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.
[Illustration: 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.
[Illustration: 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 Zoellner'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.
[Illustration: 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.
[Illustration: Fig. 31.--The reversible "open book" (after Mach).]
[Illustration: 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.
[Illustration: Fig. 33.--Reversible perspective of a group of rings or of
a tube.]
[Illustration: Fig. 34.--Schroeder'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.
[Illustration: Fig. 35.--Thiery'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.
[Illustration: 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 Zoellner 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. Zoellner 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. Zoellner 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.
[Illustration: Fig. 37.--Zoellner'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 Zoellner 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
Zoellner'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 Zoellner 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 Zoellner 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.
[Illustration: Fig. 38.--Parallel lines which do not appear so.]
[Illustration: Fig. 39.--Wundt's illusion of direction.]
[Illustration: Fig. 40.--Hering's illusion of direction.]
Many investigations of the Zoellner illusion are recorded in the
literature. From these it is obvious that the result is due to the
additive effects of many simple illusions of angle. In order to give an
idea of the manner in which such an illusion may be built up the reasoning
of Jastrow[1] will be presented in condensed form. When two straight lines
such as _A_ and _B_ in Fig. 41 are separated by a space it is usually
possible to connect the two mentally and to determine whether or not, if
connected, they would lie on a straight line. However, if another line is
connected to one, thus forming an angle as _C_ does with _A_, the lines
which appeared to be continuous (as _A_ and _B_ originally) no longer
appear so. The converse is also true, for lines which are not in the same
straight line may be made to appear to be by the addition of another line
forming a proper angle. All these variations cannot be shown in a single
figure, but the reader will find it interesting to draw them. Furthermore,
the letters used on the diagram in order to make the description clearer
may be confusing and these can be eliminated by redrawing the figure. In
Fig. 41 the obtuse angle _AC_ tends to tilt _A_ downward, so apparently if
_A_ were prolonged it would fall below _B_. Similarly, _C_ appears to fall
to the right of _D_.
[Illustration: Fig. 41.--Simple effect of angles.]
This illusion apparently is due to the presence of the angle and the
effect is produced by the presence of right and acute angles to a less
degree. The illusion decreases or increases in general as the angle
decreases or increases respectively.
Although it is not safe to present simple statements in a field so complex
as that of visual illusion where explanations are still controversial, it
is perhaps possible to generalize as Jastrow did in the foregoing case as
follows: If the direction of an angle is that of the line bisecting it and
pointing toward the apex, the direction of the sides of an angle will
apparently be deviated toward the direction of the angle. The deviation
apparently is greater with obtuse than with acute angles, and when obtuse
and acute angles are so placed in a figure as to give rise to opposite
deviations, the greater angle will be the dominant influence.
Although the illusion in such simple cases as Fig. 41 is slight, it is
quite noticeable. The effect of the addition of many of these slight
individual influences is obvious in accompanying figures of greater
complexity. These individual effects can be so multiplied and combined
that many illusory figures may be devised.
In Fig. 42 the oblique lines are added to both horizontal lines in such a
manner that _A_ is tilted downward at the angle and _B_ is tilted upward
at the angle (the letters corresponding to similar lines in Fig. 41). In
this manner they appear to be deviated considerably out of their true
straight line. If the reader will draw a straight line nearly parallel to
_D_ in Fig. 41 and to the right, he will find it helpful. This line
should be drawn to appear to be a continuation of _C_ when the page is
held so _D_ is approximately horizontal. This is a simple and effective
means of testing the magnitude of the illusion, for it is measured by the
degree of apparent deviation between _D_ and the line drawn adjacent to
it, which the eye will tolerate. Another method of obtaining such a
measurement is to begin with only the angle and to draw the apparent
continuation of one of its lines with a space intervening. This deviation
from the true continuation may then be readily determined.
[Illustration: Fig. 42. The effect of two angles in tilting the horizontal
lines.]
[Illustration: Fig. 43. The effect of crossed lines upon their respective
apparent directions.]
A more complex case is found in Fig. 43 where the effect of an obtuse
angle _ACD_ is to make the continuation of _AB_ apparently fall below
_FG_ and the effect of the acute angle is the reverse. However, the net
result is that due to the preponderance of the effect of the larger angle
over that of the smaller. The line _EC_ adds nothing, for it merely
introduces two angles which reinforce those above _AB_. The line _BC_ may
be omitted or covered without appreciably affecting the illusion.
[Illustration: Fig. 44.--Another step toward the Zoellner illusion.]
In Fig. 44 two obtuse angles are arranged so that their effects are
additive, with the result that the horizontal lines apparently deviate
maximally for such a simple case. Thus it is seen that the tendency of the
sides of an angle to be apparently deviated toward the direction of the
angle may result in an apparent divergence from parallelism as well as in
making continuous lines appear discontinuous. The illusion in Fig. 44 may
be strengthened by adding more lines parallel to the oblique lines. This
is demonstrated in Fig. 38 and in other illustrations. In this manner
striking illusions are built up.
[Illustration: Fig. 45.--The two diagonals would meet on the left vertical
line.]
[Illustration: Fig. 46.--Poggendorff's illusion. Which oblique line on the
right is the prolongation of the oblique line on the left?]
If oblique lines are extended across vertical ones, as in Figs. 45 and 46,
the illusion is seen to be very striking. In Fig. 45 the oblique line on
the right if extended would meet the upper end of the oblique line on the
left; however, the apparent point of intersection is somewhat lower than
it is in reality. In Fig. 46 the oblique line on the left is in the same
straight line with the lower oblique line on the right. The line drawn
parallel to the latter furnishes an idea of the extent of the illusion.
This is the well-known Poggendorff illusion. The upper oblique line on the
right actually appears to be approximately the continuation of the upper
oblique line on the right. The explanation of this illusion on the simple
basis of underestimation or overestimation of angles is open to criticism.
If Fig. 46 is held so that the intercepted line is horizontal or vertical,
the illusion disappears or at least is greatly reduced. It is difficult to
reconcile this disappearance of the illusion for certain positions of the
figure with the theory that the illusion is due to an incorrect appraisal
of the angles.
[Illustration: Fig. 47.--A straight line appears to sag.]
According to Judd,[2] those portions of the parallels lying on the
obtuse-angle side of the intercepted line will be overestimated when
horizontal or vertical distances along the parallel lines are the subjects
of attention, as they are in the usual positions of the Poggendorff
figure. He holds further that the overestimation of this distance along
the parallels (the two vertical lines) and the underestimation of the
oblique distance across the interval are sufficient to provide a full
explanation of the illusion. The disappearance and appearance of the
illusion, as the position of the figure is varied appears to demonstrate
the fact that lines produce illusions only when they have a direct
influence on the direction in which the attention is turned. That is, when
this Poggendorff figure is in such a position that the intercepted line is
horizontal, the incorrect estimation of distance along the parallels has
no direct bearing on the distance to which the attention is directed. In
this case Judd holds that the entire influence of the parallels is
absorbed in aiding the intercepted line in carrying the eye across the
interval. For a detailed account the reader is referred to the original
paper.
Some other illusions are now presented to demonstrate further the effect
of the presence of angles. Doubtless, in some of these, other causes
contribute more or less to the total result. In Fig. 47 a series of
concentric arcs of circles end in a straight line. The result is that the
straight line appears to sag perceptibly. Incidentally, it may be
interesting for the reader to ascertain whether or not there is any doubt
in his mind as to the arcs appearing to belong to circles. To the author
the arcs appear distorted from those of true circles.
[Illustration: Fig. 48.--Distortions of contour due to contact with other
contours.]
In Fig. 48 the bounding figure is a true circle but it appears to be
distorted or dented inward where the angles of the hexagon meet it.
Similarly, the sides of the hexagon appear to sag inward where the corners
of the rectangle meet them.
The influences which have been emphasized apparently are responsible for
the illusions in Figs. 49, 50 and 51. It is interesting to note the
disappearance of the illusion, as the plane of Fig. 49 is varied from
vertical toward the horizontal. That is, it is very apparent when viewed
perpendicularly to the plane of the page, the latter being held
vertically, but as the page is tilted backward the illusion decreases and
finally disappears.
[Illustration: Fig. 49.--An illusion of direction.]
[Illustration: Fig. 50.--"Twisted-cord" illusion. These are straight
cords.]
[Illustration: Fig. 51.--"Twisted-cord" illusion. These are concentric
circles.]
The illusions in Figs. 50 and 51 are commonly termed "twisted cord"
effects. A cord may be made by twisting two strands which are white and
black (or any dark color) respectively. This may be superposed upon
various backgrounds with striking results. In Fig. 50 the straight "cords"
appear bent in the middle, owing to a reversal of the "twist." Such a
figure may be easily made by using cord and a checkered cloth. In Fig. 51
it is difficult to convince the intellect that the "cords" are not
arranged in the form of concentric circles, but this becomes evident when
one of them is traced out. The influence of the illusion is so powerful
that it is even difficult to follow one of the circles with the point of a
pencil around its entire circumference. The cord appears to form a spiral
or a helix seen in perspective.
[Illustration: Fig. 52.--A spiral when rotated appears to expand or
contract, depending upon direction of rotation.]
A striking illusion is obtained by revolving the spiral shown in Fig. 52
about its center. This may be considered as an effect of angles because
the curvature and consequently the angle of the spiral is continually
changing. There is a peculiar movement or progression toward the center
when revolved in one direction. When the direction of rotation is reversed
the movement is toward the exterior of the figure; that is, there is a
seeming expansion.