Part 6

If oblique lines are extended across vertical ones, as in Figs. 45 and 46, the illusion is seen to be very striking. In Fig. 45 the oblique line on the right if extended would meet the upper end of the oblique line on the left; however, the apparent point of intersection is somewhat lower than it is in reality. In Fig. 46 the oblique line on the left is in the same straight line with the lower oblique line on the right. The line drawn parallel to the latter furnishes an idea of the extent of the illusion. This is the well-known Poggendorff illusion. The upper oblique line on the right actually appears to be approximately the continuation of the upper oblique line on the right. The explanation of this illusion on the simple basis of underestimation or overestimation of angles is open to criticism. If Fig. 46 is held so that the intercepted line is horizontal or vertical, the illusion disappears or at least is greatly reduced. It is difficult to reconcile this disappearance of the illusion for certain positions of the figure with the theory that the illusion is due to an incorrect appraisal of the angles.

According to Judd,[2] those portions of the parallels lying on the obtuse-angle side of the intercepted line will be overestimated when horizontal or vertical distances along the parallel lines are the subjects of attention, as they are in the usual positions of the Poggendorff figure. He holds further that the overestimation of this distance along the parallels (the two vertical lines) and the underestimation of the oblique distance across the interval are sufficient to provide a full explanation of the illusion. The disappearance and appearance of the illusion, as the position of the figure is varied appears to demonstrate the fact that lines produce illusions only when they have a direct influence on the direction in which the attention is turned. That is, when this Poggendorff figure is in such a position that the intercepted line is horizontal, the incorrect estimation of distance along the parallels has no direct bearing on the distance to which the attention is directed. In this case Judd holds that the entire influence of the parallels is absorbed in aiding the intercepted line in carrying the eye across the interval. For a detailed account the reader is referred to the original paper.

Some other illusions are now presented to demonstrate further the effect of the presence of angles. Doubtless, in some of these, other causes contribute more or less to the total result. In Fig. 47 a series of concentric arcs of circles end in a straight line. The result is that the straight line appears to sag perceptibly. Incidentally, it may be interesting for the reader to ascertain whether or not there is any doubt in his mind as to the arcs appearing to belong to circles. To the author the arcs appear distorted from those of true circles.

In Fig. 48 the bounding figure is a true circle but it appears to be distorted or dented inward where the angles of the hexagon meet it. Similarly, the sides of the hexagon appear to sag inward where the corners of the rectangle meet them.

The influences which have been emphasized apparently are responsible for the illusions in Figs. 49, 50 and 51. It is interesting to note the disappearance of the illusion, as the plane of Fig. 49 is varied from vertical toward the horizontal. That is, it is very apparent when viewed perpendicularly to the plane of the page, the latter being held vertically, but as the page is tilted backward the illusion decreases and finally disappears.

The illusions in Figs. 50 and 51 are commonly termed "twisted cord" effects. A cord may be made by twisting two strands which are white and black (or any dark color) respectively. This may be superposed upon various backgrounds with striking results. In Fig. 50 the straight "cords" appear bent in the middle, owing to a reversal of the "twist." Such a figure may be easily made by using cord and a checkered cloth. In Fig. 51 it is difficult to convince the intellect that the "cords" are not arranged in the form of concentric circles, but this becomes evident when one of them is traced out. The influence of the illusion is so powerful that it is even difficult to follow one of the circles with the point of a pencil around its entire circumference. The cord appears to form a spiral or a helix seen in perspective.

A striking illusion is obtained by revolving the spiral shown in Fig. 52 about its center. This may be considered as an effect of angles because the curvature and consequently the angle of the spiral is continually changing. There is a peculiar movement or progression toward the center when revolved in one direction. When the direction of rotation is reversed the movement is toward the exterior of the figure; that is, there is a seeming expansion.

Angles appear to modify our judgments of the length of lines as well as of their direction. Of course, it must be admitted that some of these illusions might be classified under those of "contrast" and others. In fact, it has been stated that classification is difficult but it appears logical to discuss the effect of angles in this chapter apart from the divisions presented in the preceding chapters. This decision was reached because the effect of angles could be seen in many of the illusions which would more logically be grouped under the classification presented in the preceding chapters.

In Fig. 53 the three horizontal lines are of equal length but they appear unequal. This must be due primarily to the size of the angles made by the lines at the ends. Within certain limits, the greater the angle the greater is the apparent elongation of the central horizontal portion. This generalization appears to apply even when the angle is less than a right angle, although there appears to be less strength to the illusions with these smaller angles than with the larger angles. Other factors which contribute to the extent of the illusion are the positions of the figures, the distance between them, and the juxtaposition of certain lines. The illusion still exists if the horizontal lines are removed and also if the figures are cut out of paper after joining the lower ends of the short lines in each case.

In Fig. 54 the horizontal straight line appears to consist of two lines tilting slightly upward toward the center. This will be seen to be in agreement with the general proposition that the sides of an angle are deviated in the direction of the angle. In this case it should be noted that one of the obtuse angles to be considered is _ABC_ and that the effect of this is to tilt the line _BD_ downward from the center. In Fig. 55 the horizontal line appears to tilt upward toward its extremities or to sag in the middle. The explanation in order to harmonize with the foregoing must be based upon the assumption that our judgments may be influenced by things not present but imagined. In this case only one side of each obtuse angle is present, the other side being formed by continuing the horizontal line both ways by means of the imagination. That we do this unconsciously is attested to by many experiences. For example, we often find ourselves imagining a horizontal, a vertical, or a center upon which to base a pending judgment.

A discussion of the influence of angles must include a reference to the well-known Mueller-Lyer illusion presented in Fig. 56. It is obvious in _a_ that the horizontal part on the left appears considerably longer than that part in the right half of the diagram. The influence of angles in this illusion can be easily tested by varying the direction of the lines at the ends of the two portions.

In all these figures the influence of angles is obvious. This does not mean that they are always solely or even primarily responsible for the illusion. In fact, the illusion of Poggendorff (Fig. 46) may be due to the incorrect estimation of certain linear distances, but the angles make this erroneous judgment possible, or at least contribute toward it. Many discussions of the theories or explanations of these figures are available in scientific literature of which one by Judd[2] may be taken as representative. He holds that the false estimation of angles in the Poggendorff figure is merely a secondary effect, not always present, and in no case the source of the illusion; furthermore, that the illusion may be explained as due to the incorrect linear distances, and may be reduced to the type of illusion found in the Mueller-Lyer figure. Certainly there are grave dangers in explaining an illusion on the basis of an apparently simple operation.

In Fig. 56, _b_ is made up of the two parts of the Mueller-Lyer illusion. A small dot may be placed equally distant from the inside extremities of the horizontal lines. It is interesting to note that overestimation of distance within the figure is accompanied with underestimation outside the figure and, conversely, overestimation within the figure is accompanied by underestimation in the neighboring space. If the small dot is objected to as providing an additional Mueller-Lyer figure of the empty space, this dot may be omitted. As a substitute an observer may try to locate a point midway between the inside extremities of the horizontal lines. The error in locating this point will show that the illusion is present in this empty space.

In this connection it is interesting to note some other illusions. In Fig. 57 the influence of several factors are evident. Two obviously important ones are (1) the angles made by the short lines at the extremities of the exterior lines parallel to the sides of the large triangle, and (2) the influence of contrast of the pairs of adjacent parallel lines. The effect shown in Fig. 53 is seen to be augmented by the addition of contrast of adjacent lines of unequal length.

An interesting variation of the effect of the presence of angles is seen in Fig. 58. The two lines forming angles with the horizontal are of equal length but due to their relative positions, they do not appear so. It would be quite misleading to say that this illusion is merely due to angles. Obviously, it is due to the presence of the two oblique lines. It is of interest to turn to Figs. 25, 26 and various illusions of perspective.

At this point a digression appears to be necessary and, therefore, Fig. 59 is introduced. Here the areas of the two figures are equal. The judgment of area is likely to be influenced by juxtaposed lines and therefore, as in this case, the lower appears larger than the upper one. Similarly two trapezoids of equal dimensions and areas may be constructed. If each is constructed so that it rests upon its longer parallel and one figure is above the other and only slightly separated, the mind is tempted to be influenced by comparing the juxtaposed base of the upper with the top of the lower trapezoid. The former dimension being greater than the latter, the lower figure appears smaller than the upper one. Angles must necessarily play a part in these illusions, although it is admitted that other factors may be prominent or even dominant.

This appears to be a convenient place to insert an illusion of area based, doubtless, upon form, but angles must play a part in the illusions; at least they are responsible for the form. In Fig. 60 the five figures are constructed so as to be approximately equal in area. However, they appear unequal in this respect. In comparing areas, we cannot escape the influence of the length and directions of lines which bound these areas, and also, the effect of contrasts in lengths and directions. Angles play a part in all these, although very indirectly in some cases.

To some extent the foregoing is a digression from the main intent of this chapter, but it appears worth while to introduce these indirect effects of the presence of angles (real or imaginary) in order to emphasize the complexity of influences and their subtleness. Direction is in the last analysis an effect of angle; that is, the direction of a line is measured by the angle it makes with some reference line, the latter being real or imaginary. In Fig. 61, the effect of diverting or directing attention by some subtle force, such as suggestion, is demonstrated. This "force" appears to contract or expand an area. The circle on the left appears smaller than the other. Of course there is the effect of empty space compared with partially filled space, but this cannot be avoided in this case. However, it can be shown that the suggestions produced by the arrows tend to produce apparent reduction or expansion of areas. Note the use of arrows in advertisements.

Although theory is subordinated to facts in this book, a glimpse here and there should be interesting and helpful. After having been introduced to various types and influences, perhaps the reader may better grasp the trend of theories. The perspective theory assumes, and correctly so, that simple diagrams often suggest objects in three dimensions, and that the introduction of an imaginary third dimension effects changes in the appearance of lines and angles. That is, lengths and directions of lines are apparently altered by the influence of lines and angles, which do not actually exist. That this is true may be proved in various cases. In fact the reader has doubtless been convinced of this in connection with some of the illusions already discussed. Vertical lines often represent lines extending away from the observer, who sees them foreshortened and therefore they may seem longer than horizontal lines of equal length, which are not subject to foreshortening. This could explain such illusions as seen in Figs. 4 and 5. However this theory is not as easily applied to many illusions.

According to Thiery's perspective theory a line that appears nearer is seen as smaller and a line that seems to be further away is perceived as longer. If the left portion of _b_, Fig. 56, be reproduced with longer oblique lines at the ends but with the same length of horizontal lines, it will appear closer and the horizontal lines will be judged as shorter. The reader will find it interesting to draw a number of these portions of the Mueller-Lyer figure with the horizontal line in each case of the same length but with longer and longer obliques at the ends.

The dynamic theory of Lipps gives an important role to the inner activity of the observer, which is not necessarily separated from the objects viewed, but may be felt as being in the objects. That is, in viewing a figure the observer unconsciously separates it from surrounding space and therefore creates something definite in the latter, as a limiting activity. These two things, one real (the object) and one imaginary, are balanced against each other. A vertical line may suggest a necessary resistance against gravitational force, with the result that the line appears longer than a horizontal one resting in peace. The difficulty with this theory is that it allows too much opportunity for purely philosophical explanations, which are likely to run to the fanciful. It has the doubtful advantage of being able to explain illusions equally well if they are actually reversed from what they are. For example, gravity could either contract or elongate the vertical line, depending upon the choice of viewpoint.

The confusion theory depends upon attention and begins with the difficulty of isolating from illusory figures the portions to be judged. Amid the complexity of the figure the attention cannot easily be fixed on the portions to be judged. This results in confusion. For example, if areas of different shapes such as those in Fig. 60 are to be compared, it is difficult to become oblivious of form or of compactness. In trying to see the two chief parallel lines in Fig. 38, in their true parallelism the attention is being subjected to diversion, by the short oblique parallels with a compromising result. Surely this theory explains some illusions successfully, but it is not so successful with some of the illusions of contrast. The fact that practice in making judgments in such cases as Figs. 45 and 56 reduces the illusion even to ultimate disappearance, argues in favor of the confusion theory. Perhaps the observer devotes himself more or less consciously to isolating the particular feature to be judged and finally attains the ability to do so. According to Auerbach's indirect-vision theory the eyes in judging the two halves of the horizontal line in _a_, Fig. 56, involuntarily draw imaginary lines parallel to this line but above or below it. Obviously the two parts of such lines are unequal in the same manner as the horizontal line in the Mueller-Lyer figure appears divided into two unequal parts.

Somewhat analogous to this in some cases is Brunot's mean-distance theory. According to this we establish "centers of gravity" in figures and these influence our judgments.

These are glimpses of certain trends of theories. None is a complete success or failure. Each explains some illusions satisfactorily, but not necessarily exclusively. For the present, we will be content with these glimpses of the purely theoretical aspects of visual illusions.

VII

ILLUSIONS OF DEPTH AND OF DISTANCE

Besides the so-called geometrical illusions discussed in the preceding chapters, there is an interesting group in which the perception of the third dimension is in error. When any of the ordinary criteria of relief or of distance are apparently modified, illusions of this kind are possible. There are many illusions of this sort, such as the looming of objects in a fog; the apparent enlargement of the sun and moon near the horizon; the flattening of the "vault" of the sky; the intaglio seen as relief; the alteration of relief with lighting; and various changes in the landscape when regarded with the head inverted.

Although some of the criteria for the perception of depth or of distance have already been pointed out, especially in Chapter III, these will be mentioned again. Distance or depth is indicated by the distribution of light and shade, and an unusual object like an intaglio is likely to be mistaken for relief which is more common. An analysis of the lighting will usually reveal the real form of the object. (See Figs. 70, 71, 72, 73, 76 and 77.) In this connection it is interesting to compare photographic negatives with their corresponding positive prints.

Distance is often estimated by the definition and color of objects seen through great depths of air (aerial perspective). These distant objects are "blurred" by the irregular refraction of the light-rays through non-homogeneous atmosphere. They are obscured to some degree by the veil of brightness due to the illuminated dust, smoke, etc., in the atmosphere. They are also tinted (apparently) by the superposition of a tinted atmosphere. Thus we have "dim distance," "blue peaks," "azure depths of sky," etc., represented in photographs, paintings, and writings. Incidentally, the sky above is blue for the same general reasons that the atmosphere, intervening between the observer and a distant horizon, is bluish. The ludicrous errors made in estimating distances in such regions as the Rockies is usually accounted for by the rare clearness and homogeneity of the atmosphere. However, is the latter a full explanation? To some extent we judge unknown size by estimated distance, and unknown distance by estimated size. When a person is viewing a great mountain peak for the first time, is he not likely to assume it to be comparable in size to the hills with which he has been familiar? Even by allowing considerable, is he not likely to greatly underestimate the size of the mountain and, as a direct consequence, to underestimate the distance proportionately? This incorrect judgment would naturally be facilitated by the absence of "dimness" and "blueness" due to the atmospheric haze.

Angular perspective, which apparently varies the forms of angles and produces the divergence of lines, contributes much information in regard to relative and absolute distances from the eye of the various objects or the parts of an object. For example, a rectangle may appear as a rhomboid. It is obvious that certain data pertaining to the objects viewed must be assumed, and if the assumptions are incorrect, illusions will result. These judgments also involve, as most judgments do, other data external to the objects viewed. Perhaps these incorrect judgments are delusions rather than illusions, because visual perception has been deluded by misinformation supplied by the intellect.

Size or linear perspective is a factor in the perception of depth or of distance. As has been stated, if we know the size experience determines the distance; and conversely, if we know the distance we may estimate the size. Obviously estimates are involved and these when incorrect lead to false perception or interpretation.

As an object approaches, the axes of the eyes converge more and more and the eye-lens must be thickened more and more to keep the object in focus. As stated in Chapter III, we have learned to interpret these accompanying sensations of muscular adjustment. This may be demonstrated by holding an object at an arm's length and then bringing it rapidly toward the eyes, keeping it in focus all the time. The sensations of convergence and accommodation are quite intense.

The two eyes look at a scene from two different points of view respectively and their images do not perfectly agree, as has been shown in Figs. 2 and 3. This binocular disparity is responsible to some degree for the perception of depth, as the stereoscope has demonstrated. If two spheres of the same size are suspended on invisible strings, one at six feet, the other at seven feet away, one eye sees the two balls in the same plane, but one appears larger than the other. With binocular vision the balls appear at different distances, but judgment appraises them as of approximately equal size. At that distance the focal adjustment is not much different for both balls, so that the muscular movement, due to focusing the eye, plays a small part in the estimation of the relative distance. Binocular disparity and convergence are the primary factors.