By Grant Ocean

The Müller-Lyer arrow
figure (see the figure on the left below) is the most famous illusion of all, designed in 1889 by
Franz Müller-Lyer. This illusion is created by two lines of equal length, one
line being terminated by outward wings at both ends and another line being
terminated by inward wings at both ends. The line with outward wings appears to
be much longer than the line with inward wings. According to most studies and
my own experiments, the line with outward wings looks about 30% longer than the
line with inward wings to be exact. As shown in the figure on the right below, the two lines look
about the same when the line with outward wings is shortened by 30%.

This illusion is so simple, however the distortion is so compelling that it was immediately regarded by many perception researchers as the primary target for theory and experiment. It has been the subject of more than 1,250 scientific publications according to one textbook (Myers, 2003), and all sorts of theories were advanced. For example, reintroduced was the Wundt’s eye movement theory that the line with inward wings causes people to perceive it as shorter because the perception of the line is pulled back by the “turning back” of the wings; conversely, the outward wings draw our perception further making that line seem longer. The theory has been basically falsified when flashing the illusion faster than our eyes can move still produces the illusion with the same magnitude. Another example is the tip disparity theory that says that the illusion is created because people perceptually measure the illusion from the ends of the tips of the wings. Therefore, for the line with outward wings the illusion should magnify with increasing wing tip distance. However, the figure on the left below, in which two lines with outward wings of different lengths appear to be the same size, shows that the length of wings has no obvious effect on the perception of the illusion. In fact, my experiments demonstrate that altering the distance of wings does not change the proportion of 30% difference between the line with outward wings and the line of equal length with inward wings, with only one exception (which will be discussed later).

The
often-cited explanation for the Müller-Lyer
illusion was put forth by Richard Gregory (1997). According to his
perspective-constancy theory, the inward and outward facing wings on
either
side of the line segment provide distance cues that cause a viewer to
misestimate line length. The picture in the middle above is a
often-illustrated example of the theory. The figure on the right above
can help demonstrate the perspective-constancy theory better, in which
the line segment of the rug in
the front has inward facing wings and it appears closer than the bottom
line
segment of the wall in the back which has outward facing wings.
According to
Gregory, the outward wings make the wall appear further away than the
rug line
segment with inward wings. Based on the fact that the line segments
have the
same length in the figure on the right above and produce the same
retinal image, the length of the
wall in the back is interpreted as being longer than the length of the
rug. Therefore,
the theory states that the Müller-Lyer illusion is consistent with
perspective
cues. (It is also called the Size-distance Invariance Hypothesis, which
will be discussed later.) However, when we turn the figure around as
shown in the figure on the left below, and the line segment
of the rug is not in the front anymore but on top of the wall, the
illusion still persists. More importantly, the proportion of the
illusion stays
the same as well. In the figure in the middle below, the lines are
taken out, which means that the
images do not give any perspective cues, but these images still induce
the Müller-Lyer
illusion. Furthermore, removing some of the wings, as shown in the
figure on the right, still generates the illusory effect, though to a
lesser degree. But the
figures with fewer than the full wings do not resemble the distance
cues
suggested by Richard Gregory by any stretch of imagination. Therefore,
it can
be concluded that Gregory’s perspective-constancy theory cannot
adequately
explain the Müller-Lyer illusion.

Next we need to determine whether each wing plays a particular role in the illusion and how much. The logic tells us that the illusion should be reduced by half if the wings from one end of the line are taken off. The figure on the left below shows it is the case. The two lines in the figure look about the same because the bottom line with outward wings is shortened by 15%. Likewise, the two lines in the middle figure below look the same because the line with outward wing is shortened by 7.5%; and the two lines in the figure on the right seem to be the same in length because the line with outward wing is shortened by 3.75%. With these experimental results, we can say that each wing contributes 3.75% to the illusion. Each outward wing extends the line by 3.75% and each inward wing shrinks the line by 3.75% respectively. Now the task is to find out the extra force that exerts an invisible influence on the line through each angled wing.

First, we have to choose an angle for both inward and outward
wings. Based on my experiments, the illusion effect is diminished extensively
when the wing angles are closer to 90^{°}, as shown in the figure on the left below. To control the variations of the illusion
magnitude caused by the different angles, I choose 45^{°}
angle (∠*C)* for the inward wings and 135^{°} angle (∠*C)* for the outward wings,
which are the average angles (see the figure in the middle below). In fact, all the wings in the
figures of this section are angled as such.

The extra force could be *c* line in the diagram on the right above which has one
outward wing. This line is obviously longer than *b* line and as such has more force. Hence *c* line has exerted an effect on our perception of *b* line and creates an elongated
illusion. However, *c* line has the
possibility to connect anywhere on the outward wing. This fact would compromise
the 3.75% extension attained earlier because a different connection spot on the
wing will generate a line with different length. To settle the issue, I set out
to find out on exactly what point of the wing the *c* line lands. I drew many *b*
lines of different lengths with outward wings at 135^{°}
angle. Then I drew a *c* line with the
length of 103.75% of *b* line. When
this was done, I measured *a* line and
found out that the length of *a* line
is approximately 5% of *b* line without any
exception. Now let’s draw the *b* line with
the length of 5cm and then the *c* line
is measured as 5.1875cm in length when the *a*
line is approximately 0.25cm (about 5% of the *b* line), the
difference between *b *line and *c *line is:

*c - b = 5.1875cm - 5cm = 0.1875cm.
*

From this result, we know that

*0.1875cm ÷* *5cm = 0.0375cm × 100% = 3.75%.*

We can calculate the illusory effect of two, three or four outward wings by simply multiplying the number of wings with 3.75%. As for the inward wings, we should expect the opposite result because geometry (or nature) is always symmetrical. As you can see from the diagram on the right, the

*c - b = 4.8125cm - 5cm = -0.1875*

If we do the following calculations:

*-0.1875cm ÷ 5cm = -0.0375cm × 100% = -3.75%,
*

It is time to discuss the exception
mentioned earlier regarding the effect of the length of the wings on the
magnitude of the Müller-Lyer illusion. The length of the wings has no obvious
effect on the illusion as shown in the middle figure above except when the length of a wing (*a* line)
is less than 5% of *b* line (as shown in the figure on the right above). In this
case the illusory effect is diminished because *c *line has to connect to the end of a shorter wing. As such, the
shorter the wing, *a* line, is (only when it is shorter than 5% of *b* line), the shorter *c*
line is, then the less effective the Müller-Lyer illusion is. This experiment further proves that the *c*
line is possibly the influential force of distorting the length of the
shafts. We can also assume that our mind's eye in the visual cortex is
perhaps looking at the diagonal *c* line, instead of the straight *b* line. The fact that the length of *a* line
is approximately 5% of *b*
line interests me a great deal. Thus, I spent some time looking for an
answer because I believe that this five percent figure has special
meaning or has something to do with the regularity in nature. I will
report my finding in Appendix B.

It turns out that the Müller-Lyer
illusion, especially the finding that a single wing on the shaft can
make its length appear longer or shorter, plays a fundamental role in
all the line distortion illusion caused by diagonal crossing lines.
First, we will see how we can explain another famous size illusion, the Poggendorff illusion, using our understanding of the Müller-Lyer illusion.

As shown in the figure on the right above, little or no illusory misalignment occurs in this reduced Poggendorff pattern which consists of only two acute angles or inward wings. This failure of maintaining the illusion when only the acute angles or inward wings are left in the figure has been a problem for various proposed explanations of the misalignment seen in the complete standard pattern of the Poggendorff illusion. However, I can successfully explain and predict the illusion by using the Müller-Lyer illusion and especially our understanding of the effect of single wing on the distortion. Now, let's have a close look at the above figure on the right again, which has two acute angles or inward wings. As a matter of fact, we can treat each angle as having equal sides. The vertical line can be seen either as the shaft or as the wing of the oblique line. Likewise, the oblique line can be seen either as the shaft or as the wing of the vertical line. In this case, the vertical line and the oblique line can be treated as being equivalent. As such, the acute angles in the figure become ambiguous. The two lines that form the acute angle could be both shafts or both be wings at the same time. Our visual cortex must be very confused about this ambiguous situation. It cannot decide whether it should treat the oblique line as wing or as shaft; similarly, it cannot figure out what the vertical line should be in this situation, that is, a shaft or a wing. We can assume that the lines forming the acute angles are so much alike that our cortex simply cannot sort out which is the shaft and which is the wing. As a result, the oblique line and the vertical line offset each other, and no misalignment is perceived. To test my theory of the ambiguous figure of the acute angles, all we need to do is to change the ambiguous figure into unambiguous figure. If my theory is correct, the illusion should re-appear after the ambiguous vertical line is changed into an unambiguous shaft. To achieve this transformation is simply to extend the vertical line to make it look like an unambiguous shaft. As shown in the figure on the left, I have extended the vertical lines. Now I don't think anybody would doubt that the vertical lines in the left figure look like anything but a shaft. As predicted, the Müller-Lyer illusion is re-induced and the misalignment of the oblique line has occurred once again. The positive result of this experiment is a further evidence to support my approach to the Poggendorff illusion, i.e., the illusion is caused by the wings of the Müller-Lyer illusion, mostly the outward wings because the inward wings in the standard Poggendorff illusion is ambiguous. The Poggendorff illusion is a very popular phenomenon for research because it has many interesting variations. Each unique variation poses new challenge for explaining the illusion. As such, a satisfactory explanation which can account for all the variations has been eluding researchers after more than a hundred years of intensive investigation. No matter whether I have found the true cause of the Poggendorff illusion or not, the predictive power of my approach is impressive and very successful.As discussed later, I can calculate and predict all the variations of the illusion by using the understand of the Müller-Lyer illusion.

Now, we are going to use our understand of the Müller-Lyer illusion, especially the contribution of each wing to the illusion, to predict how much the misalignment in the Poggendorff illusion is and to correct the misalignment based on this knowledge. According to the Müller-Lyer illusion, discussed earlier in the first section of this article, the line with one outward wing appears 3.75% longer than a normal line without wings. To make the winged lines look the same as the lines without wings, the line with outward wing needs to be shortened by 3.75%. Since there are two outward wings that contribute to the overall illusion, we have to double the effect. Based on our understanding of the Müller-Lyer illusion, we can predict that the alignment of the oblique line is off 7.5%. In other words, the vertical line that helps form the obtuse angle in the figure look 7.5% longer than the vertical line of the same section without the wing. If this is the case, to correct the misalignment we should simply shorten the vertical line with outward wing or the vertical line that forms the obtuse angle in the figure by 7.5% to make the oblique line in the figure look aligned. That is what has exactly happened in the figure on the right when the vertical line on the right side with outward wing is shortened by 7.5%. I have experimented with the vertical bars of different lengths and widths; and the outcomes are always the same, i.e., the oblique line appears aligned when the vertical line with outward wing is shortened or the vertical line with inward wing is extended by 7.5%. This explanation of the Poggendorff illusion reinforces our understanding of the Müller-Lyer illusion, that is, the illusion is caused by a force, the elongated or shortened

Another observation of the Poggendorff illusion is that a larger width of the vertical bar causes a larger perceptual displacement. In other words, if we increase the distance between the two vertical lines, the misalignment of the oblique line will increase as well. There have been many attempts to resolve this puzzle by other researchers, but the results are disappointing. However, I can explain this variation of the Poggendorff illusion easily and with the expected predictive power. As shown in the figure on the left, the right side vertical line has been moved from the position of the dotted line to the solid line position. As you can see, the right solid line which is the section under the wing or the oblique line is longer than the same section of the dotted line. Since that section of the vertical line looks 7.5% longer because of the outward wing attached to it, a vertical line of 5cm length with the outward wing would look as long as 5.375cm in comparison to a vertical line of the same length without the wing. If we let that section of the solid right vertical line be increased to 6cm, the section of the solid line under the wing would look 6.45cm in length in contrast to a 6cm vertical line without the wing. To reduce that section of the vertical line to make the oblique line appear to be aligned, the 5cm vertical line has to be reduced by 7.5% to 4.625cm to make the oblique line look aligned. On the other hand, the 6cm section of the newly widened vertical line has to be shortened to 5.55cm to make the oblique line appear aligned. The longer vertical line section has to be shortened by 0.45cm versus 0.375cm for the shorter dotted vertical line section in the figure if we assume the solid line section is 6cm and the dotted line section is 5cm. The longer solid line section has to be reduced 20% more than the shorter dotted line section to make the oblique line appear aligned. Accordingly, the longer the vertical line section under the outward wing is, the more misaligned the oblique line appears to be in the Poggendorff illusion. Since the wider the vertical bar, the longer the vertical line section under the outward wing, therefore the wider vertical bar makes the illusory effect more distorted and perceptible.

There is another phenomenon
discovered about the Poggendorff illusion, i.e., increasing the angle
of the oblique line away from the horizontal causes the illusion to
increase. This is a variation of the illusion which has puzzled many researchers, and
it is doggedly confound any attempt to find an adequate explanation of
the illusion. As a matter of fact, any theory attempted to explain the
Poggendorff illusion which cannot adequately account for this
phenomenon would be deemed incomplete. My
approach to explaining this phenomenon is very similar to that of
widening vertical bar above. Actually, there is a great similarity
between these two variations of the illusion. As you can see in the
figure on the right, when you rotate the oblique line from the dotted
position counterclockwise to the solid line position, the right side vertical line below the oblique line has lengthened. As such, we
can use exactly the same logic that we used earlier for the widening of
the vertical bar to explain the effect of the tilting oblique line on
the increased illusion. (To save the space, I am going to forgo the
calculations.) Since the
longer right
vertical line section below the solid oblique line, as illustrated in
the figure on the right, has to be reduced more than the shorter
vertical line section below the dotted oblique line to make the solid
oblique line
appear aligned. Accordingly, the longer the vertical line section below
the oblique line or outward wing is, the more misaligned the oblique
line appears to be
in the Poggendorff illusion. For the more tilted the oblique line, the
longer the vertical line section below the outward wing, therefore the
more tilted oblique line makes the illusory effect more obvious and
salient.

A popular explanation for the Poggendorff illusion is a so-called depth-processing theory. I
don't want to spend time and space to go into details about this theory
(if you are interested in the theory, check out this site.) since it can be dismissed due
to a fatal flaw of the theory, which is that it cannot account for the
observation that when the standard Poggendorff figure is tilted so that
the interrupted oblique lines are set vertically at 90º or
horizontally at 180º, the illusory misalignment effect weakens or
disappears. (The figure on the left shows the standard Poggendorff figure is tilted vertically at 90º.) Merely
tilting the figure should not change any presumed perspective features
of the figure. Thus, the depth-processing theory is flawed. However,
using the winged force found in the Müller-Lyer
illusion
I can adequately explain this delinquency of the standard Poggendorff
illusion. When the vertical bar is tilted, the former vertical lines
have now become the oblique lines in the figure, and the former oblique
line has become the straight vertical line which is the shaft instead
of the previous wings, as shown in the figure on the left. Now, the
shaft or the vertical line, no matter whether it is above the tilted
obscuring bar or under, has both an inward wing and an outward wing
attached to its end. Since the wings only affect the shaft to make them
either appear longer or shorter, a longer or shorter shafts will not be
misaligned with each other because the wings
only affect the shafts vertically in the figure on the left. As a
result, the vertical lines in the figure on the left do not look
displaced from each other and the illusory effect of the Poggendorff
illusion has been severely weakened if not totally eliminated. However,
those researchers who found the variant of the Poggendorff illusion in
the figure on the left did not realize that the same variant could have
its own variant as well. As shown in the figure on the right, the
ninety degree tilted standard Poggendorff illusion figure is now again
misaligned for the vertical lines. How come the tilted Poggendorff figure
on the left has weakened vertical line misalgnment and the same tilted
Poggendorff figure on the right has the illusory vertical line
misalignment? For those who don't have the knowledge of the winged force found in the Müller-Lyer
illusion discussed earlier, it is hard to comprehend why the same figure can have two different perceptual effects. Whether
the vertical lines in the figure is misaligned or not depends on the
length of the vertical lines relative to the oblique lines. If the
vertical line is longer than the left side and right side of the
oblique line, the vertical lines are not misaligned in the figure
because the vertical lines are treated by our visual cortex as the
shafts and the oblique lines as the inward or outward wings; therefore,
as explained earlier, the
wings only affect the shaft to make them either appear longer or
shorter, a longer or shorter shafts will not be misaligned with each
other. On
the other hand, in the figure on the right the vertical lines are
shorter than the either side of the oblique lines; thus the vertical
lines in the figure are not treated by our cortex as the shafts but as
the wings attached to the oblique lines. The
effect of this arrangement would be similar to the standard Poggendorff
illusion figure in which the oblique line sections attached to the
outward wings, which are the vertical lines now, are elongated. As
such, the upper vertical line in the figure on the right looks
misaligned from the lower vertical line. These experiments further
prove the explanatory and predictive power of the winged force found in
the Müller-Lyer
illusion.

## The Sander illusion

The Sander illusion is another famous size illusion. In
the Sander illusion as shown in the figure on the left, there are two
smaller parallelograms within a larger parallelogram. In the larger
parallelogram the sub-parallelogram on the left side is much bigger
than the sub-parallelogram on the right side. There is a diagonal line
bisecting each of the two sub-parallelograms. Even though the diagonal
lines bisecting the both sub-parallelograms are in fact the same
length, the diagonal line within the larger left-side sub-parallelogram appears to be considerably longer
than the diagonal line within the smaller right-side sub-parallelogram. This simple illusion has been evading and frustrating researchers for many years. The best explanation they can come up with is that the sub-parallelograms around the bisecting lines perhaps give a perception of
depth, and when the bisecting lines are included in that depth, they
are perceived as different lengths.
This is basically just a wild guess and nobody knows how this
perception of depth actually works. However, the illusion can be
easily explained based on the understanding of the mechanism of
the Müller-Lyer illusion; and the magnitude of the illusory effect can be predicted and corrected.

First of all, we have to look at the two sub-parallelograms separately and find out why the diagonal line in the smaller sub-parallelogram appears to be shorter than the line of the same length in the larger sub-parallelogram. The figure on the left below is the enlarged image of the smaller right-side sub-parallelogram in the Sander illusion. The diagonal line is blackened; and the dotted lines can help us see the shape of the sub-parallelogram but can be easily ignored. If we ignore the dotted lines in the figure on the left below and focus our attention only on the solid lines, we will see a familiar figure, which is almost the same as the figure of a shaft being connected by four inward wings at its ends in the Müller-Lyer illusion which is shown as the lower figure in the diagram on the right. The figure on the right below is the enlarged image of the larger left-side sub-parallelogram in the Sander illusion. The dotted lines in the diagram help to show the shape of the sub-parallelogram and can be easily ignored. The top of the diagonal line is connected to an inward wing on the right side and a ninety degree wing on the left side which has no illusory effect on the shaft. The bottom of the diagonal line is a little bit more complicated. It is connected to an inward wing on its left and connected to three wings on its right. Out of these three wings, one is ninety degree wing which has no illusory effect on the diagonal line and can be ignored. The remaining two wings are both outward wings. These two outward wings on the same side of the diagonal line would have the same effect on the shaft. Thus, I have dotted one line and left the line which I think is more likely to be chosen by our brain as the influential outward wing solid. If we ignore the dotted lines in the right figure below, we will see a shaft being connected to a ninety degree wing, two inward wings, and an outward wing.

Having identified the wings connected to the diagonal lines in both sub-parallelograms, we
should be able to calculate the illusory magnitude of the wings for
each diagonal line and compare the difference between the two diagonal
lines in the Sander illusion. As we know, the outward wing
contributes 3.75% to the perceived length of a line and the inward wing
contributes -3.75% to the perceived length of a line. The right-side diagonal
line has four inward wings attached to its ends; thus this diagonal line looks
15% shorter than a line of the same length (-3.75% × 4 = -15%). On the other hand, the left-side
diagonal line has two inward wings and one outward wing. Therefore, the
left-side diagonal line should appear 3.75% shorter than a line of the same
length based on the calculation: (-3.75% × 2) + 3.75% = -3.75%. The difference between
the two diagonal lines is: (-15%) – (-3.75%) = -11.25%. Accordingly,
the diagonal line in the right-side sub-parallelogram appears 11.25%
shorter than the diagonal
line in the left-side sub-parallelogram. If our understanding of the
illusion and calculations are correct, the two diagonal lines should
look the
same length when the left-side diagonal line is shortened by 11.25%. As
shown
in the figure on the left, the two diagonal lines appear to be the same
length when the left-side diagonal line is actually 11.25% shorter than
the right-side diagonal line. This result demonstrates that the mechanism of the Müller-Lyer illusion can be used to explain and calculate any line
illusion caused by the wings attached to its ends. The predictive power of the winged force in the Müller-Lyer illusion has been proven once again.

## The Zöllner illusion

First of all, we have to look at the two sub-parallelograms separately and find out why the diagonal line in the smaller sub-parallelogram appears to be shorter than the line of the same length in the larger sub-parallelogram. The figure on the left below is the enlarged image of the smaller right-side sub-parallelogram in the Sander illusion. The diagonal line is blackened; and the dotted lines can help us see the shape of the sub-parallelogram but can be easily ignored. If we ignore the dotted lines in the figure on the left below and focus our attention only on the solid lines, we will see a familiar figure, which is almost the same as the figure of a shaft being connected by four inward wings at its ends in the Müller-Lyer illusion which is shown as the lower figure in the diagram on the right. The figure on the right below is the enlarged image of the larger left-side sub-parallelogram in the Sander illusion. The dotted lines in the diagram help to show the shape of the sub-parallelogram and can be easily ignored. The top of the diagonal line is connected to an inward wing on the right side and a ninety degree wing on the left side which has no illusory effect on the shaft. The bottom of the diagonal line is a little bit more complicated. It is connected to an inward wing on its left and connected to three wings on its right. Out of these three wings, one is ninety degree wing which has no illusory effect on the diagonal line and can be ignored. The remaining two wings are both outward wings. These two outward wings on the same side of the diagonal line would have the same effect on the shaft. Thus, I have dotted one line and left the line which I think is more likely to be chosen by our brain as the influential outward wing solid. If we ignore the dotted lines in the right figure below, we will see a shaft being connected to a ninety degree wing, two inward wings, and an outward wing.

Now we are going to look closely at how the crossing lines can tilt the vertical line and why this illusion is related and can be explained by the mechanism of the Müller-Lyer illusion. The figure on the right is a cut-out from the standard Zöllner illusion figure, representing a single diagonal line crossing the vertical line. It is the top right corner of the Zöllner illusion figure with a diagonal line going downwards from left to right across a vertical line. As we look at the figure closely, we should notice the similarity between the figure on the right and the Müller-Lyer illusion, especially the figure on the left. First we will analyze the top half of the figure which is above the diagonal line, as shown in the figure on the left below. When we look at the figure on the left below, let's ignore the dotted line and focus on the solid lines only. The diagonal line on the left-side of the vertical line can be regarded as the inward wing of the solid vertical line, and the diagonal line on the right-side of the vertical line can be treated as the outward wing of the solid vertical line. The only difference between the figure on the left which is the Müller-Lyer illusion and the figure on the left below is that in the figure on the left below the two wings are sharing the same vertical line or the shaft and in the figure on the left they do not share the same shaft. Because of the fact that the inward and outward wings share the same shaft, the winged forces on both sides of the shared shaft are unequal, resulting in two different forces working on both sides of the vertical line and pulling on the shared vertical line. The force on the left side of the top half vertical line is 3.75% less than the vertical line because the left wing is an inward wing which has an invisible line (the

For
the bottom half of the figure, as shown in the figure on the right with
the dotted line ignored, the diagonal line on the left side of the
bottom half vertical
line can be regarded as the outward wing and the diagonal line on the right
side of the vertical line is the inward wing. As a result, the bottom half of the line tilts toward the left, which is the side with the outward wing because the left
side of the bottom half vertical line has 7.5% more force than the left side of the vertical line. However, the vertical line as a whole tilts or
rotates in the same direction, which is clockwise. When all the diagonal lines slant the
same way on a vertical line as in the Zöllner illusion, the vertical
line
will tilt consistently toward the side where the outward wings
are located. If all the diagonal lines on a vertical line slant from
left downward to right, the whole vertical line will rotate clockwise;
if all the diagonal lines on a vertical line slant from right downward
to left, the whole vertical line will rotate counterclockwise, as shown
in the figure on the left, which is the standard Sander illusion.

If our understanding of the Zöllner illusion is correct, when the forces on both sides of the vertical lines are sufficiently balanced, the vertical lines should appear parallel to each other and are not distorted. In the figure on the left below, either both the outward wings or both the inward wings are on both sides of the vertical line section. In the figure on the right below, three vertical lines have the same wing arrangement as the figure on the left below and other three vertical lines crossed by diagonal lines in opposite directions alternatively. As you can see, the outward wing on one side is balanced by an outward wing on the opposite side; the inward wing on one side is balanced by an inward wing on the opposite side of the vertical line. And the downward slanting line to the right is balanced by a downward slanting line to the left underneath it, and vice versa. Therefore, the forces on both sides of the vertical line is the same so that the vertical lines appear to be straight and upright. These results prove that the Zöllner illusion has nothing to do with the perspective or any other factors, but is determined by the unbalanced forces on two sides.

If our understanding of the Zöllner illusion is correct, when the forces on both sides of the vertical lines are sufficiently balanced, the vertical lines should appear parallel to each other and are not distorted. In the figure on the left below, either both the outward wings or both the inward wings are on both sides of the vertical line section. In the figure on the right below, three vertical lines have the same wing arrangement as the figure on the left below and other three vertical lines crossed by diagonal lines in opposite directions alternatively. As you can see, the outward wing on one side is balanced by an outward wing on the opposite side; the inward wing on one side is balanced by an inward wing on the opposite side of the vertical line. And the downward slanting line to the right is balanced by a downward slanting line to the left underneath it, and vice versa. Therefore, the forces on both sides of the vertical line is the same so that the vertical lines appear to be straight and upright. These results prove that the Zöllner illusion has nothing to do with the perspective or any other factors, but is determined by the unbalanced forces on two sides.

The first shape illusion we are going to analyze by using the above principle is a famous illusion called the Hering illusion, which was discovered by the German physiologist Ewald Hering in 1861. In this shape illusion, as shown in the figure on the left, the two vertical lines are both straight, but they look as if they were bowing outwards. The generally accepted explanation is that the distortion is produced by the lined pattern on the background, that simulates a perspective design, and creates a false impression of depth. Specifically, the illusion figure on the left looks like bike spokes around a central point, with vertical lines on either side of this central, so-called vanishing point. The illusion tricks us into thinking we are moving forward. Since we aren't actually moving and the figure is static, we misperceive the straight lines as curved ones. I am not going to critique this explanation, but simply use the principle above to analyze the illusion. I let the readers decide which explanation is more possible and acceptable. But I need to remind you that it is almost impossible to determine the true cause of these illusions; all we can hope for is to find regularities of the illusions and have a better prediction of their happenings. Again, the principle is: If all the diagonal lines on a vertical line slant from left downward to right, the whole vertical line will rotate clockwise; if all the diagonal lines on a vertical line slant from right downward to left, the whole vertical line will rotate counterclockwise.

Now let's have a close look at the figure on the right. It is the same figure as the standard Hering illusion, shown in the figure on the left; and I have only shortened the diagonal lines crossing the left side vertical line. Let's pay attention only on the left side vertical line and its crossing diagonal lines, and forget about the so-called vanishing point. (When we look at exclusively the left side vertical line ignoring the vanishing point, the vertical line is still curved as before. This means that the illusory effect has nothing to do with the vanishing point and perspective.) This vertical line and its crossing diagonal lines can be treated as being consisted of two Zöllner illusion line sections. In the middle of the vertical line there is a horizontal line crossing it. This horizontal line can be regarded as the dividing point, above which all the diagonal lines slant from left downward to right and below which all the diagonal lines slant from right downward to left. According to the principle, the top half section of the vertical line rotates clockwise; and the bottom half section of the vertical line rotates counterclockwise. Imagine that you are holding a flexible stick, and you are bending it clockwise from the top and counterclockwise from the bottom; as a result of these two forces, you will get a stick shaped like the left vertical line in the Hering illusion, i.e., bowing toward left. The situation for the right vertical line is the opposite. The top half section of the vertical line rotates counterclockwise because all the diagonal lines slant from left downward to right; and the bottom half section of the vertical line rotates clockwise because all the diagonal lines slant from left downward to right. I believe that the principle can explain the illusion adequately. Now let's turn our attention to some more shape illusions.

The second shape illusion we are going to analyze is the Wundt illusion, which is named for German psychologist Wilhelm Wundt who discovered the illusion in the nineteenth century. The illusion, as shown in the figure on the left, produces a similar, but inverted effect of the Hering illusion. The distorted appearance of the two parallel red lines is created by the angled lines in the background. The exact mechanism by which we perceive the illusion is claimed to be not fully understood by the perception researchers, but it is speculated that the illusion might have something to do with the way the brain and visual system perceive the angles that surround the two red lines; and it might be enhanced by the impression of depth created by linear perspective. Put this explanation aside, let's use the principle to analyze the Wundt illusion. I have turned the standard Wundt illusion figure by ninety degrees to resemble the Zöllner illusion figure. If we pay attention only on the right side vertical line and its crossing diagonal lines as shown in the figure on the right in which the diagonal lines are shortened, we see a figure which is exactly the same as the figure on the right above. Now we can use the same way that we have analyzed the Hering illusion earlier to dissect the figure on the right. The vertical line and its crossing diagonal lines can be seen as two Zöllner illusion line sections with a horizontal line crossing it in the very middle. This horizontal line is the dividing line, above which all the diagonal lines slant from left downward to right and below which all the diagonal lines slant from right downward to left. Accordingly, the top half section of the vertical line rotates clockwise; and the bottom half section of the vertical line rotates counterclockwise. The left vertical line in the figure on the right has the opposite condition.

The Ehrenstein illusion is a shape illusion discovered by the German psychologist Walter Ehrenstein. As shown in the figure on the left, the sides of a square placed inside a pattern of concentric circles take an apparent curved shape, however the square is perfect with perfectly straight sides. So far there is no generally acceptable explanation for the illusion. Some people suggest that the circle as a whole acts like a lens, probably a concave lens, so that when we look at the square through this lens, the square looks distorted. Whoever came up with this explanation has a very good imagination but has no regard for scientific rigorousness. Nevertheless, we can turn a part of the illusion into a Zöllner illusion figure and use the same principle to analyze it. As shown in the figure on the right, I have isolated one side of the square from the rest of the square. If we ignore the rest of the square and concentrate only on the isolated line and the diagonal lines crossing it, we will find that it is similar to the isolated lines for the Hering illusion and the Wundt illusion discussed above. (By the way, I have failed to turn the figure around so that the square would sit squarely in the circle and would have two vertical lines and two horizontal lines. Now, you have to tilt your head slightly to the right and regard the isolated line as a vertical line.) Again, the vertical line (as you look at it when tilting your head to the right) and its crossing diagonal lines, even though they are not straight and slightly curved, can be viewed as two Zöllner illusion line sections. The top half section of the vertical line rotates clockwise because all the diagonal lines slant from left downward to right; and the bottom half section of the vertical line rotates counterclockwise because all the diagonal lines slant from right downward to left. Hence, the line is curved this way, so do other three sides.

The Orbison illusion is a shape illusion that was first described by the psychologist Roy Orbison in 1939. As shown in the figure on the left below, the bounding rectangle and inner square both appear distorted in the presence of the radiating lines. The common explanation offered is that the background gives us the impression there is some sort of perspective; as a result, our brain sees the shape distorted. However, it is correctly considered by many researchers as a variant of the Hering and Wundt illusions. Since there are two distortions in the illusion -- the larger bounding rectangle and the smaller inner square, we have to analyze them separately. First, let's look at the bounding rectangle. In order to apply the principle more easily to the rectangle distortion, I have turned the standard Orbison illusion figure by ninety degrees to make the longer sides as the vertical lines as shown in the middle figure below. As a matter of fact, it is these vertical lines in the middle figure below that are distorted. The left side vertical line in the middle figure below is connected by the diagonal lines slanting from right downward to left; therefore, the whole vertical line rotates counterclockwise. As a result, the left side vertical line appears to tilt toward left. On the other hand, the right side vertical line in the middle figure below is connected by the diagonal lines slanting from left downward to right; therefore, the whole vertical line rotates clockwise. As such, the right side vertical line appears to tilt toward right. It is necessary to point out that these diagonal lines, however, do not cross the vertical lines; but our brain treats them as crossing the vertical lines somehow. (Remember that the principle is:

The illusion figure below is called the Kmunicek illusion. A perfect square in the figure is distorted by the crossing diagonal lines. This illusion is much simpler in comparison to the Orbison illusion. I am not going to analyze this illusion. I leave this illusion to you, the reader, to analyze by using the principle:

But, why is the imposed invisible line in the Müller-Lyer illusion connected at a point on

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Appendix B

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