EQUIVOCAL FIGURESby@matthewluckiesh

# EQUIVOCAL FIGURES

April 15th, 2023

MANY figures apparently change in appearance owing to fluctuations in attention and in associations. A human profile in intaglio (Figs. 72 and 73) may appear as a bas-relief. Crease a card in the middle to form an angle and hold it at an arm’s length. When viewed with one eye it can be made to appear open in one way or the other; that is, the angle may be made to appear pointing toward the observer or away from him. The more distant part of an object may be made to appear nearer than the remaining part. Plane diagrams may seem to be solids. Deception of this character is quite easy if the light-source and other extraneous factors are concealed from the observer. It is very interesting to study these fluctuating figures and to note the various extraneous data which lead us to judge correctly. Furthermore, it becomes obvious that often we see what we expect to see. For example, we more commonly encounter relief than intaglio; therefore, we are likely to think that we are looking at the former.

Visual Illusions: Their Causes, Characteristics and Applications by Matthew Luckiesh is part of the HackerNoon Books Series. You can jump to any chapter in this book here. EQUIVOCAL FIGURES

## V. EQUIVOCAL FIGURES

MANY figures apparently change in appearance owing to fluctuations in attention and in associations. A human profile in intaglio (Figs. 72 and 73) may appear as a bas-relief. Crease a card in the middle to form an angle and hold it at an arm’s length. When viewed with one eye it can be made to appear open in one way or the other; that is, the angle may be made to appear pointing toward the observer or away from him. The more distant part of an object may be made to appear nearer than the remaining part. Plane diagrams may seem to be solids. Deception of this character is quite easy if the light-source and other extraneous factors are concealed from the observer. It is very interesting to study these fluctuating figures and to note the various extraneous data which lead us to judge correctly. Furthermore, it becomes obvious that often we see what we expect to see. For example, we more commonly encounter relief than intaglio; therefore, we are likely to think that we are looking at the former.

Proper consideration of the position of the dominant light-source and of the shadows will usually provide the data for a correct conclusion. However, habit and probability are factors whose influence is difficult to overcome. Our perception is strongly[Pg 65] associated with accustomed ways of seeing objects and when the object is once suggested it grasps our mind completely in its stereotyped form. Stairs, glasses, rings, cubes, and intaglios are among the objects commonly used to illustrate this type of illusion. In connection with this type, it is well to realize how tenaciously we cling to our perception of the real shapes of objects. For example, a cube thrown into the air in such a manner that it presents many aspects toward us is throughout its course a cube.

Fig. 27.—Illustrating fluctuation of attention.

The figures which exhibit these illusions are obviously those which are capable of two or more spatial relations. The double interpretation is more readily accomplished by monocular than by binocular vision. Fig. 27 consists of identical patterns in black and white. By gazing upon this steadily it will appear[Pg 66] to fluctuate in appearance from a white pattern upon a black background to a black pattern upon a white background. Sometimes fluctuation of attention apparently accounts for the change and, in fact, this can be tested by willfully altering the attention from a white pattern to a black one. Incidentally one investigator found that the maximum rate of fluctuation was approximately equal to the pulse rate, although no connection between the two was claimed. It has also been found that inversion is accompanied by a change in refraction of the eye.

Fig. 28.—The grouping of the circles fluctuates.

Another example is shown in Fig. 28. This may appear to be white circles upon a black background or a black mesh upon a white background. However, the more striking phenomenon is the change in the grouping of the circles as attention fluctuates. We may be conscious of hollow diamonds of circles, one inside the other, and then suddenly the pattern may change to groups of diamonds consisting of four[Pg 67] circles each. Perhaps we may be momentarily conscious of individual circles; then the pattern may change to a hexagonal one, each “hexagon” consisting of seven circles—six surrounding a central one. The pattern also changes into parallel strings of circles, triangles, etc.

Fig. 29.—Crossed lines which may be interpreted in two ways.

The crossed lines in Fig. 29 can be seen as right angles in perspective with two different spatial arrangements of one or both lines. In fact there is quite a tendency to see such crossed lines as right angles in perspective. The two groups on the right represent a simplified Zöllner’s illusion (Fig. 37). The reader may find it interesting to spend some time viewing these figures and in exercising his ability to fluctuate his attention. In fact, he must call upon his imagination in these cases. Sometimes the[Pg 68] changes are rapid and easy to bring about. At other moments he will encounter an aggravating stubbornness. Occasionally there may appear a conflict of two appearances simultaneously in the same figure. The latter may be observed occasionally in Fig. 30. Eye-movements are brought forward by some to aid in explaining the changes.

Fig. 30.—Reversible cubes.

In Fig. 30 a reversal of the aspect of the individual cubes or of their perspective is very apparent. At rare moments the effect of perspective may be completely vanquished and the figure be made to appear as a plane crossed by strings of white diamonds and zigzag black strips.

[Pg 69]The illusion of the bent card or partially open book is seen in Fig. 31. The tetrahedron in Fig. 32 may appear either as erect on its base or as leaning backward with its base seen from underneath.

Fig. 31.—The reversible “open book” (after Mach).

Fig. 32.—A reversible tetrahedron.

The series of rings in Fig. 33 may be imagined to form a tube such as a sheet-metal pipe with its axis lying in either of two directions. Sometimes by closing one eye the two changes in this type of illusion are more readily brought about. It is also interesting to close and open each eye alternately, at the same time trying to note just where the attention is fixed.

The familiar staircase is represented in Fig. 34. It is likely to appear in its usual position and then[Pg 70] suddenly to invert. It may aid in bringing about the reversal to insist that one end of a step is first nearer than the other and then farther away. By focusing the attention in this manner the fluctuation becomes an easy matter to obtain.

33.—Reversible perspective of a group of rings or of a tube.

Fig. 34.—Schröder’s reversible staircase.

In Fig. 35 is a similar example. First one part will appear solid and the other an empty corner, then suddenly both are reversed. However, it is striking to note one half changes while the other remains [Pg 71]unchanged, thus producing momentarily a rather peculiar figure consisting of two solids, for example, attached by necessarily warped surfaces.

Fig. 35.—Thiéry’s figure.

Perhaps the reader has often witnessed the striking illusion of some portraits which were made of subjects looking directly at the camera or painter. Regardless of the position of the observer the eyes of the portrait appear to be directed toward him. In fact, as the observer moves, the eyes in the picture follow him so relentlessly as to provoke even a feeling of uncanniness. This fact is accounted for by the absence of a third dimension, for a sculptured model of a head does not exhibit this feature. Perspective plays a part in some manner, but no attempt toward explanation will be made.

[Pg 72]In Fig. 36 are two sketches of a face. One appears to be looking at the observer, but the other does not. If the reader will cover the lower parts of the two figures, leaving only the two pairs of eyes showing, both pairs will eventually appear to be looking at the observer. Perhaps the reader will be conscious of mental effort and the lapse of a few moments before the eyes on the left are made to appear to be looking directly at him. Although it is not claimed that this illusion is caused by the same conditions as those immediately preceding, it involves attention. At least, it is fluctuating in appearance and therefore is equivocal. It is interesting to note the influence of the other features (below the eyes). The perspective of these is a powerful influence in “directing” the eyes of the sketch.

In the foregoing only definite illusions have been presented which are universally witnessed by normal persons. There are no hallucinatory phases in the conditions or causes. It is difficult to divide these with definiteness from certain illusions of depth as discussed in Chapter VII. The latter undoubtedly are sometimes entwined to some extent with hallucinatory phases; in fact, it is doubtful if they are not always hallucinations to some degree. Hallucinations are not of interest from the viewpoint of this book, but illusions of depth are treated because they are of interest. They are either hallucinations or are on the border-line between hallucinations and those illusions which are almost universally experienced by normal persons under similar conditions. The latter statement does not hold for illusions of depth in [Pg 73]which objects may be seen alternately near and far, large and small, etc., although they are not necessarily pure hallucinations as distinguished from the types of illusions regarding which there is general perceptual agreement.

Larger Image

Fig. 36.—Illustrating certain influences upon the apparent direction of vision.
By covering all but the eyes the latter appear to be drawn alike in both sketches.

In explanation of the illusory phenomena pertaining to such geometrical figures as are discussed in the foregoing paragraphs, chiefly two different kinds of hypotheses have been offered. They are respectively psychological and physiological, although there is more or less of a mixture of the two in most attempts toward explanation. The psychological hypotheses introduce such factors as attention, imagination, judgment, and will. Hering and also Helmholtz claim that the kind of inversion which occurs is largely a matter of chance or of volition. The latter holds that the perception of perspective figures is influenced by imagination or the images of memory. That is, if one form of the figure is vividly imagined the perception of it is imminent. Helmholtz has stated that, “Glancing at a figure we observe spontaneously one or the other form of perspective and usually the one that is associated in our memory with the greatest number of images.”

The physiological hypotheses depend largely upon such factors as accommodation and eye-movement. Necker held to the former as the chief cause. He has stated that the part of the figure whose image lies near the fovea is estimated as nearer than those portions in the peripheral regions of the visual field. This hypothesis is open to serious objections. Wundt contends that the inversion is caused by changes in the points and lines of fixation. He says, “The image[Pg 75] of the retina ought to have a determined position if a perspective illusion is to appear; but the form of this illusion is entirely dependent on motion and direction.” Some hypotheses interweave the known facts of the nervous system with psychological facts but some of these are examples of a common anomaly in theorization, for facts plus facts do not necessarily result in a correct theory. That is, two sets of facts interwoven do not necessarily yield an explanation which is correct.

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This book is part of the public domain. Matthew Luckiesh (2011). Visual Illusions: Their Causes, Characteristics and Applications. Urbana, Illinois: Project Gutenberg. Retrieved October 2022 https://www.gutenberg.org/cache/epub/36297/pg36297-images.html

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