Chapter 6 - The Influence of Angles:
More Angle Optical Illusions

In this connection it is interesting to note some other optical 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.

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

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 optical 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 Fig. 25, Fig. 26 and various optical illusions of perspective.

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 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 optical illusions, although it is admitted that other factors may be prominent or even dominant.

Fig. 59. - An optical illusion of area.

This appears to be a convenient place to insert an optical illusion of area based, doubtless, upon form, but angles must play a part in the optical 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.

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. In the last analysis, direction is an effect of angle. In other words, the direction of a line is measured by the angle it makes with a 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.

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 been convinced of this in connection with some of the optical 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 optical illusions as seen in Fig. 4 and Fig. 5. However this theory is not as easily applied to many optical 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 Muller-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 optical 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 optical illusions successfully, but it is not so successful with some of the optical illusions of contrast. The fact that practice in making judgments in such cases as Fig. 45 and Fig. 56 reduces the optical 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 Muller-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 optical illusions satisfactorily, but not necessarily exclusively. For the present, we will be content with these glimpses of the purely theoretical aspects of optical illusions.

Continue on to Chapter 7 on Distance and Depth Perception

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