Fig. 4.—The vertical line appears longer than the equal horizontal line in each case.
This type of illusion persists in geometrical figures and may be found on every hand. A perfect square when viewed vertically appears too high, although the illusion does not appear to exist in the circle. In Fig. 4 the vertical line appears longer than the horizontal line of the same length. This may be readily demonstrated by the reader by means of a variety of figures. A striking case is found in Fig. 5, where the height and the width of the diagram of a silk hat are equal. Despite the actual equality the height appears to be much greater than the width. A pole or a tree is generally appraised as of greater length when it is standing than when it lies on the ground. This illusion may be demonstrated by placing a black dot an inch or so above another on a white paper. Now, at right angles to the original dot place another at a horizontal distance which appears equal to the vertical distance of the first dot above the original. On turning the paper through ninety degrees or by actual measurement, the extent of the illusion will become apparent. By doing this several times, using various distances, this type of illusion becomes convincing.
Fig. 5.—The vertical dimension is equal to the horizontal one, but the former appears greater.
The explanation accepted by some is that more effort is required to raise the eyes, or point of sight, through a certain vertical distance than through an equal horizontal distance. Perhaps we unconsciously appraise effort of this sort in terms of distance, but is it not logical to inquire why we have not through experience learned to sense the difference between the relation of effort to horizontal distance and that of effort to vertical distance through which the point of sight is moved? We are doing this continuously, so why do we not learn to distinguish; furthermore, we have overcome other great obstacles in developing our visual sense. In this complex field of physiological psychology questions are not only annoying, but often disruptive.
As has been pointed out in Chapter II, images of objects lying near the periphery of the visual field are more or less distorted, owing to the structure and to certain defects of parts of the eye. For example, a checkerboard viewed at a proper distance with respect to its size appears quite distorted in its outer regions. Cheap cameras are likely to cause similar errors in the images fixed upon the photographic plate. Photographs are interesting in connection with visual illusions, because of certain distortions and of the magnification of such aspects as perspective. Incidentally in looking for illusions, difficulty is sometimes experienced in seeing them when the actual physical truths are known; that is, in distinguishing between what is actually seen and what actually exists. The ability to make this separation grows with practice but where the difficulty is obstinate, it is well for the reader to try observers who do not suspect the truth.
Illusions of Interrupted Extent.—Distance and area appear to vary in extent, depending upon whether they are filled or empty or are only partially filled. For example, a series of dots will generally appear longer overall than an equal distance between two points. This may be easily demonstrated by arranging three dots in a straight line on paper, the two intervening spaces being of equal extent, say about one or two inches long. If in one of the spaces a series of a dozen dots is placed, this space will appear longer than the empty space. However, if only one dot is placed in the middle of one of the empty spaces, this space now is likely to appear of less extent than the empty space. (See Fig. 7.) A specific example of this type of illusion is shown in Fig. 6. The filled or divided space generally appears greater than the empty or undivided space, but certain qualifications of this statement are necessary. In a the divided space unquestionably appears greater than the empty space. Apparently the filled or empty space is more important than the amount of light which is received from the clear spaces, for a black line on white paper appears longer than a white space between two points separated a distance equal to the length of the black line. Furthermore, apparently the spacing which is the most obtrusive is most influential in causing the divided space to appear greater for a than for b. The illusion still persists in c.
Fig. 6.—The divided or filled space on the left appears longer than the equal space on the right.
An idea of the magnitude may be gained from certain experiments by Aubert. He used a figure similar to a Fig. 6 containing a total of five short lines. Four of them were equally spaced over a distance of 100 mm. corresponding to the left half of a, Fig. 6. The remaining line was placed at the extreme right and defined the limit of an empty space also 100 mm. long. In all cases, the length of the empty space appeared about ten per cent less than that of the space occupied by the four lines equally spaced. Various experimenters obtain different results, and it seems reasonable that the differences may be accounted for, partially at least, by different degrees of unconscious correction of the illusion. This emphasizes the desirability of using subjects for such experiments who have no knowledge pertaining to the illusion.
Fig. 7.—The three lines are of equal length.
Fig. 8.—The distance between the two circles on the left is equal to the distance between the outside edges of the two circles on the right.
As already stated there are apparent exceptions to any simple rule, for, as in the case of dots cited in a preceding paragraph, the illusion depends upon the manner in which the division is made. For example, in Fig. 7, a and c are as likely to appear shorter than b as equal to it. It has been concluded by certain investigators that when subdivision of a line causes it to appear longer, the parts into which it is divided or some of them are likely to appear shorter than isolated lines of the same length. The reverse of this statement also appears to hold. For example in Fig. 7, a appears shorter than b and the central part appears lengthened, although the total line appears shortened. This illusion is intensified by leaving the central section blank. A figure of this sort can be readily drawn by the reader by using short straight lines in place of the circles in Fig. 8. In this figure the space between the inside edges of the two circles on the left appears larger than the overall distance between the outside edges of the two circles on the right, despite the fact that these distances are equal. It appears that mere intensity of retinal stimulation does not account for these illusions, but rather the figures which we see.
Fig. 9.—Three squares of equal dimensions which appear different in area and dimension.
In Fig. 9 the three squares are equal in dimensions but the different characters of the divisions cause them to appear not only unequal, but no longer squares. In Fig. 10 the distance between the outside edges of the three circles arranged horizontally appears greater than the empty space between the upper circle and the left-hand circle of the group.
Fig. 10.—The vertical distance between the upper circle
and the left-hand
one of the group is equal to the overall length of the group of three circles.
Illusions of Contour.—The illusions of this type, or exhibiting this influence, are quite numerous. In Fig. 11 there are two semicircles, one closed by a diameter, the other unclosed. The latter appears somewhat flatter and of slightly greater radius than the closed one. Similarly in Fig. 12 the shorter portion of the interrupted circumference of a circle appears flatter and of greater radius of curvature than the greater portions. In Fig. 13 the length of the middle space and of the open-sided squares are equal. In fact there are two uncompleted squares and an empty “square” between, the three of which are of equal dimensions. However the middle space appears slightly too high and narrow; the other two appear slightly too low and broad. These figures are related to the well-known Müller-Lyer illusion illustrated in Fig. 56. Some of the illusions presented later will be seen to involve the influence of contour. Examples of these are Figs. 55 and 60. In the former, the horizontal base line appears to sag; in the latter, the areas appear unequal, but they are equal.
| Fig. 11.—Two equal semicircles. | Fig. 12.—Arcs of the same circle. |
Fig. 13.—Three incomplete but equal squares.
Illusions of Contrast.—Those illusions due to brightness contrast are not included in this group, for “contrast” here refers to lines, angles and areas of different sizes. In general, parts adjacent to large extents appear smaller and those adjacent to small extents appear larger. A simple case is shown in Fig. 14, where the middle sections of the two lines are equal, but that of the shorter line appears longer than that of the longer line. In Fig. 15 the two parts of the connecting line are equal, but they do not appear so. This illusion is not as positive as the preceding one and, in fact, the position of the short vertical dividing line may appear to fluctuate considerably.
Fig. 14.—Middle sections of the two lines are equal.
Fig. 15.—An effect of contrasting areas (Baldwin’s figure).
Fig. 16 might be considered to be an illusion of contour, but the length of the top horizontal line of the lower figure being apparently less than that of the top line of the upper figure is due largely to contrasting the two figures. Incidentally, it is difficult to believe that the maximum horizontal width of the lower figure is as great as the maximum height of the figure. At this point it is of interest to refer to other contrast illusions such as Figs. 20, 57, and 59.
Fig. 16.—An illusion of contrast.
A striking illusion of contrast is shown in Fig. 17, where the central circles of the two figures are equal, although the one surrounded by the large circles appears much smaller than the other. Similarly, in Fig. 18 the inner circles of b and c are equal but that of b appears the larger. The inner circle of a appears larger than the outer circle of b, despite their actual equality.
Fig. 17.—Equal circles which appear unequal due to contrast (Ebbinghaus’ figure).
Fig. 18.—Equal circles appearing unequal owing to contrasting concentric circles.
In Fig. 19 the circle nearer the apex of the angle appears larger than the other. This has been presented as one reason why the sun and moon appear larger at the horizon than when at higher altitudes. This explanation must be based upon the assumption that we interpret the “vault” of the sky to meet at the horizon in a manner somewhat similar to the angle but it is difficult to imagine such an angle made by the vault of the sky and the earth’s horizon. If there were one in reality, it would not be seen in profile.
Fig. 19.—Circles influenced by position within an angle.
Fig. 20.—Contrasting angles.
If two angles of equal size are bounded by small and large angles respectively, the apex in each case being common to the inner and two bounding angles, the effect of contrast is very apparent, as seen in Fig. 20. In Fig. 57 are found examples of effects of lines contrasted as to length.
Fig. 21.—Owing to perspective the right angles appear oblique and vice versa.
The reader may readily construct an extensive variety of illusions of contrast; in fact, contrast plays a part in most geometrical-optical illusions. The contrasts may be between existing lines, areas, etc., or the imagination may supply some of them.
Fig. 22.—Two equal diagonals which appear unequal.
Illusions of Perspective.—As the complexity of figures is increased the number of possible illusions is multiplied. In perspective we have the influences of various factors such as lines, angles, and sometimes contour and contrast. In Fig. 21 the suggestion due to the perspective of the cube causes right angles to appear oblique and oblique angles to appear to be right angles. This figure is particularly illusive. It is interesting to note that even an after-image of a right-angle cross when projected upon a wall drawn in perspective in a painting will appear oblique.
Fig. 23.—Apparent variations in the distance between two parallel lines.
A striking illusion involving perspective, or at least the influence of angles, is shown in Fig. 22. Here the diagonals of the two parallelograms are of equal length but the one on the right appears much smaller. That AX is equal in length to AY is readily demonstrated by describing a circle from the center A and with a radius equal to AX. It will be found to pass through the point Y. Obviously, geometry abounds in geometrical-optical illusions.
Fig. 24.—A striking illusion of perspective.
The effect of contrast is seen in a in Fig. 23; that is, the short parallel lines appear further apart than the pair of long ones. By adding the oblique lines at the ends of the lower pair in b, these parallel lines now appear further apart than the horizontal parallel lines of the small rectangle.
The influence of perspective is particularly apparent in Fig. 24, where natural perspective lines are drawn to suggest a scene. The square columns are of the same size but the further one, for example, being apparently the most distant and of the same physical dimensions, actually appears much larger. Here is a case where experience, allowing for a diminution of size with increasing distance, actually causes the column on the right to appear larger than it really is. The artist will find this illusion even more striking if he draws three human figures of the same size but similarly disposed in respect to perspective lines. Apparently converging lines influence these equal figures in proportion as they suggest perspective.
Fig. 25.—Distortion of a square due to superposed lines.
Although they are not necessarily illusions of perspective, Figs. 25 and 26 are presented here because they involve similar influences. In Fig. 25 the hollow square is superposed upon groups of oblique lines so arranged as to apparently distort the square. In Fig. 26 distortions of the circumference of a circle are obtained in a similar manner.
Fig. 26.—Distortion of a circle due to superposed lines.
It is interesting to note that we are not particularly conscious of perspective, but it is seen that it has been a factor in the development of our visual perception. In proof of this we might recall the first time as children we were asked to draw a railroad track trailing off in the distance. Doubtless, most of us drew two parallel lines instead of converging ones. A person approaching us is not sensibly perceived to grow. He is more likely to be perceived all the time as of normal size. The finger held at some distance may more than cover the object such as a distant person, but the finger is not ordinarily perceived as larger than the person. Of course, when we think of it we are conscious of perspective and of the increase in size of an approaching object. When a locomotive or automobile approaches very rapidly, this “growth” is likely to be so striking as to be generally noticeable. The reader may find it of interest at this point to turn to illustrations in other chapters.
The foregoing are a few geometrical illusions of representative types. These are not all the types of illusions by any means and they are only a few of an almost numberless host. These have been presented in a brief classification in order that the reader might not be overwhelmed by the apparent chaos. Various special and miscellaneous geometrical illusions are presented in later chapters.
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 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 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 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 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.
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 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 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.
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 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.
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 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.
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 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.
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 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.