THE INFLUENCE OF ANGLESby@matthewluckiesh

# THE INFLUENCE OF ANGLES

April 16th, 2023

As previously stated, no satisfactory classification of visual illusions exists, but in order to cover the subject, divisions are necessary. For this reason the reader is introduced in this chapter to the effects attending the presence of angles. By no means does it follow that this group represents another type, for it really includes many illusions of several types. The reason for this grouping is that angles play an important part, directly or indirectly, in the production of illusions. For a long time many geometrical illusions were accounted for by “overestimation” or “underestimation” of angles, but this view has often been found to be inadequate. However, it cannot be denied that many illusions are due at least to the presence of angles.

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## VI. THE INFLUENCE OF ANGLES

As previously stated, no satisfactory classification of visual illusions exists, but in order to cover the subject, divisions are necessary. For this reason the reader is introduced in this chapter to the effects attending the presence of angles. By no means does it follow that this group represents another type, for it really includes many illusions of several types. The reason for this grouping is that angles play an important part, directly or indirectly, in the production of illusions. For a long time many geometrical illusions were accounted for by “overestimation” or “underestimation” of angles, but this view has often been found to be inadequate. However, it cannot be denied that many illusions are due at least to the presence of angles.

Apparently Zöllner was the first to describe an illusion which is illustrated in simple form in Fig. 29 and more elaborately in Figs. 37 to 40. The two figures at the right of Fig. 29 were drawn for another purpose and are not designed favorably for the effect, although it may be detected when the figure is held at a distance. Zöllner accidentally noticed the illusion on a pattern designed for a print for dress-goods. The illusion is but slightly noticeable in Fig. 29, but by multiplying the number of lines (and angles) the long parallel lines appear to diverge in[Pg 77] the direction that the crossing lines converge. Zöllner studied the case thoroughly and established various facts. He found that the illusion is greatest when the long parallel lines are inclined about 45 degrees to the horizontal. This may be accomplished for Fig. 37, by turning the page (held in a vertical plane) through an angle of 45 degrees from normal. The illusion vanishes when held too far from the eye to distinguish the short crossing lines, and its strength varies with the inclination of the oblique lines to the main parallels. The most effective angle between the short crossing lines and the main parallels appears to be approximately 30 degrees. In Fig. 37 there are two illusions of direction. The parallel vertical strips appear unparallel and the right and left portions of the oblique cross-lines appear to be shifted vertically. It is interesting to note that steady fixation diminishes and even destroys the illusion.

Fig. 37.—Zöllner’s illusion of direction.

[Pg 78]The maximum effectiveness of the illusion, when the figure is held so that the main parallel lines are at an inclination of about 45 degrees to the horizontal was accounted for by Zöllner as the result of less visual experience in oblique directions. He insisted that it takes less time and is easier to infer divergence or convergence than parallelism. This explanation appears to be disproved by a figure in which slightly divergent lines are used instead of parallel ones. Owing to the effect of the oblique crossing lines, the diverging lines may be made to appear parallel. Furthermore it is difficult to attach much importance to Zöllner’s explanation because the illusion is visible under the extremely brief illumination provided by one electric spark. Of course, the duration of the physiological reaction is doubtless greater than that of the spark, but at best the time is very short. Hering explained the Zöllner illusion as due to the curvature of the retina, and the resulting difference in the retinal images, and held that acute angles appear relatively too large and obtuse ones too small. The latter has been found to have limitations in the explanation of certain illusions.

This Zöllner illusion is very striking and may be constructed in a variety of forms. In Fig. 37 the effect is quite apparent and it is interesting to view the figure at various angles. For example, hold the figure so that the broad parallel lines are vertical. The illusion is very pronounced in this position; however, on tilting the page backward the illusion finally disappears. In Fig. 38 the short oblique lines do not cross the long parallel lines and to make the[Pg 79] illusion more striking, the obliquity of the short lines is reversed at the middle of the long parallel lines. Variations of this figure are presented in Figs. 39 and 40. In this case by steady fixation the perspective effect is increased but there is a tendency for the parallel lines to appear more nearly truly parallel than when the point of sight is permitted to roam over the figures.

Fig. 38.—Parallel lines which do not appear so.

Fig. 39.—Wundt’s illusion of direction.

Fig. 40.—Hering’s illusion of direction.

Many investigations of the Zöllner illusion are recorded in the literature. From these it is obvious that the result is due to the additive effects of many simple illusions of angle. In order to give an idea of the manner in which such an illusion may be built up the reasoning of Jastrow[1] will be presented in condensed form. When two straight lines such as A and B in Fig. 41 are separated by a space it is usually possible to connect the two mentally and to determine whether or not, if connected, they would lie[Pg 81] on a straight line. However, if another line is connected to one, thus forming an angle as C does with A, the lines which appeared to be continuous (as A and B originally) no longer appear so. The converse is also true, for lines which are not in the same straight line may be made to appear to be by the addition of another line forming a proper angle. All these variations cannot be shown in a single figure, but the reader will find it interesting to draw them. Furthermore, the letters used on the diagram in order to make the description clearer may be confusing and these can be eliminated by redrawing the figure. In Fig. 41 the obtuse angle AC tends to tilt A downward, so apparently if A were prolonged it would fall below B. Similarly, C appears to fall to the right of D.

Fig. 41.—Simple effect of angles.

This illusion apparently is due to the presence of[Pg 82] the angle and the effect is produced by the presence of right and acute angles to a less degree. The illusion decreases or increases in general as the angle decreases or increases respectively.

Although it is not safe to present simple statements in a field so complex as that of visual illusion where explanations are still controversial, it is perhaps possible to generalize as Jastrow did in the foregoing case as follows: If the direction of an angle is that of the line bisecting it and pointing toward the apex, the direction of the sides of an angle will apparently be deviated toward the direction of the angle. The deviation apparently is greater with obtuse than with acute angles, and when obtuse and acute angles are so placed in a figure as to give rise to opposite deviations, the greater angle will be the dominant influence.

Although the illusion in such simple cases as Fig. 41 is slight, it is quite noticeable. The effect of the addition of many of these slight individual influences is obvious in accompanying figures of greater complexity. These individual effects can be so multiplied and combined that many illusory figures may be devised.

In Fig. 42 the oblique lines are added to both horizontal lines in such a manner that A is tilted downward at the angle and B is tilted upward at the angle (the letters corresponding to similar lines in Fig. 41). In this manner they appear to be deviated considerably out of their true straight line. If the reader will draw a straight line nearly parallel to D in Fig. 41 and to the right, he will find it helpful.[Pg 83] This line should be drawn to appear to be a continuation of C when the page is held so D is approximately horizontal. This is a simple and effective means of testing the magnitude of the illusion, for it is measured by the degree of apparent deviation between D and the line drawn adjacent to it, which the eye will tolerate. Another method of obtaining such a measurement is to begin with only the angle and to draw the apparent continuation of one of its lines with a space intervening. This deviation from the true continuation may then be readily determined.

Fig. 42. The effect of two angles in tilting the horizontal lines.

Fig. 43. The effect of crossed lines upon their respective apparent directions.

A more complex case is found in Fig. 43 where the effect of an obtuse angle ACD is to make the continuation[Pg 84] of AB apparently fall below FG and the effect of the acute angle is the reverse. However, the net result is that due to the preponderance of the effect of the larger angle over that of the smaller. The line EC adds nothing, for it merely introduces two angles which reinforce those above AB. The line BC may be omitted or covered without appreciably affecting the illusion.

Fig. 44.—Another step toward the Zöllner illusion.

In Fig. 44 two obtuse angles are arranged so that their effects are additive, with the result that the horizontal lines apparently deviate maximally for such a simple case. Thus it is seen that the tendency of the sides of an angle to be apparently deviated toward the direction of the angle may result in an apparent divergence from parallelism as well as in making continuous lines appear discontinuous. The illusion in Fig. 44 may be strengthened by adding more lines parallel to the oblique lines. This is demonstrated in Fig. 38 and in other illustrations. In this manner striking illusions are built up.

Fig. 45.—The two diagonals would
meet on the left vertical line.

Fig. 46.—Poggendorff’s illusion.
Which oblique line on the right is the
prolongation of the oblique line on the left?

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.

Fig. 47.—A straight line appears to sag.

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[Pg 87] 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.

Fig. 48.—Distortions of contour due to contact with other contours.

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. 4950 and 51. It is interesting to note the disappearance of the illusion, as the plane of Fig. 49 is varied from[Pg 88] 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.

Fig. 49.—An illusion of direction.

Fig. 50.—“Twisted-cord” illusion. These are straight cords.

Fig. 51.—“Twisted-cord” illusion. These are concentric circles.

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[Pg 90] 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.

Fig. 52.—A spiral when rotated appears to expand or contract, depending upon direction of rotation.

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[Pg 91] 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.

Fig. 53.—Angles affect the apparent length of lines.

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[Pg 92] are cut out of paper after joining the lower ends of the short lines in each case.

Fig. 54.—The horizontal line appears to tilt downward toward the ends.

Fig. 55.—The horizontal line appears to sag in the middle.

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.

[Pg 93]A discussion of the influence of angles must include a reference to the well-known Müller-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.

Fig. 56.—The Müller-Lyer illusion.

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[Pg 94] 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 Müller-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 Müller-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 Müller-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[Pg 95] 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. 2526 and various illusions of perspective.

Fig. 57.—Combined influence of angles and contrasting lengths.

Fig. 58.—Two equal oblique lines appear unequal because of their different positions.

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[Pg 96] 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.

Fig. 59.—An illusion of area.

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[Pg 97] 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.

Fig. 60.—Five equal areas showing the influence of angles and contrasting lengths.

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[Pg 98] 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.

Fig. 61.—Showing the effect of directing the attention.

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[Pg 99] 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 Thiéry’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 Müller-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.[Pg 100] 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[Pg 101] the two parts of such lines are unequal in the same manner as the horizontal line in the Müller-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.

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