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THREE-DIMENSIONAL MÜLLER-LYER ILLUSION

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Investigation of a Three-Dimensional Variant of the Classical Müller-Lyer Illusion

Word Count: 2938


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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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References

Brentano, F., 1892 "Ueber ein optisches Paradoxen" Journal of Psychology, 3, 349-358.

Day, R. H. (Ross Henry) & International Congress of Psychology (24th : 1988 : Sydney,

N.S.W.) (1988). Natural and artificial cues, perceptual compromise and the basis of

verdical and illusory perception. Monash University, Clayton, Vic

Day, R. H., & Parks, T. E. (1989). To exorcize a ghost from the perceptual machine. The

moon illusion, 343-350.

Delboeuf, M. J. (1892). De l'appreciation du temps par les somnambules. Proc. Soc. Psychical

Res, 8, 414-421.

DeLucia, P., & Hochberg, J. (1985). Illusions in the real world and in the mind’s eye.

Proceedings of the Eastern Psychological Association, 56, 38.

Deregowski, J. B. (2013). On the Müller-Lyer illusion in the Carpentered World. Perception,

42, 790-792.

Gordon, I. E., Day, R. H., & Dorwood, F. (1993). A Müller‐Lyer‐type illusion of egocentric

distance with 3D convex and concave objects. Scandinavian journal of psychology,

34(2), 97-106.

Gregory, R. L., (1963). "Distortion of visual space as inappropriate constancy scaling" Nature

(London) 199 678-680

Gregory, R. L., (1966). Eye and Brain.

Gregory, R.L., (1968). "Visual illusions". Scientific American, 219(11), 66-76.

Gregory, R.L. (1997). Eye and Brain: The Psychology of Seeing. (5), 277.

Happé, F. G. (1996). Studying weak central coherence at low levels: Children with autism do

not succumb to visual illusions. A research note. Journal of Child Psychology and

Psychiatry, 37(7), 873-877.


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Lewis, E. O. (1909). Confluxion and contrast effects in the Mueller‐Lyer illusion. British

Journal of Psychology, 1904-1920, 3(1‐2), 21-41.

Müller-Lyer, F.C. (1889) Optische Urteilstäuschungen. Archiv für Physiologie Suppl. 263-

270.

Nijhawan, R. (1991). Three-dimensional Müller-Lyer illusion. Perception & psychophysics,

49(4), 333-341.

Nijhawan, R. (1995). " Reversed' illusion with three-dimensional Müller-Lyer shapes.

Perception, 24(11), 1281.

Segall, M. H., Campbell, D. T., & Herskovits, M. J. (1963). Cultural differences in the

perception of geometric illusions. Science.

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