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?
[Illustration: 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.
[Illustration: 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_.
[Illustration: 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.
[Illustration: Fig. 7.--The three lines are of equal length.]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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
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.
[Illustration: Fig. 11.--Two equal semicircles.]
[Illustration: Fig. 12.--Arcs of the same circle.]
[Illustration: 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.
[Illustration: Fig. 14.--Middle sections of the two lines are equal.]
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 17.--Equal circles which appear unequal due to
contrast (Ebbinghaus' figure).]
[Illustration: 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.
[Illustration: Fig. 19.--Circles influenced by position within an angle.]
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: 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.
[Illustration: Fig. 27.--Illustrating fluctuation of attention.]