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Investigation of a Three-Dimensional Variant of the Classical Müller-Lyer Illusion
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Abstract
Extending previous findings that the Müller-Lyer illusion still occurs n a three-dimensional
format, the aim of the present study was to investigate the relationship between the classical
two-dimensional Müller-Lyer illusion and a three-dimensional variant. Magnitude of illusion
in both formats was measured as a function of the internal angle of the illusion-inducing
arrowheads. It was hypothesised that the three-dimensional variant would conform to the
well-established pattern in the classical figure showing that smaller arrowhead angles induce a
stronger illusion. Results supported our hypothesis as no significant main effect of dimension
was found. The implications of the proposed homology between the two formats are discussed
in light of popular explanations of the Müller-Lyer illusion.
Keywords: Müller-Lyer illusion, constancy scaling, confusion hypothesis, perception
Investigation of a Three-Dimensional Variant of the Classical Müller-Lyer Illusion
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For decades, geometric illusions have served as useful tools with which to reveal the
elaborate processes occurring in the visual pathway. Despite the simplicity of many of these
illusions, consensus as to what mechanisms underlie them is still lacking. One such instance is
that of the Müller-Lyer illusion (Müller-Lyer, 1889). It is one of the most extensively
investigated optical illusions with research spanning 125 years. Yet to date, no theory has
been able to satisfactorily explain the mechanisms behind the Müller-Lyer illusion.
One factor contributing to the extent of research around the Müller-Lyer illusion is the
number of variations it has to its name. The original illusion consists of two equal-length
horizontal shafts presented one above the other, each bounded by arrowheads (Figure 1).
Inward-facing arrowheads placed at each end of one shaft induces a shortening effect, whilst
outward-facing arrowheads induce a lengthening effect. Side by side this gives the illusion
that one shaft is considerably longer than the other to the average observer; the lines appear
unequal by more than 20% (Nijhawan, 1991). Numerous variations of the classical figure
have been put forward in an attempt to detect the origin of the illusion. The two components
of the original illusion, the shaft and the inducing elements (IE) (i.e. the arrowheads) have
been manipulated in numerous ways; the IEs have been presented in the shape of semicircles
(Delboeuf, 1892) and brackets (Brentano, 1892), and the shafts have been removed (Brentano,
Figure 1. The original Muller-Lyer illusion (Muller-Lyer, 1889).
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1892). All these variations still induce the illusion originally observed, although its strength is
reduced.
The extent of the diversity shown by the Müller-Lyer pattern on first glance calls for a
theory that is general enough to explain the illusions that exist in all these variations. Gregory
(1963, 1966, 1968) proposed a theory in which the visual mechanisms that produce size
constancy of objects in the real-world are misapplied to the two-dimensional patterns,
resulting the brain making adjustments to the line lengths that are advantageous in the three-
dimensional environment but produce an illusionary effect on the two-dimensional Müller-
Lyer figures. According Gregory (1963), the inward-facing arrowheads produce retinal
images that mimic those of the edges of a closer cuboidan structure, whereas the outward-
facing arrowheads produce images that mimic the edges of a farther cuboidan structure
(Figure 2). The visual system therefore perceives the line that is ‘further away’ as larger than
the one perceived as ‘nearer’. The ‘Carpentered World’ hypothesis (Segall, 1963) is in close
agreement with this approach, proposing that these mechanisms are reinforced by the right
angle dominated-world we live in; Deregowski (2013) found that people were exceptionally
Figure 2. Gregory’s (1963) explanation of the Muller-Lyer illusion. Wall A is the closer wall
as has inward facing arrow heads, whereas the farther Wall B has outward facing arrowheads.
According to Gregory (1963), the perceptual system then applies these patterns in the physical
world to the two-dimensional figures.
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good at replicating the angle between a pair of callipers in a photograph using real-life
callipers when the angle was 90º. However at 60º and 120º this skill diminished. He concludes
that this is what one would expect if experience in carpentered environments were to affect
perception.
Gregory’s (1963) theory fails to give due to consideration to the aforementioned
prevalent findings that even when shapes conveying no distance cues replace the arrowheads,
the illusion still occurs. It therefore hinges on the original Müller-Lyer stimuli, for which the
shape of the patterns and the shape of the projected retinal image are more or less identical.
Further evidence that is incongruent with Gregory’s (1963, 1966, 1968) theory of size
constancy is that the illusion has been found using three-dimensional figures (Day and Parks,
1989; DeLucia & Hochberg, 1985; Gordon, Day & Dorwood, 1993; Happe, 1993; Nijhawan
1991; Nijhawan 1995), yet three-dimensional representations contain explicit distance cues
indicating that the two lines are of equal distance away. Nijhawan (1991) used the well-
established finding that smaller angles between the fins of the arrowheads tend to yield a
stronger illusion than larger angles (Lewis, 1909). The results confirmed his hypothesis that if
such a variation in magnitude were to be found with the three-dimensional pattern as well, he
could conclude that a common causal mechanism underlies the classic and three-dimensional
Müller-Lyer illusions. These findings are yet more that cannot be explained by Gregory’s
theory, as the angles of the three-dimensional illusions did not contain perspective information
that could trigger misapplied size constancy scaling, that would in turn induce an illusion.
Thus Nijhawan (1991) concluded that since this three-dimensional illusion does not lend itself
to the constancy scaling interpretation, it is unlikely that constancy scaling is an important
contributor to the to the production of the classical illusion as well. A better-suited theory for
these findings perhaps lies in the ‘Confusing Cues’ hypothesis (Day, 1989), which suggests
that the illusion is a result of the overall length of each figure being different depending on
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whether the arrowheads are facing in or out. The perceived length of the line becomes subject
to an averaging process that takes into account the overall length of the entire stimulus.
Nijhawan’s (1991) study is limited in the lack of direct comparison between the two-
and three-dimensional Müller-Lyer variations. Furthermore, opposing data was found by
Gordon, Day and Dorwood (1985), who using a three-dimensional structure found a mean
illusion magnitude of only 2%. When compared to the typical magnitude of 20-25% in the
two-dimensional Muller-Lyer, this suggests that the small distance distortions they found
cannot be direct homologues of the classical illusion. Thus the collective evidence still
remains inconclusive. The present study aims to revisit these hypotheses using a directly
controlled comparison of the two- and three-dimensional Müller-Lyer variations. The basis of
comparison, magnitude of illusion as a function of angle size, will be the same used by
Nijhawan (1991), since this is the most reliable and consistent characteristic of the two-
dimensional Müller-Lyer illusion (Lewis, 1909). A two-way repeated-measures design will be
used, with dimension (2D, 3D) and angle (30º, 60º, 90º, 120º, 150º, 180º, 210º, 240º, 270º,
300º, 330º) serving as the independent variables. The dependent variable will be the
magnitude of illusion. It is predicted that there will be no significant difference between the
variation in magnitude of illusion as a factor of angle size between the two-dimensional and
three-dimensional formats.
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Method
Participants
Thirty-three participants (18 male, 15 female), aged 18-59 (M = 29.30, SD = 2.31)
were opportunistically sampled for this experiment. All had normal or corrected vision.
Participants were not reimbursed for their participation.
Design
A repeated-measures design was adopted in which participants took part in both the
two-dimensional and three-dimensional conditions and all eleven angle conditions. The
independent variables were the dimensions of the Müller-Lyer illusion (two-dimensional or
three-dimensional) and the angle between the planes of the effect-inducing arrows (30º, 60º,
90º, 120º, 150º, 180º, 210º, 240º, 270º, 300º or 330º). The dependent variable of interest was
the magnitude of the illusion, measured as the distance in millimetres that the participants’
inferred midpoint was from the true midpoint. Counterbalancing was achieved by randomly
determining the order of conditions for each participant.
Stimuli
a
Figure 3. Brentano version of the Müller-Lyer illusion. Participants were asked to slide the
central arrow to where they perceived it to correctly bisect the horizontal line. The illusion
was presented in 2D (a) and 3D (b) format. Within these formats the angle between the arrow
fins was varied at 30º intervals between 30º and 330º. The examples shown here have an
angle of 90º.
b
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The experimental stimuli used Brentano versions of the Müller-Lyer illusion (Figure
2). Eleven different angle configurations were presented in two formats (two dimensional and
three-dimensional) resulting in a total of 22 experimental stimuli. These were generated and
presented on a Windows 7 computer with a 23” monitor, in Microsoft Powerpoint 2010. The
page size settings were set to ‘On Screen Show (4:3)’. Using the built-in ruler, each stimulus
was created to have a 120mm long shaft and 20mm long arrowheads. They were arranged so
that according to the page ruler the true midpoint of the shaft (i.e. 60mm along) was
positioned at 0mm and each end of the shaft terminated at -60mm and +60mm. During the
procedure, the ‘Ruler’ option was then deselected, as were all the options under the ‘Guides’
option, and display was set to full screen. This was to ensure that participants had no external
cues with which to measure distance.
Procedure
All participants were tested individually in lab cubicles, after signing informed consent
forms and providing demographic information. Seated with their eyes 30cm away from the
monitor, participants were presented with all 22 of the experimental stimuli, each on its own
Powerpoint slide. Participants saw the Brentano version of the Müller-Lyer illusion with each
of the eleven angle configurations twice, once in two-dimensional format (see Figure 3a) and
once in three-dimensional format (see Figure 3b). Participants were asked to slide the
intermediate arrow, using the left and right arrow keys, to where they believed the halfway
point of the horizontal shaft was, using only inference from their vision. The initial position of
the sliding arrow was randomly varied to control for any effects this may produce (as reported
by DeLucia & Hochberg, 1985). The order of the conditions was randomly counterbalanced,
as was the order of the slides within each condition. Overall the experiment took five minutes
to complete.
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Results
Figure 4 below shows the magnitude of illusion across the 11 angle conditions. Magnitude of
illusion was measured using the ruler tool in Powerpoint to find as the distance in millimetres
that the participants’ inferred midpoint was from the true midpoint. This was then averaged
for each of the 22 stimuli, divided by the total length of each bisection of the horizontal line
(60mm) then multiplied by 100 to give a percentage. The reported means are estimated
marginal means obtained after conducting a 2 (dimension, within) x 11 (angle, within)
repeated measures ANOVA.
Overall, the mean magnitude of illusion was higher in the 2D condition (M = 14.21,
SD = 7.77), than in the 3D condition (M = 12.92, SD = 7.36). However as Figure 4 shows, this
pattern was not found at 30º, 150º or 180º. Across conditions the magnitude of illusion
conformed to a decreasing pattern from 30º to 180º and an increasing pattern from 180º to
330º, being strongest at 30º (M = 14.79, SD = 4.71) and weakest at 180º (M = 0.15, SD =
2.37).
A 2 (dimension: 2D or 3D, within subjects) x 11 (angle: 30º, 60º, 90º, 120º, 150º, 180º,
210º, 240º, 270º, 300º, 330º, within subjects) repeated measured ANOVA was conducted on
the magnitude of illusion, with dimension and angle size as the independent variables and
magnitude of illusion as the dependent variable. Mauchly’s test indicated that the assumption
of sphericity had been violated (χ2(54) = 279.62, p < .001), therefore degrees of freedom were
corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.22). The predicted main
effect of angle size was found, F(2.17, 69.67) = 377.82 , p < .001, η² = 0.25%, such that the
average magnitude of illusion was significantly higher as the angle moved further away from
180º in either direction. The main effect of dimension was non-significant. However, the
interaction between angle size and dimension was significant, F(4.92, 157.42) = 3.10 , p = .01,
η² = 0.09%. A simple effects analysis of dimension at each of the eleven angles using paired t-
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tests Bonferroni-adjusted to an alpha of α = .0045 revealed that the only significant difference
Figure 4. Left column: Example two-dimensional stimuli showing the 11 different angle
configurations. Increasing the angle α between the 2 fins in the arrow in the center
decreased the angle ß between the fins of the arrows to the right and to the left. Right
column: Bars indicate the magnitude of illusion (i.e. the shift of the apparent line center
from the real line center), averaged for all subjects separately for different angle
configurations (shown in the left column), with exact values displayed adjacently.
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between magnitude of illusion with 2D and 3D stimuli is to be found at 90º, t(32) = 2.69, p =
.01 and 270º, t(32) = -3.14, p = .004.
In summary, angle size significantly influenced the magnitude of illusion in the
anticipated direction, conforming to a decreasing pattern from 30º to 180º and an increasing
pattern from 180º to 330º. The main effect of dimension was non-significant, however there
was a significant Angle Size x Dimension interaction. Paired t-tests found the magnitude of
illusion to be significantly different at angles 90º and 270º (i.e. when α = 90º and ß = 90º).
Discussion
The aim of the present study was to investigate the relationship between the classical
two-dimensional Müller-Lyer illusion and a three-dimensional variant. Magnitude of illusion
in both formats was measured as a function of the internal angle of the illusion-inducing
arrowheads. It was hypothesised that the three-dimensional variant would conform to the
well-established pattern in the classical figure of smaller arrowhead angles inducing a stronger
illusion. Results supported our hypothesis, with no significant main effect of dimension being
found and a significant main effect of angle. Therefore these data support our hypothesis.
These data support the view that the 3-D Müller-Lyer illusion being studied is indeed
homologous with, rather than analogous to, the classical illusion. Gregory’s (1997)
Misapplied Size Constancy Scaling theory cannot account for the three-dimensional variations
of the Müller-Lyer illusion. One would expect for the misapplied size constancy effect to
disappear when the image is already three-dimensional complete with distance cues to
indicate that all elements are of equal distance from the viewer. Yet the overall magnitude of
illusion differed by 1% between the two conditions, suggesting a common underlying causal
mechanism that is not predicted by this theory. These findings can be better explained by
general theories such as the ‘Confusing Cues’ hypothesis (Day, 1983), which states that the
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illusion occurs because the overall length of the stimuli is different depending on whether it is
of an inward-arrows or outward-arrows formation, and this affects how long we perceive the
length of the line to be.
One would perhaps expect that a three-dimensional Muller-Lyer presented on a screen
might have a stronger magnitude effect than when presented in the form of a real-life
structure. However the mean magnitude of illusion for the present study was 18%, compared
to Najhawan (1991), DeLucia and Hochberg (1985) and Gordon, Day and Dorwood (1993),
who using actual three-dimensional structures found mean magnitudes of illusion of 19%,
15% and 2% respectively. The first two authors used the same angle-varying method between
the angles of 10º to 120º to obtain their mean, whereas Gordon, Day and Dorwood (1993)
used a set angle of 90º each time. The fact that the present study’s findings are most
synonymous with those using a similar methodology is unsurprising, but it is interesting that
the presentation on a screen (the present study) versus as a real object (Najhawan, 1991;
DeLucia & Hochberg, 1985) seems to only make marginal differences to the magnitude. This
is promising for the ease with which future research can be obtained, where if dimension is
the only focal-factor (i.e. not on haptic influences) then using 3D imaging can serve as a
substitute for creating real-life structures.
What is equally notable is that in the present study, a 90º was the only angle to
produce significantly differing magnitudes of illusion between the two-dimensional and three-
dimensional formats, and this is the angle that produced such a low magnitude of illusion in
Gordon et al’s study. The fact that this pattern is present in both α = 90º and ß = 90º (see
Figure 4) suggests that this is not coincidence, being a true function of the angle itself as
opposed to random variation. It will be necessary to replicate these results before any real
inferences can be made but considering this effect is present in a relatively small sample size,
it urges further investigation. This finding is particularly interesting in the light of the
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‘Carpented World’ hypothesis, an extension of size-constancy theory proposing that in a
world full of right-angles (in buildings, furniture etc.), our brain engages heuristics to
distinguish how the right-angle patterns projected on the flat surface of the retina differ from
the real world- a heuristic that only makes sense in environments with many right angles.
Deregowski (2013) found that people were exceptionally good at replicating the angle
between a pair of callipers in a photograph using real-life callipers when the angle was 90º but
not 60º or 120º, concluding that this is what one would expect if experience in carpentered
environments were to affect perception. It is tangible that the difference in magnitude of
illusion between two-dimensional and three-dimensional figures at 90º was significantly
different because it is the “characteristic angle of the carpentered world” (Deregowski, 2013),
and thus familiarity with this angle in a three-dimensional format made it particularly
incongruent with the rest of the Muller-Lyer structure and therefore reduced its illusion-
inducing effects.
One limitation of the present study is that of participant bias, something which is hard
to avoid with such a widely known optical illusion. It is possible that some of the participants
knew the purpose of the lines was to create an illusion attempted to compensate as such. A
method to avoid this problem in future studies would be to establish unfamiliarity with the
Müller-Lyer phenomenon as a requirement for participation.
The present study aimed to investigate the relationship between the classical two-
dimensional Müller-Lyer illusion and a three-dimensional variant. It was hypothesised that the
three-dimensional variant would conform to the well-established pattern of smaller arrowhead
angles inducing a stronger illusion. Results supported our hypothesis, as no significant main
effect of dimension was found. The proposed homology, best predicted by Conflicting Cues
hypothesis (Day, 1989), requires further work to gain a full understanding of the potential
anomaly, the three-dimensional right-angle.
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