Chapter 6 - The Influence of Angles:
Zollner Optical Illusions

As previously stated, no satisfactory classification of optical 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 optical illusions of several types. The reason for this grouping is that angles play an important part, directly or indirectly, in the production of optical 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 optical illusions are due at least to the presence of angles.

Fig. 37. - Zollner's optical illusion of direction.

Apparently Zollner was the first to describe an optical illusion which is illustrated in simple form in Fig. 29 and more elaborately in Fig 37 through Fig 40. The two cross hatched lines in the right side 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. Zollner accidentally noticed the optical illusion on a pattern designed for a print for dress-goods. The optical 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 the direction that the crossing lines converge. Zollner studied the case thoroughly and established various facts. He found that the optical 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 optical 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 optical illusion.

The maximum effectiveness of the optical 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 Zollner 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 Zollner's explanation because the optical 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 Zollner optical 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 optical illusions.

This Zollner optical illusion is striking and can be constructed in a many different forms. In Fig. 37 the effect is quite easily seen. It is interesting to view the figure at various angles. For example, hold the figure so that the broad parallel lines are vertical. The optical illusion is very pronounced in this position; however, on tilting the page backward the optical illusion finally disappears. In Fig. 38 the short oblique lines do not cross the long parallel lines and to make the optical illusion more striking, the obliquity of the short lines is reversed at the middle of the long parallel lines.

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

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. 39. - Wundt's optical illusion of direction.

Fig. 40. - Hering's optical illusion of direction.

Many investigations of the Zollner optical illusion are recorded in the literature. From these it is obvious that at the result is due to the additive effects of many simple optical illusions of angle. In order to give an idea of the manner in which such an optical illusion may be built up the reasoning of Jastrow [A Study of Zollner's Figures and Other Related Figures, J. Jastrow, Amer. Jour. of Psych. 1891, 4, p. 381.] 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 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.

Fig. 41. - Simple effect of angles.

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.

This optical illusion apparently is due to the presence of the angle and the effect is produced by the presence of right and acute angles to a less degree. The optical 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 optical 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 optical 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.

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