Part 9

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.