Part 4

Wheatstone discovered that the dissimilar pictures of an object or scene, when united by means of optical systems, produce a visual effect similar to that produced by the actual solid object or scene provided the dissimilarity is the same as that between two retinal images of the solid object or scene. This is the principle upon which the familiar stereoscope is founded. Wheatstone formulated a theory which may be briefly stated as follows: In viewing a solid object or a scene two slightly dissimilar retinal images are formed in the two eyes respectively, but the mind completely fuses them into one "mental" image. When this mental fusion of the two really dissimilar retinal images is complete in this way, it is obvious that there cannot exist a mathematical coincidence. The result is a perception of depth of space, of solidity, of relief. In fact the third dimension is perceived. A stereoscope accomplishes this in essentially the same manner, for two pictures, taken from two different positions respectively corresponding to the positions of the eyes, are combined by means of optical systems into one image.

Lack of correct size and position of the individual elements of stereoscopic pictures are easily detected on combining them. That is, their dissimilarity must exactly correspond to that between two views of an object or scene from the positions of the two eyes respectively (Fig. 2). This fact has been made use of in detecting counterfeit notes. If two notes made from the same plate are viewed in a stereoscope and the identical figures are combined, the combination is perfect and the plane of the combined images is perfectly flat. If the notes are not made from the same plate but one of them is counterfeit, slight variations in the latter are unavoidable. Such variations will show themselves in a wavy surface.

The unwillingness of the visual sense to combine the two retinal images, if they are dissimilar to the extent of belonging to two different objects, is emphasized by means of colors. For example, if a green glass is placed over one eye and a red glass over the other, the colors are not mixed by the visual sense. The addition of these two colors results normally in yellow, with little or no suggestion of the components--red and green. But in the foregoing case the visual field does not appear of a uniform yellow. It appears alternately red and green, as though the colors were rivaling each other for complete mastery. In fact, this phenomenon has been termed "retinal rivalry."

The lenses of the stereoscope supplement eye-lenses and project on the retina two perfect images of a near object, although the eyes are looking at a distant object and are therefore not accommodated for the near one (the photographs). The lenses enlarge the images similar to the action of a perspective glass. This completes the illusion of an object or of a scene. There is a remarkable distinctness of the perception of depth of space and therefore a wonderful resemblance to the actual object or scene. It is interesting to note the effect of taking the two original photographs from distances separated by several feet. The effect is apparently to magnify depth. It is noteworthy that two pictures taken from an airplane at points fifty feet or so apart, when combined in the stereoscope, so magnify the depth that certain enemy-works can be more advantageously detected than from ordinary photographs.

Stereoscopic images such as represented in Fig. 2 may be combined without the aid of the stereoscope if the optical axes of the eye can be sufficiently converged or diverged. Such images or pictures are usually upon a card and are intended to be combined beyond the plane of the card, for it is in this position that the object or scene can be perceived in natural perspective, of natural size, of natural form, and at natural distance. But in combining them the eyes are looking at a distant object and the axes are parallel or nearly so. Therefore, the eyes are focally adjusted for a distant object but the light comes from a very near object--the pictures on the card. Myopic eyes do not experience this difficulty and it appears that normal vision may be trained to overcome it. Normal eyes are aided by using slightly convex lenses. Such glasses supplement the lenses of the eye, making possible a clear vision of a near object while the eyes are really looking far away or, in other words, making possible a clear image of a near object upon the retina of the unadjusted eye. Stereoscopic pictures are usually so mounted that "identical points" on the two pictures are farther apart than the interocular distance and therefore the two images cannot be combined when the optical axes of the eyes are parallel or nearly so, which is the condition when looking at a distant object. In such a case the two pictures must be brought closer together.

In Figs. 2 and 3 are found "dissimilar" drawings of the correct dissimilarity of stereoscopic pictures. It is interesting and instructive to practice combining these with the unaided eyes. If Fig. 2 is held at an arm's length and the eyes are focused upon a point several inches distant, the axes will be sufficiently converged so that the two images are superposed. It may help to focus the eyes upon the tip of a finger until the stereoscopic images are combined. In this case of converging axes the final combined result will be the appearance of a hollow tube or of a shell of a truncated cone, apparently possessing the third dimension and being perceived as apparently smaller than the actual pictures in the background at arm's length. If the two stereoscopic pictures are combined by looking at a point far beyond the actual position of Fig. 2, the combined effect is a solid truncated cone but perceived as of about the same size and at about the same distance from the eye as the actual diagrams. In the latter case the smaller end of the apparent solid appears to be nearer than the larger end, but in the former case the reverse is true, that is, the smaller end appears to be at a greater distance. The same experiments may be performed for Fig. 3 with similar results excepting that this appears to be a shell under the same circumstances that Fig. 2 appears to be a solid and vice versa. A few patient trials should be rewarded by success, and if so the reader can gain much more understanding from the actual experiences than from description.

The foregoing discussion of vision should indicate the complexity of the visual and mental activities involved in the discrimination, association, and interpretation of the data obtained through the eye. The psychology of visual perception is still a much controverted domain but it is believed that the glimpses of the process of vision which have been afforded are sufficient to enable the reader to understand many illusions and at least to appreciate more fully those whose explanations remain in doubt. Certainly these glimpses and a knowledge of the information which visual perception actually supplies to us at any moment should convince us that the visual sense has acquired an incomparable facility for interpreting the objective world for us. Clearness of vision is confined to a small area about the point of sight, and it rapidly diminishes away from this point, images becoming dim and double. We sweep this point of sight backward and forward and over an extensive field of view, gathering all the distinct impressions into one mental image. In doing this the unconscious interpretation of the muscular activity attending accommodation and convergence of the eyes aids in giving to this mental picture the appearance of depth by establishing relative distances of various objects. Certainly the acquired facility is remarkable.

IV

SOME TYPES OF GEOMETRICAL ILLUSIONS

No simple classification of illusions is ample or satisfactory, for there are many factors interwoven. For this reason no claims are made for the various divisions of the subject represented by and in these chapters excepting that of convenience. Obviously, some divisions are necessary in order that the variegated subject may be presentable. The classification used appears to be logical but very evidently it cannot be perfectly so when the "logic" is not wholly available, owing to the disagreement found among the explanations offered by psychologists. It may be argued that the "geometrical" type of illusion should include many illusions which are discussed in other chapters. Indeed, this is perhaps true. However, it appears to suit the present purpose to introduce this phase of this book by a group of illusions which involve plane geometrical figures. If some of the latter appear in other chapters, it is because they seem to border upon or to include other factors beyond those apparently involved in the simple geometrical type. The presentation which follows begins (for the sake of clearness) with a few representative geometrical illusions of various types.

_The Effect of the Location in the Visual Field._--One of the most common illusions is found in the letter "S" or figure "8." Ordinarily we are not strongly conscious of a difference in the size of the upper and lower parts of these characters; however, if we invert them (8888 SSSS) the difference is seen to be large. The question arises, Is the difference due fundamentally to the locations of the two parts in the visual field? It scarcely seems credible that visual perception innately appraises the upper part larger than the lower, or the lower smaller than the upper part when these small characters are seen in their accustomed position. It appears to be possible that here we have examples of the effect of learning or experience and that our adaptive visual sense has become accustomed to overlook the actual difference. That is, for some reason through being confronted with this difference so many times, the intellect has become adapted to it and, therefore, has grown to ignore it. Regardless of the explanation, the illusion exists and this is the point of chief interest. For the same reason the curvature of the retina does not appear to account for illusion through distortion of the image, because the training due to experience has caused greater difficulties than this to disappear. We must not overlook the tremendous "corrective" influence of experience upon which visual perception for the adult is founded. If we have learned to "correct" in some cases, why not in all cases which we have encountered quite generally?

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.

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_.

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.

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.

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.

_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 Mueller-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.

_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. 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.

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.

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.

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.

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.

_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.