mirrored file at http://SaturnianCosmology.Org/ 
For complete access to all the files of this collection
	see http://SaturnianCosmology.org/search.php 
==========================================================
The Moon Illusion, An Unsolved Mystery.

by Donald E. Simanek

Disclaimer. The following review of this long-standing problem is
not a research paper. The author is not trained in psychology or
cognitive science. Consider it a semi-critical review of the common
competing hypotheses about the moon illusion, and some experiments
relevant to that problem. Consider it merely an opinion piece if
you wish.

What is the moon illusion?

The so-called "moon illusion" or "moon effect" has perplexed people
since earliest historical times, at least as early as the 7th century
BCE. It is described in early Greek and Chinese writings. Aristotle
mentions it in 350 BCE.

The moon seems larger in angular size when it is near the horizon than
when it is high in the sky. Some people judge it to be as much as
twice as large, but the average estimate is 50% to 75% larger.

It's not a small effect. The Sun displays the same illusion, but
viewing the Sun directly with the eye is hazardous without proper
precautions, so let's leave that out of this discussion. The same
illusion is observed with any extended object in the sky, such as a
constellation.

What it's not.

We know that this phenomenon is not a physical or atmospheric effect,
as may be easily demonstrated by photographing the moon in the sky at
various elevations and comparing its size on the negatives. The moon's
angular size is nearly constant as it moves across the sky, the moon
subtending about 0.5 degree in the sky. One may also verify this with
sighting instruments.

The moon does vary in angular size due to the eccentricity of its
orbit. When the moon is closest to earth it's angular size is about
11% larger than when it is most distant. The moon illusion describes
the variation of apparent size during a much shorter time period of a
few hours betwen its maximum elevation in the sky and its rising and
setting.

Another real physical effect causes the angular size of the moon's
image on the retina to be about 2% smaller when the moon is near the
horizon, compared to its size near the zenith. This is due to the fact
that the moon is one earth radius farther away when observed on the
horizon. This size change from zenith to horizon is much smaller, and
in the opposite sense to the moon illusion. A change of such a small
amount is not large enough to be noticed with our unaided visual
system.

Some people suppose the moon illusion to be due to atmospheric
refraction. Refraction effects can be measured with instruments or
cameras, and we find that refraction actually makes the moon's disk
subtend a smaller angle in the sky than it would have if the
atmosphere were not present. And these refraction effects make the
moon's apparent horizontal angular diameter still smaller (by about
1.7 percent) when the moon is near the horizon! These physical effects
can be confirmed with telescopes and cameras.

The moon's vertical angular diameter at the horizon is even smaller,
causing the moon to appear "flattened." At the horizon the light must
pass through a greater distance in our atmosphere than when the moon
is higher in the sky. This size change is also opposite (and much
smaller than) the psychological moon illusion!
_________________________________________________________________

The reason that the atmosphere makes the moon's disk appear smaller is
interesting in itself. Consider looking at a star directly overhead.
The ray from this star takes the path AO to the observer at O. It
passes through the atmosphere without refraction, along the radial
line AC. Another star at lower elevation sends light along the line
DF. This light is continuously refracted along the curved line FO, its
path bending toward the radial line BC, finally reaching the observer
at O. But the observer judges the direction of this ray to be along
the line OE, which makes a smaller angle with AO than does the line
DF. Therefore the angular separation of the stars seems smaller than
it would be if the atmosphere were not present. This diagram is
exaggerated. In the case of the moon, the angular separation of the
edges of the moon is much smaller (about half a degree), but the same
principle applies.
_________________________________________________________________

Astronomers understand these atmospheric refraction effects quite
well, and must precisely account for them as they interpret data from
telescopes aimed at various elevations in the sky.

Inadequate proposed explanations.

To return to the moon illusion, many arm-chair speculators have
proposed explanations for it. Some of these explanations, long ago
discredited, still appear in textbooks. Many of these invoke visual or
cognitive mechanisms that do have some relevance to the moon illusion.
Some of these mechanisms modify the moon illusion, making it stronger
and weaker, but are not in themselves sufficient to account for the
full and strong moon illusion that most people experience.

Note: Some people report that they do not experience the moon
illusion at all. The literature on the illusion is largely silent
on the reasons for this fact.

Explanations of illusions must be taken with appropriate skepticism.
Many are of the nature of "plausible hypotheses." Most are (as yet)
not such that they can be independently verified in terms of physical
processes in the brain. Also, we know that our visual perceptions
arise because our brain synthesizes multiple cues. Our brain weights
these cues; some dominate in certain conditions, while others are
"weaker" and are ignored. But the weightings shift in strength
according to the nature of the stimuli. Many classic visual illusions
arise from conflicting sensory cues of nearly equal "strength". By
controlled experiments, we may often rule out some of the inadequate
hypotheses about particular illusions. But when one devises a test for
one specific illusion hypothesis, it's difficult to avoid introducing
some other type of illusion into the procedure.

Let's consider some of the proposed hypotheses about the moon
illusion. Each will be presented, followed by evidence and arguments
for and against it.

A Ponzo illusion.
_________________________________________________________________

Illusion of shape. The two figures are exactly the same size, yet many
people judge the upper one to be smaller. But not all persons do. This
illusion illustrates how our judgment of shape and size of an object
can be influenced by the shapes and sizes of other objects nearby in
the field of view.
_________________________________________________________________

Illusions of shape are well-known as in the above example. Multiple
cues in the field of vision can modify our judgments of shape, size,
alignment, and even straightness of straight lines. Some claim the
moon illusion is nothing more than this. One particular version of
this explanation invokes the Ponzo illusion. I wouldn't even consider
this worthy of mention here, but for the fact that is still commonly
seen in popular books on this subject.
_________________________________________________________________

The Ponzo illusion. The two circles are the same angular diameter. Yet
many people judge the right one to be smaller. The two converging
straight lines nearby influence our judgment. If the lines are
replaced with solid black areas, the illusion remains.
_________________________________________________________________

When we view a drawing of, say, a straight railway track receding to
the horizon in the distance, we interpret it as stretching from near
to far even though it is only a pattern of radial lines drawn on a
flat paper at constant distance from our eyes. This is relevant here
because our experience with railway tracks and roadways has
conditioned us to so interpret these patterns in this way. This is the
mental map or model against which we interpret other things. We can
draw two small identical figures on such a drawing, say a couple of
plain (featureless) disks of equal size, the one placed where the
tracks are well-separated on the paper (interpreted as near) and the
other placed where the tracks converge to the vanishing point on the
horizon (interpreted as far). The brain tells us that the disk where
the tracks are narrow is the larger disk. Some writers liken this to
the Ponzo illusion, but it is more than that.
_________________________________________________________________

The figure shows convergent lines that we imagine as parallel lines
converging to a common "vanishing point" on the horizon. The two black
bars are the same angular size, but we think that the one labeled B is
farther away, and therefore many people think it is "larger". Not all
persons perceive this size difference, but very vew think A is larger.
Remove the convergent lines, and nearly everyone judges the bars to be
equal length.
_________________________________________________________________

In my opinion this illusion is not sufficient in magnitude to account
for the moon illusion, only to modify it (and confound attempts to
experimentally study the moon illusion). But consideration of such
illusions does remind us that other objects in the local field of view
may affect judgments of size and shape. By "local" I mean "visually
contiguous or adjacent".

Contextual effects; reference cues in the field of vision.

When we judge the size of an object near the horizon our perception is
influenced by familiar terrestrial objects in the field of view
(trees, houses, roads). We know from everyday experience that many of
the recognizable things we see in the distance are quite far away. But
when our gaze is upwards, we have no reference cues for distance, and
judge things near the zenith to be closer than those on the horizon.
Ibn Alhazan proposed this explanation for the moon illusion around
1000 CE.

Some experiments seem to support this explanation.

* After-images of bright objects may be produced on the retina.
Their apparent size depends upon where you look, being smaller
when you look at a blank wall, or up into the sky, but larger when
there are comparison objects in your visual field.
* When one looks at the moon near the horizon through a hole in a
piece of paper held some distance in front of the eye the moon
appears smaller. The "horizon effect" largely disappears. One
interpretation of this experiment is that the paper tube obscures
any familiar reference objects. It seems to show that the moon
illusion is due to direct comparison with reference objects of
known size. Another interpretation is that the paper tube provides
a visually dominating reference object already judged to be
"near".

Some experiments cast doubt that these explanations are complete.

* The moon illusion persists even when viewed on a dark night on a
featureless plain, on the ocean, and even by airline pilots flying
high above clouds. So reference objects of known size aren't the
only basis for the illusion.
* The moon illusion disappears (for most people) when they bend down
and look at the moon between their legs. [Is it merely coincidence
that this is the position used for "mooning"? Disregard that.] Or,
if so inclined, one can view the moon by hanging from one's heels!
At least two hypotheses have been proposed to account for this.
(1) Familiar objects in the field of view may become useless as
distance references because of their unfamiliar appearance when
viewed upside-down. (2) The illusion may have something to do with
our inner-ear's balance mechanism that tells us whether the head
is upright or upside-down.
* The method of viewing the moon through a hole in a piece of paper
may also be used to view other objects. They too, appear smaller
when viewed through the hole. Is our brain using the paper and the
hole as a strongly dominant reference object? The nearby reference
object is the paper; distant reference objects are hidden.

Hypotheses with some merit.

Some proposed explanations clearly have merit and are based on
experimentally verified visual phenomena. But they also have
"problems" that prevent any one of them being pronounced the "final
answer".

A memory map.

Perceptions are influenced by our past experience. One model of visual
perception postulates that when we perceive a new and unfamiliar
phenomenon our brain interprets it by comparing it to a mental map or
model of our memory of previous sensory experiences. Of course, this
represents just one of the cues that the brain must sort out, weighing
it against other cues. Conflict between sensory cues is the basis of
many common visual illusions.

When we observe the horizon moon and think, "My that moon seems
unusually large," we are comparing it to past expericnec of moons seen
higher in the sky. The moon illusion, seen "in the wild" always
involves a comparison of the horizon moon being seen now with a memory
of an overhead moon seen earlier, or vice versa. There's about six
hours between the horizon moon and the overhead moon (more or less,
depending on the tilt of the moon's orbital plane and the season). Too
often discussions bury this important point in a lot of moonshine.
Perhaps it's more accurate to say that we are comparing our immediate
perception of the moon with all of our memories of moons seen in the
past, under a variety of conditions. We have only one moon, and cannot
observe it in two positions at the same time.

Most of the studies done under controlled conditions ask human
subjects to compare sizes of two objects seen simultaneously. And Yet
these studies clearly show that the illusion of size difference
between overhead objects and objects on a level with the head is still
present. This alone does not discredit the mental map idea, but does
show that if such a map, cognitive mechanism, or mental model is
involved it does not

require

long-term memory. But it also is perceived even without simultaneously
present comparison objects. The distinction between these two
manifestations of the moon illusion is often blurred in the
literature.

If there's a mental map or model, it need not be assumed to be
located in a particular place in the brain. Very likely it is an
interactive linking of specific visual functions distributed over
various places in the physical brain.

The mental sky-model.

This commonly-seen explanation postulates that we have in our minds a
mental map or model of the shape of the sky. We use this model as a
real-time reference surface for objects too distant for stereoscopic
distance cues, even when there are no other cues in the visual field
to serve as distance benchmarks. This mental model does not picture
the sky in the shape of a hemisphere, but is more like a relatively
shallow, inverted soup-bowl. When we view something on the horizon we
perceive that it's located on a portion of the sky farther away than
an object of the same real angular size at the zenith.
_________________________________________________________________

The figure shows the comparison between a spherical sky dome and the
shallow apparent sky bowl. When an object of constant size is seen
near the horizon at A, and then near the zenith at B, we judge that
both are on the apparent sky bowl, and therefore the object seems to
be at A' and B', with A' being farther away than B', and is therefore
judged to be larger.
_________________________________________________________________

This diagram is the one usually seen in books that champion this
hypothesis. But the diagram can be misleading. The hemispherical
sky-dome in the picture is quite irrelevant, and ought to be omitted,
for it falsely suggests a mental process of "projecting" the moon onto
that dome. That's not a necessary part of the process.

The important point of the argument is that our judgment starts with
the stimulus of the retinal image of the moon, which is very nearly
the same size for horizon moons or overhead moons. This is because the
moon has nearly the same actual (physical) angular size wherever it is
in the sky. So we assume an object of this retinal size lies on the
perceptual sky dome. That dome is perceived as more distant at the
horizon than overhead.

This is the same as saying "What is our judgment of actual size of two
things at different perceived distances, even though they have the
same angular size." The answer is that the one assumed nearer is
judged to be smaller. This conclusion is consistent with the
conclusion that the horizon moon is farther from us.

This process operates even (especially) in the absence of any other
visual cues. But the process gets confused when we have our heads in
an unusual position. This may be the result of our knowledge of the
orientation of our head, from visual cues, and perhaps from
information from the balance-sensing mechanisms of our inner ear. When
there are competing sensory cues, our judgment of angular size can be
altered by them, which may account for the confusing results of
experiments designed to show that visual cues are the sole reason for
the moon effect.

Any hypothesis that depends on a mental model of the sky requires that
we have some way to know, at least approximately, which direction is
"up".

What's wrong with this mental sky dome model as an explanation of the
moon illusion?

One reader of the earlier version of this document mistakenly assumed
I was promoting and defending the sky-dome model as an explanation of
the moon illusion. That was not my intent at all, so I must increase
the wordiness of this document by pointing out the obvious objection.
Let me be very clear. The evidence for the illusion of the sky an
inverted bowl is, in my view, abundant and undeniable, and must be
dealt with if we are ever to understand the moon illusion. However, to
use it as above, as an explanation of the moon illusion is specious.

Consider an equally plausible argument using the sky-dome. We know
that the full moon displays constancy of appearance no matter where it
is in the sky. It is the same moon wherever we see it. Few persons,
when asked, would claim that the moon actually changes its physical
size like a deflating balloon as it moves up in the sky. We adults
agree that the moon has size constancy (children may not have formed
such a conclusion). If we observe the moon in a starry sky, most
persons say that the apparent distance of moon and surrounding stars
is the same. They are all located on this flattened "sky dome." If
that is so, then the overhead moon should be judged larger in angular
size, since it is on a portion of the sky that we judge is nearer to
us.

Comparing these two hypotheses with contradictory conclusions leads us
to see the inadequacy of both of them, and the emptiness of this sort
of argument. It forces us to examine just what we mean by "distance",
"size", and "angular size". We will return to this point later. This
confusion does not invalidate the perception of a flattened sky dome,
but does reveal the dangers in trying to use one illusion (the sky
dome) to explain another (the moon illusion).

I hate to suggest this, but many of the arguments about moon and sky
illusions are similarly flawed, and seem only to be playing verbal
games with the reader. They "sound good" but fall apart on close
examination.

The flattened sky-dome hypothesis examined.

So why should we have this shallow-bowl model of the sky? Or, to put
the question another way, why do we have a cognitive processing
mechanism (however complex) that, in effect, gives us a perception of
the sky distorted to conform to such a model? Two suggestions have
been made. (1) The mechanism is hard-wired into our brains from birth.
(2) The mechanism is built up through experience, by daylight sensory
experiences, from a host of visual cues in everyday life.

Studies of children from age 4 to adult have suggested that the moon
illusion is present in children, and is stronger than it is in adults.
It decreases in strength with age. [Liebowitz, H. and Hartman, T.
"Magnitude of the Moon illusion as a Function of the Age of the
Observer." Science, 130,, 569-570.] This study was done indoors in a
large darkened room (no other visual cues) with artificial moons at
distances of 85 feet.

Studies with children can be confusing, for a child will often reach
out to touch a distant object, like the moon on the horizon. Children
haven't yet developed the same mental model of visual space that
adults have. This study suggests that visual experience as one matures
to adulthood modifies the illusion by improving our judgment of
horizontal distances, thereby decreasing the illusion. This casts
doubt on any explanations that assume that the moon illusion is a
result of visual experience and suggests that the illusion itself may
be innate and present at birth.

A result of innate cognitive processes?

The hard-wired hypothesis supposes that natural-selection has shaped
those brain mechanisms that process and interpret sensory data,
devoting more resources to those things that are important to
survival. This results in brain resources being biased toward things
seen in front of us, less resources to things overhead. Similar
imbalance of perception details are seen in animals.

Running great risk of over-simplifying a complex problem, we might say
that things seen overhead appear smaller in angular size because our
brain never evolved adequate resources to interpret data we judge as
"overhead". That data was not important to our species' survival. It
is rendered with less detail and with poorer judgment of relative
distances.

Anisotropy of visual space.

Luneberg (1947) proposed a theory of vision that attempted to relate
physical space with virtual (perceptual) space. He concluded that
virtual space is non-Euclidean, that it is a Riemannian space of
negative curvature: a hyperbolic space. Visual space does not have the
same metric properties in all directions. (This idea is mentioned here
for completeness, not to claim that Luneberg developed the idea fully
and successfully, nor that he applied it to the moon illusion.)

This hypothesis is most helpful in understanding the fact that all
lines of a set of physically parallel lines are also perceived as
straight. Certainly they are not rendered straight on our retinas,
and their curvature on the retina changes in complicated ways as we
move our eyes right/left and up/down. Yet at any instant, they seem
perceptually straight. Our brain is continually recalculating the
retinal image to give a perception of straightness. Is our
cognitive apparatus biased to render "straightness", or is this
merely a by-product of cognitive processes which "correct" the
"warped" retinal image to resolve visual contradictions when our
eyes scan the real visual world? We live in a world with many
straight and parallel lines: streets, walls, railroad tracks, etc.
Yet the mechanism for dealing with the visual world evolved to its
present form long before our ancestors ever experienced such
geometric regularities. Clearly this process has more fundamental
importance to our vision than rendering straight lines "correctly".
I am not arguing that the curved retina is the reason for the
anisotropy of visual space. I'm simply pointing out that the brain
has mechanisms for dealing with the retinal shape, and dynamically
changing retinal image (as the eyes scan a scene), to produce a
stable, and reasonably consistent geometry of visual space.

The larger visual space.
Spherical perspective of parallel lines
perceved from wide-ranging sweeping of the eyes
over more than a whole hemisphere.
This is similar to a fish-eye lens photograph.
Z (zenith), N (nadir), V (left and right vanishing points).
The insets show how parallel lines are
rectified as straight in the smaller visual field.

Only when we scan our eyes around this geometric world, consciously
trying to get the "bigger picture", do we become aware how our
eye/brain mechanism handle this problem. Try this experiment. Look
at a long straight wall. The wall seems to have straight and
parallel lines when our line of sight is perpendicular to the wall.
But when looking "down" the wall at an angle, those same lines
appear to be straight lines converging to a "vanishing point" on
the horizon. Shifting our gaze from one end to the other we
integrate all of these views and finally perceive that those
parallel lines appar as curved lines diverging from a point at the
horizon, becoming nearly parallel, then converging to a point at
the other horizon. This is the geometric world of our wider visual
field, one of curved lines, a "Remannian" space. But when we fix
our gaze in one direction, our brain straightens out those curves,
producing a result like the "Euclidean" space of an artist's strict
perspective rendering.

This process does not, however, consistently "correct" angles
between lines. Right angles in a perspective drawing are usally not
right angles on the paper. Nor are angles correctly perceived by
the eye when the plane of the angle is tilted with respect to our
line of sight.

Once we recognize that visual space may not be anisotropic, that it is
a somewhat consistent distortion of real space, we have a new way of
thinking about the moon illusion and related sky illusions.

Anyone can appreciate the character of this anisotropy by looking at
the daylight cloud-covered sky in relatively flat terrain. Physically
the cloud canopy is nearly a flat plane, as is the earth under our
feet, because the radius of their curvature is so large compared the
distance to the visual horizon. Think of observing the sky from a ship
in the middle of an ocean. The ocean does appear nearly flat, but the
cloud cover appears as an inverted shallow bowl. This shape can be
appreciated even better if the observer slowly moves his eyes to scan
around the horizon and up from horizon to zenith, with conscious
intent to get a feeling for the shape of things in this larger visual
space. Some people even perceive that the ocean curves somewhat upward
toward the distant horizon just as the cloud cover curves downward. In
any case, most people judge the sky at the horizon to be the same
distance as the horizon.

How does this impression square with the situation in real (physical)
space? The physical distance to the most distant object one can see on
the horizon depends on the elevation of the observer's eye above the
water. One can derive the formula for it, in terms of the earth's
radius. For an eye elevation of six feet, the horizon seems about 3
miles away. Alto-cumulous clouds are about 2 to 3.5 miles overhead. So
physically the distances are nearly the same, yet the overhead clouds
seem much closer to most people than those near the horizon. This
calculation may not seem quite fair, for we can see clouds that are
physically well beyond the surface horizon, around 100 miles away, due
to their height above the earth surface. But can any reader and
observer honestly claim that the clouds at the horizon seem farther
away than the water at the horizon? I've never found anyone who would
make that claim.

Since the clouds at the horizon are about 100 miles away, the physical
dome of a cloud covered sky is a very flattened bowl (or saucer), 100
miles in radius and only a few miles high. Yet when this calculation
is done, even the most seasoned mathematician or physicist must admit
"I don't see it that way!" Need we further belabor the point that the
geometry of our visual space of such distant things is severely warped
compared to physical space.

We have a strong impression that the cloud cover "joins" the horizon.
Can this simply be that there's absolutely no visual cue to suggest
that they are at different distances? Our brain may be making the
simplest reconciliation of the situation.

Why does the cloud cover appear to curve downward at all? This is seen
even when there are individual clouds with cues of shape, size, and
shadowing. Beyond a certain distance, a distance set by other visual
cues, our brain refuses to place objects at a greater apparent
distance, especially if there are no local cues indicating objects
between the horizon and the cloud cover. By "local" I mean, visually
contigouous objects near the horizon line. Our cognitive mechanism
seems to prioritize decisions, favoring reconciliation of local cues
at the expense of those displaced from one another by larger visual
angles.

These facts illustrate that when we are dealing with these great
distances, our usual geometric logic about distances is essentially
useless to describe what we perceive. Once this fact of perception is
realized, one also realizes that many of the experiments and theories
of the moon and sky illusions and many of the published papers on the
subject of visual illusions are simply irrelevant our judgments of
very distant objects.

Why is visual space anisotropic? Specifically how does the anisotropy
work? Why does the "metric" of this anisotropy shift in response to
visual cues, such as objects at various distances in the field of
view, and even on past visual experience? And why does some anisotropy
remain even if nearly all visual distance cues are absent? Those are
questions that need to be addressed. Little progress has been made in
that direction. But we can't solve the moon and sky illusions until we
deal with those questions.

Space-filling

The research clearly indicates that nearby objects in the field of
view do influence our judgment of the distance and size of more
distant objects. In a general way, it seems that

* Objects in the field that are contiguous to the distant objects
have the greatest effect.
* When there are many objects in the field of view for which we can
readily and unambiguously assign sizes and distances, this can
give a strong impression of greater distance for objects merely
judged "beyond" the others.
* When there are few contiguous objects and few strongly located
nearer objects in the field of view, distant objects appear
nearer.
* We tend to perceive an object with fewer distance cues as "just
beyond" the farthest object for which we have mentally assigned a
concrete "distance".
* Easily recognized objects, and those for which distance cues are
strong, are probably processed by the brain first, leaving more
ambiguous objects for later resolution. Of course the whole
process is so rapid we think the judments are instantaneous.

It's as if our visual space becomes "filled" with objects for which
our brain can readily assign distances, leaving those with fewer (or
ambiguous) distance cues to occupy the "more distant" space, located
just beyond the more confidently located objects.

Even within the visual space of relatively unambiguous distances and
sizes, our brain has placed objects in this space by resolving visual
conflicts and by using some visual cues to modify and reinforce other
cues.

Trehub's retinoid model.

Arnold Trehub, in his book The Cognitive Brain (MIT Press, 1991)
proposed a comprehensive theoretical model of cognition, postulating
neuronal mechanisms and systems that are responsible for cognition. He
hypothesizes a neuronal structure that acts as a dynamic buffer for
processing information from the retina, which he calls a retinoid.
This retinoid performs many functions, some of which determine how
visual sensory information is interpreted, before it is passed on to
higher levels of visual processing. The retinoid organizes and
integrates visual information into a unified model of visual space.
Trehub calls it a "visual scratch pad" that stores spatially organized
information as short-term memory.

One need not delve into the details of how this might work on the
cellular level in order to recognize the importance of this general
idea to the study of perception and to illusions of perception.

The visual process begins with sensory data, primarily from the
retina. The retinal images represent a crude map (in two dimensions)
of the real space of luminous points. The cognitive processes
synthesize this data, reconciling ambiguities, correcting for
deficiencies of the eyes themselves and creating a final "veridical"
image that we perceive as the space we are looking at. Past experience
plays a role, when we observe familiar objects.

For example, we see three faces of a cube, but our experience with
cubes generates a veridical judgment that the cube has six faces,
including the ones that produce no retinal sensation until we rotate
the cube. Our brains have established sub-processing systems for
cubes, spheres, trees, and other familiar objects, treating them as
entire objects even when we see them only partially.

The veridical image is in some ways better than the retinal images at
any instant. Yet it can also have features that simply aren't an
accurate representation of real space, as demonstrated by the many
visual illusions studied by psychologists. Many such illusions have
contradictory visual cues, and the brain does the best it can at
reconciling those contradictions.

This cognitive bias of veridical space depends upon our prior
cognitive judgment of up and down. We have a sub-processing brain
mechanism for the space surrounding us. But the data funneled into
this system has already subconsciously been realigned at an earlier
stage of cognitive processing that made a judgment about up and down.
Various cues, including our vestibular balance mechanism tell us at
all times how our head is oriented. Then the rest of the cognitive
apparatus further interprets visual cues based on this judgment of
"up". [The older "sky dome" hypothesis depends upon this up/down
judgment also.] Tilt of the head can slightly alter this judgment, but
if there are enough cues, it won't destroy the sky and moon illusions.
However, hanging by one's heels or looking at the sky with head
inverted greatly reduces the illusion for many people. But this
orientation also confuses one's interpretation of many visual spatial
relationships. It's worth noting that the quality of many visual
judgments is much reduced when viewing the world from this unusual
position, even when viewing a picture or photograph that is also
inverted. Things don't seem quite right. The quality of cognitive
processing is severely degraded.

The moon illusion is consistent with what would be expected from
evolutionary considerations. We have evolved cognitive processes that
provide high quality visual information from nearby things, and things
on our level that we can walk to and experience from various angles.
These are all important to survival. Things seen high above, in the
sky, or even those seen below, as when looking over the edge of a
cliff, are less important, and distance discrimination and judgment of
these is compromised.

The size-distance paradox.

Most people have a cognitive model that relates angular size to
distance. This is called the "size-distance invariance law". When one
observes two persons, one subtending a small angle, one a larger
angle, one judges the person subtending the small angle to be more
distant. This works for objects that we know or have already judged
(from memory and other cues) to be "really" about the same size.

As usually seen, the sky-dome model "explains" the moon illusion in
this manner: The retina receives a certain size stimulus, the same
angular size for horizon or overhead moons. The brain has a model of
the sky that causes us to judge that the overhead moon is the nearer
one. Therefore since the two moon images have the same retinal size
(stimulus), we judge the overhead moon to have smaller actual size. As
indicated earlier, this over-simplified "explanation" seems to me
empty and misleading.

Some who object to the "sky-dome" model in any of its variants point
out that when subjects are asked which moon is larger, they answer
"The horizon moon." When asked which moon is nearer, most will say
"The horizon moon." This, critics say, contradicts the simple sky-dome
explanation, and therefore invalidates it. They call this the
"size-distance paradox". This objection, in my opinion, carries no
weight.

In these discussions we must distinguish three levels of sensory
processing that play a role in the final judgment.

1. The sensation itself. In this case, the diameter of the image on
the retina, stimulating light receptors there. All commentators
agree that the moon illusion does not occur on this level.
2. The unconscious judgment of some "physical" property, such as
brightness, angular size, color. This may be influenced by past
experience, or by quirks of innate cognitive functioning, but is
done without us consciously "thinking about the matter" or
deliberately "analyzing" something.
3. The reasoned consideration of the sensation, by questioning and by
conscious analysis. Example: "Those two bars seem connected, but
that can't be so, for it violates some laws of math or physics."
On this level, judgments may be expected to be quite different for
untrained and trained observers, and also different for those
unschooled in math and physics compared to those who are. The moon
illusion literature is largely silent on these possible
differences of interpretation.

Mixing these levels has confused many discussions of the moon effect.

What is the meaning of 'size'? When dealing with physical measuring
instruments such as cameras and telescopes, the relation between
angular size, linear size and distance is a straightforward
application of trigonometry.

tan(angular size) = (linear size)/(distance)

Or tan(a ) = S/D 

But visual judgments of linear size and distance are clearly
confounded by visual illusion effects of various kinds. So "apparent
linear size" and "apparent distance" were introduced, and the
size-distance-invariance hypothesis was written:

tan(angular size) = (apparent linear size)/(apparent distance)

Or tan(a ) = S'/D' 

On this view, if distance cues predominate, altering D to D', then the
apparent size S' will be "computed" by the brain as necessary to
interpret the angular size A of the image on the retina. But if size
cues predominate (as when you are looking at a person whose size is
familiar from long experience), the apparent distance D' will be
computed by the brain.

This hypothesis has an underlying assumption. That the angular size of
importance is simply the physical size of the retinal image, the
visual stimulus. But is it possible that perceived angular size is
also subject to illusory effects? If so we should write:

tan(a ' ) = S'/D' 

This view, and this equation, was first proposed by Don McCready
[1965]. He developed this more fully in his [1985] paper, and applied
it to the moon illusion [1986]. His website provides a very thorough
presentation of these ideas.

This raises the question: What are we doing when we judge the "size"
or "distance" of something (like the moon) for which we've had no
experience close-up, and whose physical linear size and distance are
too large to perceptually comprehend anyway? When we say a moon is
"larger", are we talking about angular size or linear size?

Some argue that the basic moon illusion is one of angular size.
Certainly all the experiments that ask subjects to compare the moon's
image (or a simulated moon) with a physical object (or a simulated
physical object) are asking for a comparison of perceived angular
size. Also, the interpretation of experimental studies may be
confounded by ambiguity of the meaning of the word "size", as
understood by the subjects and by the experimenters. Even if the
experimenter carefully explains to naive subjects the difference
between angular size and linear size, and between "real" size and
"perceived size", a subject may be incapable of making those
distinctions when describing an observed object.

I will not belabor the three size equations given above, for they may
not even be relevant to the question of the moon illusion. The moon
illusion operates at real distances essentially infinite, and real
sizes beyond direct apprehension. The only relevant variables are
angular size, real and apparent. However we must admit that
experimental studies of illusions seen in nearby space may help us
understand the cognitive mechanisms that may also apply to the moon
illusion.

Some studies have asked subjects "which moon seems nearer", which
seems an unfair question, but still one cannot deny that we have
mental processes that produce at least an approximate subconscious
judgment of "near and far" for things in the sky, and this is
certainly relevant to the sky illusions. The cognitive judgment of
apparent angular size is made at a very early level of cognitive
processing, without "thinking about it" consciously (deliberately).
When asked which moon is larger, we then must consciously process this
question (without being consciously aware of our previous unconscious
determination of its apparent angular size).

Suppose an observer has subconsciously determined that the horizon
moon has larger angular size than the zenith moon. This is the
"perceived" angular size, A'. When asked which moon is nearer, the
observer may consciously reason that the horizon moon must therefore
be nearer. Never in this process is this observer aware of the
unconscious process by which his brain already concluded that the
horizon moon had larger angular size. No contradiction or paradox is
present here, for these two levels of mental processing are not
simultaneous. The contradiction comes only when we think about what we
said. 

Most visual illusions contain similar judgment contradictions. When
the Penrose impossible tribar is seen, a subject may judge that the
right side is more distant than the left, based on the way they
connect at the top vertex. But then, looking at the constant
angular width of the lower horizontal bar, the subject is forced to
conclude that the two bottom corners are the same distance away.
Contradiction. Yet each individual judgment seems "right" and is
based on habitual visual associations of everyday experience. This
is what makes illusions tantalizing.

Observers schooled in physics and math, physicists, astronomers, and
amateur sky-gazers, have learned somewhat different conscious models.
When asked "Which moon is nearer" they may respond "That's an unfair
question; it's still the same moon, and in both cases the distance and
size are too great to make a distance judgment" If a follow-up
question is asked, "Which one seems nearer?" the answer is often
"Well, you can't trust your eyes in such matters." Many scientists
have learned (by their mistakes) not to trust their eyes in certain
kinds of informal observations. As Helmholtz said, "I would never
believe anything solely on the evidence of my eyes." Unlike Helmholtz,
most physical scientists do not inquire into the reasons behind visual
deceptions, they simply try to stick to observations and instruments
that can be shown to produce reliable and observer-independent
results.

Oculomotor micropsia and macropsia

Don McCready's web documents emphasize the importance of angular size
illusions. He also proposes an explanation of the moon illusion based
on angular size illusions due to oculomotor micropsia, an illusion
that accompanies the accommodation of the eye lens and the convergence
of the two eyes. Oculomotor micropsia has been known since it was
discovered by Charles Wheatstone (1852), and has been amply
demonstrated experimentally for nearby objects, those for which
convergence and accommodation play a dominant role in perception of
distance. McCready's detailed website document describes this effect
quite well.

Briefly oculomotor micropsia is this: Normally, if there's only one
object, or a dominant object, in the field of view, our eyes try to
converge and accommodate to its distance. The muscles that control
convergence and accommodation send signals to our brain, which are
important distance cues. But when one's eyes converge and accommodate
to a smaller distance than an object, that object appears to subtend a
smaller angle than if one's eyes were converged and accommodated upon
it. Micropsia means "appearing small", and here refers to the visual
angle subtended by an object. The reverse effect, macropsia, occurs
when the eyes converge or accommodate at a distance greater than the
object being judged.

Why would the eyes converge or accommodate at distances different than
the object of interest in the field of view? Several reasons. A very
large number of nearby objects spread over the field of view may bias
the brain towards convergence at their distance. For most people, the
functions of convergence (aiming the eyes to a common point) and
accommodation (focus setting of the eye's lenses) are "locked
together" so if one converges to nearby objects, the accommodation
adjusts to that distance also, and vice-versa.

Proponents of this hypothesis as explanation of the moon illusion
argue that if there are a number of distant objects in the field of
view, as there would be when observing the rising or setting full
moon, the brain adjusts accommodation and convergence to them. But
when viewing the full moon directly overhead, in a clear sky, there
are no other distance cues, and the eye adjusts to its resting focus a
distance of 1 or 2 meters. This makes the perceived angular size of
the overhead moon seem smaller. Further support of this hypothesis is
the effect of dark surroundings which biases the eyes to adjust to the
dark focus distance of about 1 meter.

But, these micropsia and macropsia illusions cause angular size
differences of less than 10%, nowhere near large enough to account for
the moon illusion seen by most persons.

Also, if accommodation were involved in the moon illusion, you'd think
that elderly people who have lost nearly all accommodation should not
perceive the illusion. Yet they do. Persons with eye lens implants
have no accommodation, and they do perceive the moon illusion.
Covering one eye removes convergence from consideration, but that
doesn't make the moon illusion go away.

As usual in these matters it's not so simple. Experiments have shown
that even when one eye is covered, or blind, the muscles and the eye
lens still accommodate to the distance perceived by visual information
supplied by the other eye. Also when the eye lens muscle is paralyzed
in both eyes by use of atropine drops, micropsia still occurs. It
seems that micropsia and macropsia are not the result of physical
changes in the eyes, but by processes occurring in the brain, the same
processes that control the muscles of the eyes and the ciliary body
that adjusts the focus of the eye lens.

The planetarium sky.

Does the same thing happen in a planetarium? No! This is why the
planetarium is a poor simulation of what we see in the night sky. All
real-sky effects are beyond the range where our stereoscopic vision
(due to eye convergence) works (about 50 to 100 feet), so our eyes
can't triangulate objects in the sky. The planetarium, with, say a 50
foot diameter dome, has all star images just within the range of
stereoscopic vision, and this fact dominates our judgment, minimizing
other psychological effects. Also, as we walk into a planetarium we
have a visual and rather direct feeling for its size that persists
even after the lights are turned off.

Have you noticed that the moon, projected onto a planetarium dome with
exactly the correct angular size, seems the same apparent size
wherever it is located on the dome, and therefore appears "too small"
when on the horizon, compared to our memory of the setting or rising
moon in the real sky?

I've been informed (Jan 2000) that some researchers claim that the
threshold for detecting depth due to retinal disparity is 1 arc
second, so persons with normal vision can detect distance
differences between the moon and objects up to at least 100 meters
away. Such experiments are done under "ideal conditions" and
usually involve judging whether two points of light are equally far
away. My comments about the planetarium sky do not depend on the
correctness of this figure, since the planetarium dome is clearly
well within the range of stereoscopic vision, whichever figure you
accept for the maximum distance for stereo vision.

The hypothesis of the importance of eye-convergence (stereoscopic
vision) in the moon illusion can be tested. Suppose we suppress
stereoscopic cues in the planetarium by covering one eye. Experiments
related to the moon illusion, especially those where nearby objects
(nearer than about 50 feet) are in the field of view, should also be
done with one-eyed viewing to suppress stereoscopic cues. This has not
always been recognized as an important visual cue in these
experiments. However, I think they play little role in the moon
illusion itself, or any related illusion related to objects beyond the
range of stereoscopic vision, except to establish the distance of
nearby reference objects in the field of vision.

This illustrates why experimental psychology can be every bit as
difficult as physics. As McCready says (August 1999): "Clearly, no
theory has been fully accepted by the experts yet."

What's the bottom line?

In summary, it seems to me that:

1. The class of size illusions that includes the moon illusion affect
primarily objects at distances beyond those where stereoscopic and
other strong distance cues are operative. We therefore should not
expect that the size-distance invariance laws for near objects are
even relevant to the moon illusion.
2. The sky illusion (sky-dome as a shallow inverted bowl) should not
be ignored, for it is nearly universally observed. Both it and the
moon illusion may arise from the same cause. Understanding the sky
illusion may be the key to understanding the moon illusion, but to
merely invoke the sky illusion as the cause of the moon illusion
evades the fundamental issues, and is empty of content.
3. The sky illusion represents a warping of visual space for objects
so distant that other everyday cues to distance and size are
inoperative.
4. The sky illusion depends greatly on one important cue: our sense
of what direction is up. Cognitive processing of visual space may
be linked to our knowledge of the tilt of the head, from visual
cues relating to up/down and the vestibular balance mechanism in
the inner ear. Therefore we know the position of our head and eyes
relative to "up" and "down". However, some studies show that head
tilt alone plays only a small role in angular size determination
of isolated objects in an indoor laboratory environment, in the
absence of other stimuli.
5. Other objects seen in the visual field while viewing the moon can
further modify our judgment of distance and size, confusing some
of the experiments that have been done.
6. Spatial relations that are familiar, unambiguous, and easily
"computed" are processed promptly, and perceived objects are
mentally assigned sizes, distances and locations in visual space.
Ambiguous objects, and those with no strong distance cues are
processed after the near visual space has been "filled" and are
assigned to distances just beyond the "filled" space, if that can
be done without cognitive conflict.
7. Trehub postulates that we are born with a subconscious cognitive
processing mechanism that is innate. It is a result of
evolutionary necessity for devoting more visual processing
resources to nearby space, and space at eye level, but fewer
resources to things seen at higher elevation. Therefore the
angular size of objects at eye level is perceived as greater than
the angular size of the same objects seen when looking upward.
This model does not rule out the possibility that such perceptions
can be modified through experience and by particular sets of
visual cues. Such cognitive models may serve as a guidline for
experimental investigations, but direct experimental confirmation
of all of a model's details is necessary before it can be
considered a physically and biologically correct explanation.
8. We have many descriptive models of cognition that certainly are
better than nothing, but do not adequately and fully represent
what goes on in the brain. Their power to correctly predict how we
perceive particular new situations is not impressive. These are
very much like a flow chart of a computer program, which doesn't
give us a clue about what goes on at the bit and byte level when
the program is run on a particular computer.
9. The final word has not yet been written on this subject.

-- Donald Simanek

AN INCOMPLETE BIBLIOGRAPHY

The literature on the moon illusion is vast, indicating an obsession
with the problem that borders on lunacy. I'm in no position to compile
a complete and definitive bibliography. I do not have easy access to
the relevant journals, especially the older ones. Fortunately
Hershenson's book includes many literature references, as do the web
sites listed here. I have not consulted all of these. Some were
recommended by others. I welcome suggestions for making this
bibliography more useful.

In this document I have not attempted to give a complete history of
ideas about the moon illusion, nor have I tried to indicate in every
case who first proposed a hypothesis or who did the experimental
studies, or when, for I had no intention to write a definitive history
of the subject. Besides, it's often impossible to know who first had a
particular idea, as distinct from who was first to publish it. Bart
Borghuis' web document may be consulted for such details, and the
anthology by Herschenson [1989] contains detailed review and
bibliography of the literature of the subject.

1. Corum, M. C. "On the Moon Illusion" The Rand Corporation, June
1976.
2. Frisby, John. Seeing--Illusion, Brain and Mind. ~1980's.
3. Gregory, R. L., et. al. "Changes in Size and Shape of Visual
After-Images Observed in Complete Darkness During Changes of
Position in Space." Quart. J. Exp. Psychol. 11:54-55. (1959)
4. Herschenson, M. (Ed.) The Moon Illusion. L. Earlbaum Associates,
Publishers. Hillsdale, NJ, 1989, pp 193-234.
5. Jewell, John G. "The moon illusion: untangling the size-distance
paradox through stereoptic viewing." Bucknell University Master of
Science Thesis, May, 1994.
6. Kaufman, Lloyd. Sight and mind, an introduction to visual
perception. Oxford University Press, 1974.
7. Kaufman, L. and Rock, I. "The Moon Illusion: I," Science, 1962,
136, p. 953-961.
8. Kaufman, L. and Rock, I. "The Moon Illusion: II," Science, 1962,
136, p. 1023-1031.
9. Kaufman, L. and Rock, I. "The Moon Illusion." Scientific American,
July 1962, 207(1) Cover and p. 120-130.
10. Kaufman, L. and Kaufman, J. "Explaining the Moon illusion."
Proceedings of the National Academy of Sciences, 97, 1, Jan 4,
2000, pp 500-504.
11. Liebowitz, H. and Hartman, T. "Magnitude of the Moon illusion as a
Function of the Age of the Observer." Science, 130,, 569-570.
12. Luckiesh, M. Visual Illusions, their Causes, Characteristics and
Applications. 1922, Dover 1965.
13. Luneberg, R. K. Mathematical Analysis of Binocular Vision,
Princeton, New Jersey. Princeton University Press, 1947.
14. McCready, D. [1965] Size-distance perception and
accommodation-convergence micropsia: A critique. Vision Research,
5, 189-206.
15. McCready, D. [1985] On size, distance and visual angle perception.
Perception & Psychophysics, 37, 323-334.
16. McCready, D. [1986] Moon illusions redescribed. Perception &
Psychophysics, 39, 64-72.
17. Minnaert, M. The Nature of Light and Color in the Open Air. Dover,
1954.
18. Quarterly Journal of the Royal Astronomical Society, Vol. 27, p.
205. (1986)
19. Rock, I. Perception Scientific American Library.
20. Ross, Helen E. & Plug, Cornelius. The mystery of the moon
illusion: Exploring size perception Oxford University Press, 2002.
21. Suzuki, Kotaro. "Moon illusion simulated in complete darkness:
Planetarium experiment reexamined." Perception & Psychophysics,
49, 4 (1991) p.349-354.
22. Taylor, F. V. "Change in Size of the After-Image Induced in Total
Darkness." J. Exp. Psychol. 29:75-80. (1941)
23. Trehub, A. The Cognitive Brain. MIT Press, 1991. Chapter 14, p.
342-347.
24. Walker, B. "The Moon Illusion: A Review," Parts 1 and 2. Optical
Spectra, January, February, 1978.
25. Wenning, Carl J. The Planetarian, Vol. 14, #4 (December 1985). Web
copy.

SOME INTERNET SITES

Listing these sites here does not constitute endorsement of the
validity of their content. I, personally, have reservations about many
of them.

1. The Moon Illusion, A literature thesis by Bart Borghuis. A very
extensive review of the published research, and the theories of
the moon illusion.
2. The Moon Illusion.
3. The Moon Illusion by Carl J. Wenning.
4. The Moon Illusion, by Hendrik Ball.
5. The Moon Illusion Explained, by Don McCready.
6. Sensation & Perception; Ponzo Illusion.

Input and suggestions are welcome at dsimanek at lhup.edu. When
commenting on a specific document, please reference it by name or
content.

Revised 11 Jan 2002

Return to top of this page.
Return to illusions page
Return to front page and main menu.