortar line, the harder it is for this given force to move it. As shown in the figure below, the illusion is significantly diminished when the mortar lines are thickened to the degree that the mortar line is strong enough to resist the force pushing on it. As a result, the prediction comes true. The application of the principle discovered for the Helmhotlz illusion to the Cafè wall illusion can be claimed to be successful.

              

The Bulging Checkerboard Illusion

        The figure below is called the Bulging checkerboard illusion. The illusion looks like the checkerboard bulging right off the screen. But this bulge is illusory because the checkerboard is fully regular and each check is a regular square. So far there is not any credible explanation out there for the illusion. Akiyoshi Kitaoka, an illusion researcher who collects this kind of illusions,  classifies the phenomenon as a “geometrical illusion with 3D impression”; in other words, it has something to do with distance and perspective (Kitaoka, 1998, 2004, 2007). As a rule, whenever someone mentions the perspective in the explanation of an illusion, it will induce a knee-jerk like agreement. It seems that most people would accept that all the illusions are caused by the interpretation of our brain about the perspective without a second thought. However, I am here to challenge this widely accepted explanation. Besides, nobody has offered any explanation for the mechanism of the illusion. I am going to rely on the following principle to explain the mechanism of the illusion: A brighter object or relatively brighter object with its darker surroundings exerts a stronger force on its surroundings and on our cortex which in turn produces a larger image in our brain.

          When all the small squares (also called the ticks) are removed from the larger squares (also called the checks), the bulging illusion is eliminated. Therefore, the ticks, both white and black ones, in the checks must play an essential role in the illusion. Now we have to ascertain how much of a role either the white ticks or the black ticks play in the illusion. In the figure on the left below, I have taken out all the black ticks from the right side of the bulging globe. Without the black ticks on its right side the bulging effect of the globe is not weakened very much. In the figure on the right below, all the white ticks are removed from the right side of the globe. Now the bulging effect of the right side of the globe has been severely reduced. These experiments demonstrate to us that the white ticks play a much more important role in the bulging illusion.

        At this point, we have to take a closer look at the bulging illusion figure. I have cut out a small portion, which is shown as the red rectangle in the figure on the left below, from the left side of the bulging globe. As you can see, the red rectangle is not regular. Its right side is slightly wider than its left side; thus its top side slants downwards from right to left and its bottom side slants upwards from right to left. Also, both the left side and right side of the rectangle are bent toward left. Furthermore, all the vertical lines inside the red rectangle are bent like the sides; and all the horizontal lines within the rectangle are shaped like the top and bottom sides. The figure on the right below is the enlarged cut-out, which shows the same patterns as the red rectangle that I have just described. 

        The three figures below are the same cut-out; but I have taken out all the black ticks for the one on the left, all the white ticks for the one in the middle, and all the ticks for the one on the right. As we have already known, the white ticks play a more important role than the black ticks in the illusion since the vertical lines with the white ticks are more bent and the horizontal lines with the white ticks are more tilted than those lines in the figure with the black ticks only, and the checks become perfectly regular without these ticks. 

        The next puzzle we have to solve is why the checkerboard with the ticks appears bulging out. If we use the concept of perspective to explain that the center of the globe seems to be closer to us and its edge seems to be further away from us, then it would be incoherent. According to the perspective and the size constancy theory derived from it, for the objects that project the same size retinal image the farther ones are perceived as bigger. If this is the case, the central part of the checkerboard should be perceived as farther away rather than closer since the checks there are perceived as bigger even though they are the same size as those on the edge. As a result, the checkerboard with the ticks would look like caving in rather than bulging out if the perspective is the cause of the illusion. Then, what is the cause of the bulging effect? The answer lie in the next article which offers some far more important discoveries. In brief, the bulging effect has something to do with the visual field volume; an object that occupies a larger part of the visual field seems to be closer than those occupying a smaller visual field. The checks in the central part of the illusory checkerboard is perceived as larger, therefore occupying a larger portion of the visual field and perceived as closer.
        Thus far you may be impressed with the effectiveness of the principle discovered from the Helmhotlz illusion figure in explaining both the Cafè wall and Bulging checkerboard illusions. However, the understanding of these illusions does not have as far-reaching implications as the illusions we are going to encounter next. It merely proves that these illusions are influenced by a force beyond our control and subjective interpretation and understanding. Now fasten your belt. We are going to have quite a ride in the next articles.  

References

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Gregory R. L. (1997). Eye and brain: The psychology of seeing (5^th ed.). Princeton University Press.

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     Kitaoka, A. (1998). Apparent contraction of edge angles. Perception, 27, 1209-1219.

Myers, D. G. (2003). Psychology (7th ed.). New York: Worth Publishers.

Pritchard, R. M. (1961, June). Stabilized images on the retina. Scientific American, 72-78.

Shepard, R. N. (1990). Mind sights. New York: Freeman.

Wenderoth, P. (1992). Perceptual illusions. Australian Journal of Psychology, 44, 147-151.