A New Theory of the Moon Illusion
by Ken Amis
The moon illusion has confounded philosophers and scientists since ancient times.  Why does the moon look larger when it is near the horizon, but smaller when it is high overhead? While observers don't agree on the relative sizes, most judge the horizon moon to be at least 1.5 times the size of the moon seen high overhead. There is now no doubt that this is a cognitive illusion. Photographs of the moon clearly demonstrate that the moon's angular size is a bit more than 1/2 degree at all times, whether near the horizon or elsewhere. It is not a trick of atmospheric refraction, for that actually makes the moon's image slightly smaller at the horizon. It is not due to varying distance of the moon, for the moon at the horizon is actually farther from us by 1 earth radius, and this also makes it's angular size very slightly smaller there. Many theories have been devised in an attempt to explain this.  All of them are largely failures.
Certain facts are indisputable. The angular size of the moon is nearly constant at approximately 1/2 degree for any position in the sky. Small variations in this angular size aren't usually noticed, and are certainly nowhere mear large enough to account for the moon illusion that we see. The size of the moon's image on our retinas is nearly constant for all viewing conditions.
The commonest explanation seen in textbooks is the "apparent sky dome" theory. It supposes that we have in our brains a distorted perception of the sky, not hemispherical as it "ought to be" but flattened like an inverted soup bowl. We mentally "project" our retinal perception of the moon's disk onto that dome, and, as the illustration shows, the projection of the high elevation moon seems smaller because that portion of the apparent sky-dome is nearer to us.
It seems incredible that this loony "explanation" has persisted so long in textbooks. It is based on questionable and undemonstrated premises, shaky logic and a lot of hand-waving. So accustomed are students to reading simplistic explanations of science, that they easily swallow this fraudulent nonsense without burping.
The argument is subtly circular. It assumes that every distant thing we see is mentally projected onto a flattened dome. Then it tries to argue that particular things seem to vary in size because they are projected onto different portions of that dome. This can be summed up: Perceptions of sizes of things in the sky create the mental construct of a flattened dome, then the flattened dome is used to explain the perceptions of other things seen in the sky. One illusion (the sky-dome illusion) is being used to explain another (the moon illusion).
We should ask "Why does the brain construct this mental sky dome model, and just how does the mind "project" anything onto it?" The usual handwaving commences. One is "The model is built up from our perceptions of cloudy skies." Then why don't overhead clouds seem smaller than distant ones on the horizon, just as the overhead moon seems smaller? Yet the reverse is perceived with clouds. In fact, that kind of sky, with fluffy separated clouds over the whole sky, is not the only sky we experience. There are hazy skies, perfectly cloudless skies, and heavy overcast skies. And they, too, seem to be something like the flattened dome. A starry night seems like a flattened dome also. Even a featureless sky, if you scan your eyas around and think about it, appears to be a flattened dome. The flattened sky-dome is not an explanation of any of these observations, but is a phenomenon still begging for an explanation.
If an explanation has not been found, at least a better model is in order.
In most situations, we view the rising or setting moon by looking forward at it. The sky above the horizon is otherwise devoid of visual reference, though it may be star-filled. We are accustomed to viewing things with our bodies upright and looking horizontally. Terrestrial objects, like trees, mountains and doorways rise straight up from the earth below, and we have a strong sense of what is up, relative to the ground below. We can imagine that our visual field is a theater screen, like a giant IMAX ® movie screen. We can draw this as a rectangle bent to cylindrical form, as shown in the figure.
When we look at a particular portion of the sky, including the horizon, we do not have an impression of a domed sky. We perceive that everything in that portion of the sky lies on a flat or slightly curved "canvas" whose bottom is the horizon. On this canvas, the top edge is "directly above" the bottom edge, and the top (AB) and bottom (CD) edges are perceived to be of equal length.
When we stand still and look at the moon, this is a more realistic model of our perception than a sky dome. Try as you might, you cannot force your brain to interprest a view like this as a portion of a dome of any kind. The distant sky looks either flat, or cylindrically curved around a portion of the horizon. (The horizontal extent of our field of vision is larger than its vertical extent.) It is only when we scan our eyes around the whole sky, and even look up, that we can synthesize a mental picture of anything like the sky dome pictured in the textbooks. But this is not what we do when we observe the moon.
If we are looking straight at the horizon, our field of view spans an angle that extends, say, 45° above the horizon. Let's suppose the giant screen in the diagram is square, spanning a field of view of 45° vertically and the bottom edge spanning 45° horizontally. But the top of the screen will span more than that horizontally. Simple geometry gives the result, it spans (1/cos(θ))45° = (1.41)45° = 63.64°. Remember that the angular size of the moon on the retina is about 0.5°. Therefore along the bottom edge of the screen one could fit 90 moons, but along the top of the screen one could fit about 135 of them. Relative to the screen, which we naturally assume is square, the moons aligned along the top seem smaller that those aligned along the bottom. Therefore we judge the horizon moon to be about 1.41 times the diameter of the ones along the top of our field of view, at 45° elevation from the horizon.
This result is in good agreement with the vast amount of experimental data, of observations on the real moon in a real sky, which tell us that the horizon moon seems at least 1.5 times as large as an overhead moon.
Gometrical calculation shows, then, that the apparent angular size of the moon should relate to elevation angle above the horizon as:
where So is the angular size of the moon when seen at the horizon and α is the moon's elevation above the horizon. The formula is valid for angles up to about 40° to 50° since that's about the vertical limit of one's visual field when it includes the horizon. [The vertical extent of the visual field is about 45° to 50° on either side of the optic axis, but its limits are not sharply defined.] When viewing the moon at higher angles, the horizon is either not in the visual field, or is so near the edge of the visual field that it does not affect one's visual jugment appreciably. It does not provide a lower "anchor point" for that virtual backdrop screen onto which we the objects being viewed. This formula is in agreement with existing data as well as, or better than, any of the other theories of the moon illusion. At those larger angles, the visual field includes mostly empty sky, with no visual clues to judge relative sizes and distances.
You may have noticed that books that "explain" the moon illusion using the sky-dome model never give you the raw data (observations) the model is supposed to account for. Those observations of the real moon, by real observers in the real night sky, may be found in the literature [5,6]. The data from Holloway and Boring  are singularly unimpressive, as data goes, consisting of observations by three observers, resulting in a couple dozen data points with considerable variability. And the data doesn't include any lunar elevations greater than 50° above the horizon. The proponents of the sky-dome model seldom show an equation for the shape of their dome, nor do they show how well, or how poorly, their dome fits the observations.
A surprising prediction.
Through most of the history of moon illusion experiments, human subjects were asked only about the apparent relative size of moons at different elevations. When subjects were also asked to judge which moon appeared to be closer, the answers were surprising. Most subjects judged the horizon moon to be nearer than the overhead moon. These size and distance judgments seemed contradictory.
The answers would seem contradictory to anyone who believed the "sky dome model", for that model assumes that the moons of constant angular size are projected onto a mental model of a flattened dome. But once one turns this model inside out and assumes the moons are projected onto a distant vertical backdrop, does this make sense.
Any successful model of the moon illusion must explain.
We submit that the "backdrop screen" model answers all of these.
 For the history, see The Moon Illusion, A literature thesis by Bart Borghuis. A very extensive review of the published research, and the theories of the moon illusion.
 An overlong and tedious review of various theories of the moon illusion can be found at Donald Simanek's web site: The Moon Illusion, An Unsolved Mystery.
 Kaufman, Lloyd. Sight and Mind, an introduction to visual perception. Oxford, 1974.
 Kaufman, Lloyd and James H. Kaufman. Explaining the moon illusion. Procedings of the National Academy of Sciences in the United States of America. Vol. 97, Issue 1, 500-505, January 4, 2000.
 Holway, A.H, and Boring, E.G., 1941. Determinants of apparent visual size with distance variant. American Journal of Psychology, 54, 21-37.
 Higashiyama, A. 1992. Anisotropic perception of visual angle: Implications for the horizontal-vertical illusion, overconstancy of size and the moon illusion. Perception & Psychophysics, 51, 218-230.
Document written 2004, edited for small errors, Jan 2012 and Dec 2014.