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Ecological Psychology
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Oculomotor Effects in the Size-
Distance Paradox and the Moon
Illusion
Nam-Gyoon Kim a
a Department of Psychology, Keimyung University,
Korea
Available online: 18 May 2012
To cite this article: Nam-Gyoon Kim (2012): Oculomotor Effects in the Size-Distance
Paradox and the Moon Illusion, Ecological Psychology, 24:2, 122-138
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Ecological Psychology, 24:122–138, 2012
Copyright © Taylor & Francis Group, LLC
ISSN: 1040-7413 print/1532-6969 online
DOI: 10.1080/10407413.2012.673977
Oculomotor Effects in the Size-Distance
Paradox and the Moon Illusion
Nam-Gyoon Kim
Department of Psychology
Keimyung University, Korea
Drawing on 2 concepts—the resting position of the eyes and a binocular geometry
for perceived size, the moon illusion is explained as the consequence of different
oculomotor adjustments caused by change in the direction of gaze contingent upon
the viewing conditions of the moon. Hence, each particular moon will be viewed
with a different vergence state which, in turn, yields a different amount of binocular
disparity. The vergence state will determine the perceived size of an object whereas
disparity will determine its perceived distance. It is further contended that the
perceived size of the moon is based on a new binocular information source for
size perception enabling the size of an object to be perceived even in the absence
of egocentric distance information. Discussion focuses on the paradoxical aspect
of the moon illusion and how the size-distance invariance hypothesis may have
contributed to its effect.
People observing the moon commonly report that it appears larger and closer
when it is near the horizon than when it is higher in the sky. This phenomenon
arises despite the fact that the moon itself remains constant revolving around
the earth at a distance of about 245,000 miles1with a fixed diameter of 2,200
1Because the moon orbits the earth along an elliptical path, its distance from the earth changes,
depending on its location in the orbit, from 363,104 km (225,622 miles) at its minimum to 405,696
km (252,088 miles) at its maximum. On average, the distance from the center of the earth to the
center of the moon is approximately 384,403 km (238,857 miles).
Correspondence should be addressed to Nam-Gyoon Kim, Department of Psychology, Keimyung
University, 1095 Dalgubeol Boulevard, Dalseo-Gu, Daegu, 704-701, Republic of Korea. E-mail:
nk70@kmu.ac.kr
122
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THE MOON ILLUSION 123
miles. Hence, for an observer on earth, the moon subtends the same visual angle
at every elevation (approximately half a degree at the observer’s eye).
In fact, the moon size phenomenon consists of two illusory effects: change
in the perceived size of the moon and change in its perceived distance. Each
effect is in and of itself puzzling, but a particularly puzzling aspect, especially
for the student of vision, results from the combined effects (i.e., the larger and
closer horizon moon and the smaller and more distant zenith moon). These two
effects run counter to what would be predicted by a universally accepted doctrine
of visual perception, the size-distance invariance hypothesis (SDIH; Epstein,
Park, & Casey, 1961; Kilpatrick & Ittelson, 1953). This hypothesis, essentially
a corollary of Euclid’s law of visual angle, states that the visual angle subtended
by an object determines a unique ratio of perceived size to perceived distance.
Hence, the perceived size of an object of fixed visual angle should be inversely
proportional to its distance, a relation also known as Emmert’s law (Gregory,
2008). It follows that if an object subtending a constant visual angle appears
larger (e.g., the moon on the horizon), it should appear farther away, whereas
if it appears smaller (the moon at zenith), it should appear closer. However, the
moon on the horizon appears both larger and closer than the moon higher in
the sky, thus contradicting the SDIH. For that reason, this phenomenon is not
just an illusion but also an anomaly (Epstein et al., 1961; Hershenson, 1989b).2
Indeed, of the many visual illusions discovered to date, the moon illusion is
certainly one of the most puzzling (Hershenson, 1989a).3
Of the many accounts put forward to date, two are most prominent: the
apparent distance theory (Kaufman & Rock, 1962, 1989; Rock & Kaufman,
2This and other similar anomalous effects are collectively known as the size-distance paradox
(see Ross, 2003, for a review; see also Hershenson, 1989a, 1989b). A classic demonstration of
this paradoxical effect, apart from the moon illusion, was reported by Heinemann, Tulving, and
Nachmias (1959). These authors reported decrease in apparent size with increase in the angle of
convergence, consistent with the SDIH. The perceived distance of a target, however, increased with
decreased convergence. That is, the target that appeared smaller was judged as farther away, whereas
the target that appeared larger was judged as closer, an instance of the size-distance paradox.
3Some authors denounce the secondary aspect of the moon illusion. In particular, Kaufman and
Kaufman (2000; see also Kaufman et al., 2007) reported that the binocular visual system perceives
the artificial horizon moon as if it is farther away than the elevated moon. Mon-Williams and
Tresilian (1999), on the other hand, found the paradoxical distance effect only in verbal reports but
not in the reaching responses. Their finding was interpreted as evidence for two levels of awareness
regarding perceived distance, which they contended are held separately in two streams of visual
pathways, consistent with the two visual systems theory championed by Milner and Goodale (1995,
2008; see also Goodale & Milner, 1992). These findings are controversial (Ross & Plug, 2002).
More important, the phenomenal aspect of the paradoxical distance effect remains unaccounted
for. In what follows, the moon illusion is considered only in the context of these two illusory
effects.
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124 KIM
1962)4and variants of the oculomotor theory (Enright, 1989; McCready, 1985,
1986; Roscoe, 1989; see Ross & Plug, 2002, for a review). The apparent distance
theory postulates a two-stage process of distance perception involving the un-
conscious registration of distance information followed by a conscious judgment
of distance. In the first stage, the visual system “registers” distance information
based on available contextual cues. Thus, the terrain near the horizon may present
sufficient context to cause the horizon moon to be perceived farther away than the
zenith moon, which appears in a relatively empty sky. In accordance with the
SDIH, the horizon moon appears larger than the zenith moon. In the second
stage, the moon’s distance is “judged” based on cognitive knowledge (e.g.,
knowing that larger objects are typically closer than smaller objects). Thus, it is
in the second stage that a paradox is introduced; specifically, human observers
often report that the larger horizon moon is closer than the smaller zenith moon
(see also Gregory, 2008).
By contrast, the oculomotor theory attributes the illusion to physiological
causes, notably to accommodation (i.e., automatic adjustment of the shape of
the lens of the eye to bring an object into focus) and convergence (i.e., inward
or outward turning of the eyes to fixate an object). First reported by Wheatstone
(1852), changes in vergence state have been known to have a direct effect
on perceived size and a lesser effect on perceived distance (Enright, 1989;
McCready, 1965). Specifically, a stimulus with a constant visual angle appears
both smaller and farther away with increase in convergence but larger and closer
with relaxation in convergence, both in violation of the SDIH. The former
is referred to as convergence micropsia, whereas the latter is referred to as
macropsia (Ross & Plug, 2002).5With changes in vergence state, perceived
distance is altered. As in the case with the apparent distance theory, however,
the accompanying impression of size does not conform to the SDIH, resulting
in size-distance paradox. To circumvent the paradoxical effect of perceived size,
the theory contends that the resultant change in perceived size is not of linear,
but of angular, size of the moon, again conforming to the SDIH (Enright, 1989;
McCready, 1965).
Note that the two theories resort to two different mechanisms to trigger
changes in the perceived size of the moon at varying elevations, that is, presence
or absence of contextual cues in the apparent distance theory and changes
in vergence state in the oculomotor theory. However, their accounts of the
4This position is also referred to as the taking account of distance (TAD) approach with its
emphasis on perceived distance and visual angle as the primary sources of information from which
perceived size is derived in accordance with the SDIH. See Egan, 1998; Ross, 2003; and Schwartz,
1994, for further details.
5The same effects also occur with changes in accommodation and are termed accommodative
micropsia.
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THE MOON ILLUSION 125
corresponding moon’s perceived distance are essentially identical.6As a solid
object, the size of the moon remains constant irrespective of its distance from
the observer. Its angular extent, however, varies depending on its distance from
the observer. Thus, whether contextual cues or adjustments of the oculomotor
system, the mechanisms operating on perceived distance can influence the an-
gular, but not the linear, size of the moon. In doing so, the operational meaning
of “perceived size” changes surreptitiously somewhere in the middle of each
model’s two-stage perceptual process (Ross, 2003). The modified angular size
acts on size-distance scaling once again to produce the moon’s perceived distance
and is the source of the moon illusion in each theory.
Other minor criticisms have been levied against each theory by proponents
of the opposing theory. For example, some authors point out that the notion
of registered distance in the first account is unobservable and, therefore, not
verifiable (Enright, 1989; Wagner, Baird, & Fuld, 1989). Others note that a
detailed description of how micropsia takes place leading to misperceived an-
gular size is unaccounted for in the oculomotor account (Kaufman & Kaufman,
2000). Still, the most significant problem with both theories is that they sidestep,
rather than address, the secondary aspect of the moon illusion by changing the
operational meaning of “perceived size” (Ross, 2003). Avoiding this issue is not
acceptable.
Here I propose a new rationale for the moon illusion. The account extends
the oculomotor hypothesis by incorporating the resting position of oculomotor
adjustments. My account relies on the premise that perceptions of size and
distance are independent, not interdependent as the SDIH predicts. A newly
proposed binocular source of size information that specifies the frontal size
of an object independent of its distance forms the basis for this new way of
conceptualizing the illusory effects (Kim, 2007). In the discussion that follows,
unless stated otherwise, perceived size is used strictly in reference to the linear
extent of an object, not angular extent.
A NEW ACCOUNT
In a series of studies, Leibowitz and Owens (1975; Owens & Leibowitz, 1976,
1983) reported that the tonic states of accommodation and (con)vergence systems
observed in the absence of stimulation or under degraded conditions assume an
intermediate distance, referred to as dark focus and dark vergence, respectively.
6As Enright (1989) acknowledges, the difference between these two approaches with respect
to the secondary aspect of the moon illusion is slight—the apparent distance theory calls for
“farther-larger-nearer” interpretations, whereas the oculomotor theory requires a “larger-nearer”
interpretation.
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126 KIM
Although the authors observed substantial individual differences, the distance
corresponded to 1 or 2 m on average. This finding contradicts the conventional
view that the oculomotor system shifts to optical infinity at rest. As a conse-
quence, the demand for oculomotor effort changes substantially. For example,
whereas active convergence effort is required to fixate near objects according
to the conventional view, in actuality it is distant objects that require more
oculomotor effort for proper fixation due to the dark vergence posture.7
When these facts are pulled together, a picture emerges: To observe the
moon on the horizon, the vergence system has to diverge farther beyond its
resting posture to fixate on the distant object. This effort will be facilitated by
contextual information such as linear perspective and texture gradients, among
others, provided by the terrain near the horizon. By contrast, when watching the
moon at higher elevations, the vergence system essentially maintains its resting
position. Vision, therefore, is severely overconverged and underfixated (Heuer
& Owens, 1989; Owens & Leibowitz, 1983). Indeed, the oculomotor system
has been known to become nearsighted under low illumination (night myopia)
(Leibowitz & Owens, 1975; Levene, 1965) or when watching an empty field such
as the sky (empty-field myopia; Brown, 1957; Leibowitz & Owens, 1975).8It
turns out that these phenomena, which were initially considered anomalous, were
actually manifestations of an extended resting posture of the oculomotor system
due to gaze elevation or degraded visual conditions (Owens, 1986; Owens &
Leibowitz, 1983).
Note that the present account, like the oculomotor theory, attributes changes
in vergence state as the primary cause for changes in the perceived size of
the moon. The diverged vergence state resulting in an enlarged appearance
of the horizon moon and the converged state producing a smaller appearance
of the zenith moon are comparable to convergence macropsia and micropsia,
respectively. However, the similarity stops here. For the oculomotor theory, these
oculomotor adjustments are cognitive phenomena “derived from interpretations
made at higher levels in the brain” (Enright, 1989, p. 83). Following Wheatstone
(1852), I contend that these adjustments are physiological phenomena, that is,
changes in vergence state due to binocular viewing of an object beyond the
focusing power of the ocular system, in particular, celestial objects such as the
sun and the moon.
7Note that dark vergence is known to be further influenced by elevation of gaze. In particular,
when the eyes are raised, dark vergence shifts to a nearer (i.e., more convergent) posture; when
the eyes are lowered, the opposite is true (Heuer & Owens, 1989; Jaschinski, Koitcheva, & Heuer,
1998). However, the change is small, with fixation distance varying from about 1.15 m on average
at 0 deg elevation to about 1.35 m at 30 deg elevation.
8Although the moon illusion is a nighttime phenomenon, because of empty-field myopia, it is
conceivable that similar illusory effects can occur even during the day.
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THE MOON ILLUSION 127
FIGURE 1 The binocular geometry for viewing a line segment AB. L and R refer to the
left and right eye, respectively; A and B are the two end points of the line segment; ˛and ˇ
are visual angles subtended by AB with respect to each eye; and ıare binocular parallaxes
of each end point of the segment with respect to the two eyes; and is the interocular
distance. (color figure available online)
This way of reconceptualizing oculomotor adjustments draws on the general
recognition that the moon illusion is largely a binocular phenomenon (Enright,
1989; Ross & Plug, 2002; Taylor & Boring, 1942).9The geometry of the SDIH
consists of a right triangle with distance and size as its two legs and a subtended
angle geared toward monocular vision. A new geometry such as that shown in
Figure 1 may be needed to describe the binocular viewing of an object.
Kim (2007) provides a proof that a physical extent AB can be expressed as
follows:
AB Dssin ˛
sin ı
sin ˇ
sin
The model is expressed by four angular measures and interocular distance,
explicitly excluding specific egocentric distance information. Provided the visual
system can access its interocular distance, for which there is strong evidence
(e.g., Cutting & Vishton, 1995), any frontal size can, in principle, be perceived
binocularly based on the model.
9Kaufman and Rock (1962, 1 989; Rock & Kaufman, 1962) disag reed contending instead that the
illusion is a monocular phenomenon. Kaufman and Kaufman (2000), however, manipulated binocular
disparity to explore Kaufman and Rock’s contention that the horizon moon is perceived at a greater
distance. It is also worth noting that most of the reports of the moon illusion, whether anecdotal
or part of scientific investigations, were based on binocular observations. See also Enright (1989)
or Roscoe (1989) for possible explanations of the monocular moon illusion as the consequence of
oculomotor adjustments.
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128 KIM
Conceived this way, it is easy to see how changes in vergence states affect our
perceptions of size and distance. Figure 2 illustrates how the vergence system
influences the perceived size of an object with a constant visual angle. As the
eyes diverge (from angles ˛1to ˛3), the perceived size of the object defined by
two intersecting visual angles enlarges; conversely, perceived size shrinks as the
eyes converge (from angles ˛3to ˛1). The effects are comparable to viewing the
moon at different elevations—the perception of a larger moon on the horizon
corresponding to diverging eyes and the perception of the small moon at zenith
corresponding to converging eyes.
It is worth emphasizing that the two accounts of the moon illusion reviewed
earlier employed an intermediary step that switches between two different no-
tions of perceived size, linear and angular. This ad hoc step was needed to
address the paradoxical effect of the perceived distance associated with the
moon illusion. In fact, this may be the only way to bring about one percept (i.e.,
perceived size) to be inversely correlated with the other percept (i.e., perceived
distance) without violating the SDIH. No such step is needed under this account.
FIGURE 2 A schematic depiction of the effect of vergence on perceived (linear) size. An
object casting a constant visual angle .˛1D˛2D˛3/can have its size perceived differently
depending on where the eyes converge. As the eyes diverge (from ˛1to ˛3), perceived size
enlarges (S1to S3). As the eyes converge (from ˛3to ˛1), perceived size shrinks (S3to S1).
(color figure available online)
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THE MOON ILLUSION 129
As illustrated in Figure 2, the vergence states alone can affect the apparent size
of an object, even with the visual angle kept constant.
Thus, this account regards oculomotor adjustments as the primary cause for
changes in the perceived size of the moon at different elevations. The question
remains as to why the perceived distance of the moon is altered depending on
its elevation in the sky. Before addressing this issue it is important to recognize
a point raised by Egan (1998): “We have no way of measuring or specifying the
apparent distance of the moon” (p. 621). It is unrealistic, therefore, to expect
moon observers to perceive the actual distance of the moon when it is about
245,000 miles from the earth. Neither can we expect the moon’s perceived size
to be anywhere near its actual size of 2,200 miles in diameter.10 The horizon
moon is judged to be larger relative to the contracted zenith moon and vice
versa. Likewise, the closer appearance of the horizon moon is judged relative to
its more distant counterpart when the moon is higher in the sky.
Thus, when watching a celestial object such as the moon, which is so far away,
it would be unreasonable to expect our perceptions of its size and distance to
match its actual size and distance. Similarly, it would be unreasonable to expect
the two perceived quantities to conform to the SDIH. Put differently, our ocular
devices would err to some extent when watching the moon because oculomotor
adjustments in general are underfocused and underfixated. In particular, vergence
errors, referred to as “fixation disparity,” are regularly encountered under de-
graded binocular stimulation (Francis & Owens, 1983; Heuer & Owens, 1989).
When watching the moon over the horizon, the vergence system responds to
available contextual information and therefore the degree of fixation disparity is
relatively small (left panel of Figure 3). By contrast, when watching the moon
high in an empty sky in the relative absence of contextual information, the
vergence system will remain in resting posture (right panel of Figure 3). As
a result, the eyes will be severely overconverged, resulting in larger fixation
disparity (Jaschinski, 2001; Jaschinski et al., 1998).
This account suggests that both horizon and zenith moons will be viewed
in relatively overconverged states, but the degree of overconvergence for the
horizon moon will be less than that of the zenith moon.11 Thus, the horizon moon
will be viewed near the fixation point with relatively little binocular disparity,
whereas the zenith moon will be underfixated with greater disparity. This account
10Even if the moon were observed in the context of the tallest building of the world, Burj Khalifa
in Dubai, which stands 828 m tall, its size would not be perceived as high as the building.
11Note that the oculomotor theory contends that the enlarged horizon moon is an instance of
macropsia. Macropsia occurs when the focus and convergence of the eyes shift from a nearby object
to a distant one (McCready, 1999). Thus, the theory contends that as an object’s visual angle enlarges
it appears closer, as is the case with the horizon moon. This description, however, is not applicable
to the moon because it is impossible to properly fixate on the moon let alone fixate beyond it. In
short, the appearance of the horizon moon cannot be an instance of macropsia.
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130 KIM
FIGURE 3 A schematic depiction of the effect of fixation distance and the extent of
overconvergence on perceived size and perceived distance. An object AB is underfixated,
hence, located beyond the fixation plane creating uncrossed disparity; that is, the left (l)
and the right image (r) are seen by the left and the right eye, respectively. Each case of
overconvergence (or fixation error) is depicted as the distance between the target and the
fixation plane (top panel) and as a fused image with a binocular disparity (bottom panel). A
less severe case of fixation error is shown in the left panel and a more severe case in the right
panel. The perceived size of an object (S0) is determined by the state of vergence whereas
its perceived distance (D0) is determined by the amount of disparity (the extent of lateral
separation between the two stereo images). Dand S, on the other hand, refer to the objective
distance and size of the object, respectively. With a large fixation error, stereo images become
farther apart (large disparity) causing the object to appear smaller and distant (right panel).
With a small fixation error, the disparity gets smaller, hence, an apparently larger and closer
object (left panel). (color figure available online)
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THE MOON ILLUSION 131
contends that it is these degrees of binocular disparity that would be registered
by the visual system as the corresponding moon’s perceived distance. Thus, the
horizon moon should appear larger and closer and the zenith moon smaller and
much farther away.
Even with overconvergence, the moon at varying elevations (in particular,
the zenith moon) should be fused easily. In other words, although the two
eyes are overconverged when watching the moon in the sky, these conditions
would not elicit double vision. At a fixation distance of 4 m, relative binocular
disparity—the convergence difference required to fixate the moon—would be
less than 1 deg; at a fixation distance of 13 m, the disparity will be smaller
than 17 arc-min, well below the 2 deg fusion limit determined using patchlike
stimuli (Fender & Julesz, 1967; Foley, Applebaum, & Richards, 1975; Howard
& Rogers, 1995). In short, this account maintains that the illusory effect of
change in the moon’s size is a consequence of different oculomotor adjustments
triggered by an impoverished viewing condition, which in turn yields different
amounts of binocular disparity and a perception of differing distances of the
zenith and horizon moons.
The account proposed here is not inconsistent with the SDIH. Take, for
example, any triangle depicted in Figure 3. Each and every triangle conforms to
the SDIH. However, when the two triangles are conjoined as in Figure 1, they
better depict the geometry of binocular viewing and the accompanying changes
in vergence state. A single triangle may be adequate to depict monocular vision,
but it is inadequate to describe certain uniquely binocular phenomena, perhaps
due to the effect of vergence on perception. The moon illusion is a prime example
of this, resulting in the paradox. In short, this model subsumes the SDIH.
As depicted in Figure 3, perceived distance is inversely correlated with
convergence angle; that is, perceived distance recedes as convergence angle
increases and vice versa. If convergence is indeed a depth cue as generally
assumed, this would simply be another instance of the size-distance paradox.
Instead, the perception of distance is largely determined by the resultant
disparities.12 Note that the role of convergence angle as a depth cue has always
been controversial (Brenner & van Damme, 1998; Foley, 1980; Heinemann et al.,
1959; Sedgwick, 1986).13 Perhaps the present interpretation casts further doubt
on the efficacy of convergence as a distance cue.
12This account is not unprecedented. In fact, by drawing on disparity rather than convergence for
distance estimation, this account is consistent with, and even extends, the conclusion drawn by von
Hofsten (1976; see also Blank, 1953), who stated that “perceived egocentric distances in binocular
space are determined by convergence differences and not by absolute convergences” (p. 195).
13It is interesting to note that two studies directed at the utility of convergence as an effective
cue to distance utilizing the same wallpaper illusion found the opposite results. Whereas Lie (1965)
demonstrated the efficacy of convergence with a result consistent with the SDIH, Logvinenko and
Belopolskii (1994) did not, further extending the equivocal nature of convergence as a distance cue.
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132 KIM
DEMONSTRATION
To examine this theory, I devised a demonstration by implementing the geometry
depicted in Figure 3. (To be able to quantify the phenomenon under experimental
manipulation would have been preferable but it was not feasible to explore this
option in my laboratory due to the aforementioned wide-varying resting posture
of the eyes across individuals and the lack of a device to measure the vergence
state of the eyes. Consequently this issue must be left for future investigation.)
The demonstration consisted of a virtual textured square object that appeared
behind the monitor against a white background. The disparity of the stereo
images was adjusted by varying the fixation distance. This manipulation induced
different appearances of the object’s size and distance. This manipulation was
identical to that used by Wheatstone (1852) in his demonstration of oculomotor
micropsia. Wheatstone produced the effect by mechanically adjusting the dis-
tance between two stereo images on his stereoscope; for this demonstration the
effect was accomplished in a more principled manner, that is, by adjusting the
fixation distance.
Displays were generated on a PC workstation equipped with a Wildcat4
7110 graphics card (3Dlabs, Milpitas, CA) and displayed on a 22-in. Mitsubishi
Diamond Pro 2070SB monitor refreshed at 120 Hz. The display had a resolution
of 1,280 H 1,080 V pixels and subtended a field of view of 32.3ıH
24.5ıV when viewed from a distance of 70 cm in a dimly lit room. Ten
volunteers drawn from the University of Leicester community participated in the
demonstration. All had normal or corrected-to-normal vision. The participants
watched the virtual object stereoscopically through LCD shutter glasses that
were synchronized with the monitor’s refresh rate, which alternated at 60 Hz in
stereo frame sequential mode.
The effect of fixation distance was simulated by employing three viewing
distances: 0.7, 7.0, and 14.0 m (see Tittle, Todd, Perotti, & Norman, 1995,
for a similar manipulation). For each display, the amount of vergence and the
perspective were adjusted in accordance with the viewing distance. The object
appeared behind the fixation distance by 0.1 and 0.2 m in the 0.7 m condition, 1
and 2 m in the 7 m condition, and 2 and 3 m in the 14 m condition, respectively.
The cube appeared in three different sizes for each condition of viewing distance:
0.03, 0.04, and 0.05 m in the 0.7 m viewing condition; 0.3, 0.4, and 0.5 m in the
7.0 m condition; and 0.6, 0.8, and 1.0 m in the 14.0 m condition, respectively.
Observers pressed one of two arrow keys to alter the simulated viewing
distance. Pressing one key reduced the viewing distance, whereas pressing the
other key extended it. The key pressing displaced the fixation plane toward or
away from the observer but left the target object intact at its original location.
In particular, the distance between the object and the fixation distance decreased
with increased viewing distance (left panel of Figure 3) but increased with
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THE MOON ILLUSION 133
decreased viewing distance (right panel of Figure 3). Nonetheless, disparity
changed with viewing distance and, consequently, vergence had to be adjusted
to fuse the stereo images. Observers were asked to report the size and distance
of the depicted object when pressing one of these two keys.
All observers reported that the object shrank and receded when the viewing
distance decreased but enlarged and moved closer when the viewing distance
increased. It should be noted that the pattern observed here is the same pattern
that arises when watching the moon at different elevations. This result contradicts
the SDIH and, therefore, constitutes an instance of the size-distance paradox,
exactly as the model predicted. The result is robust and unequivocal.
CONCLUSION
The moon illusion has persisted for over 25 centuries and baffled some of the
best minds in science (Egan, 1998; Ross & Plug, 2002). Of the many accounts
put forward to address this puzzling phenomenon, two general explanations are
most prominent: the apparent distance theory (Kaufman & Rock, 1962, 1989;
Rock & Kaufman, 1962) and variants of the oculomotor theory (Enright, 1989;
McCready, 1985, 1986; Roscoe, 1989). Although Kaufman and Kaufman (2000)
consider these two accounts diametrically opposite, they are, in some respects,
quite similar. In particular, the two accounts differ as to the type of mechanism
that engenders changes in the perceived size of the moon, that is, internal
computation in the apparent distance theory or oculomotor adjustments in the
oculomotor theory. However, both mechanisms are triggered by presence and
absence of contextual cues along the terrain. More important, the consequences
of these processes are the same, that is, paradoxical distance effects. When faced
with the more challenging situation, both accounts circumvent it by modifying
the operational meaning of “perceived size” to be angular rather than linear size.
This strategy is not convincing (Ross & Plug, 2002).
The alternative account I offer here draws on two concepts: the resting
position of the eyes and a binocular geometry for perceived size. In this ac-
count, the illusory effects associated with the moon at varying elevations are
the consequence of different oculomotor adjustments caused by change in the
direction of gaze contingent upon the viewing conditions of the moon. Viewing
the moon at a particular elevation demands a different vergence state which, in
turn, yields a different amount of binocular disparity. The conjecture is that the
particular vergence state determines the perceived size of the moon, whereas the
resultant disparity determines its perceived distance.
A particularly significant aspect of this account lies in the rejection of the
SDIH as an account of size perception. Both theories of the moon illusion
offer reasonable explanations with respect to the moon’s altered size at varying
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134 KIM
elevations. However, it is the moon’s altered distance for which both accounts
become arbitrary and speculative. The seeming failures of these accounts arise
because acceptability of any solution is assumed to meet the constraints imposed
by the SDIH.
The SDIH has a long history in the psychology of perception as an account
of size perception (Hatfield, 2002). The results of numerous attempts to validate
the SDIH empirically have been inconclusive at best and contradictory at worst
(Brenner & van Damme, 1998; Collewijn & Erkelens, 1990; Foley, 1980; Heine-
mann et al., 1959; Higashiyama & Adachi, 2006; Sedgwick, 1986; but see Kauf-
man et al., 2006, for evidence in support of the SDIH). Nevertheless, the SDIH
has endured, perhaps in part because alternative information sources for size
perception were lacking. After failing to explain how their subjects arrived at ac-
curate size estimations for unfamiliar objects, Haber and Levin (2001) remarked,
All we can say is that they did not do it in the same way as they did for the
distance estimations. This ignorance reflects a general ignorance about the percep-
tual variables underlying size perception. Most of the theoretical discussions about
size perception appeal to familiarity (as do we) and ignore any other variables.
But there must be some others, and size perception theorists have to identify and
demonstrate them. (p. 1150; emphasis added)
The present account of the moon illusion is based on the binocular source of
size information proposed by Kim (2007). Except for this information source,
there is virtually no known information source that can provide a metric for
binocular space perception. More important, the information for an object’s
size is directly available in optical stimulation (see Gillam, 1995, for a further
discussion regarding this issue), even in the absence of egocentric distance
information, thus assuring the independence of size and distance perceptions.
With perceptions of size and distance no longer tied together as in the SDIH,
changes in one percept (i.e., perceived size) bear little influence on its counterpart
(i.e., perceived distance).
Research on the resting posture of the eyes has demonstrated convincingly
that the oculomotor system assumes a converged posture reflecting its tonic
state, even at rest. Moreover, the vergence system adjusts itself constantly as
it converges or diverges from a resting posture to meet the demands of the
current visual task. In the case of moon watching, these adjustments result in
different degrees of overconvergence, which the proposed binocular information
source will estimate as different sizes of the moon. At the same time, changes in
vergence state will yield changes in the amount of binocular disparity, which in
turn will be used as different estimates of the perceived distance of the moon. In
short, the moon illusion may have been a paradox according to an old conceptual
scheme (i.e., the SDIH), but when the perceived quantities of size and distance
are considered independent, its paradoxical features dissolve.
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THE MOON ILLUSION 135
It would be naïve to expect that the account presented here will settle all
disputes pertaining to the moon illusion. After all, not only has the illusion
persisted for over 25 centuries (Egan, 1998; Ross & Plug, 2002) but also it was
declared to be the most puzzling of all the visual illusions discovered to date
(Hershenson, 1989a). It is conceivable, therefore, that the specific conditions
exploited to motivate this account, albeit taken from various research findings,
may turn out to be unsubstantiated under further scrutiny. Still, the unique
contribution of the proposed model (i.e., its explicit recognition of the roles
played by the vergence system and binocular disparity) remains. It is also
worth noting that the moon illusion has been reported under monocular vision.
However, its binocular effects are more robust (Enright, 1989; Taylor & Boring,
1942). Any future account of the moon illusion should factor in these two
binocular mechanisms before resorting to nonperceptual sources as causes of the
illusion. Further clarifying the role of these two mechanisms will be facilitated
by using an appropriate geometry, such as that portrayed in Figures 1 to 3.
In summary, I have proposed a new explanation for the moon illusion that
addresses, in a coherent framework, not only changes in the perceived size of the
moon at different elevations but also corresponding changes in its perceived dis-
tance. My account differs from all previous attempts to explain the phenomenon
by not being constrained by the SDIH. This is important because the effect
of altered distance is a paradox only under the SDIH. The present explanation
avoids this conundrum because the different physiological states induced by dif-
ferent viewing conditions were evaluated using a geometry suitable for binocular
viewing of an object rather than the conventional monocular geometry.
ACKNOWLEDGMENTS
This work was supported by the National Research Foundation of Korea Grant
funded by the Korean Government (NRF-2010-327-H00023). I thank Andrew
Colman, Judy Effken, Michael Joseph, and Kevin Paterson for carefully reading
and commenting on previous versions of this manuscript and Lloyd Kaufman,
Bill Mace, Helen Ross, and an anonymous reviewer for their constructive criti-
cisms for improving the manuscript.
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