Few would argue that the Moon is more wondrous and enchanting than when it is near the horizon, appearing closer and looming larger than when it is high in the sky. But this is an illusion-the "Moon Illusion," to be precise: the distance traveled by light reflected from the moon to the eye of an observer is essentially the same regardless of the Moon's elevation. A similar illusion is observed for the setting sun and for celestial distances between star points at different elevations.
But the Moon Illusion is not only captivating to behold; it also happens to be perhaps the oldest unsolved problem in science. References to it can be found on clay tablets from the royal library of Nineveh and Babylon, dating from before the sixth century BC, as well as in a collection of Chinese legends ascribed to Lieh Tzu dating from the fifth century BC. Many of history's leading scientists and mathematicians have analyzed the phenomenon: da Vinci, Kepler, Descartes, to name but a few.
For most of recorded history, the illusion was thought to be a consequence of physical processes. For example, Aristotle in the third century BC and Ptolemy in the second century AD incorrectly attributed the illusion to the magnifying properties of the atmosphere. Alhazen (Ibn al-Haytham) related the illusion to the flattened appearance of the dome of the sky.
In the nineteenth century, it became clear that the Moon Illusion is psychological, a consequence of the cognitive processes underlying the human brain's visual space perception. Many theories have since been offered to explain the illusion, but there is still little agreement among researchers.
Most modern explanations treat the illusion as a static phenomenon, in which a stationary observer views a fixed illuminated object over a ground. The perceived size of the moon is determined by two factors: the physical extent of the light falling on the eye - also known as the angular subtense or the visual angle - and the distance information provided by the ground. Some theorists include in this formulation the perceived distance of the moon (how far it appears to be from an observer).
The relationships between the stimulus (visual angle) and the perceived size and distance of the moon are summarized by the so-called "static size-distance invariance hypothesis" (SDIH): stimulus determines the ratio of perceived size to perceived distance. The information from the ground fixes the unique values of each variable in this ratio.
The SDIH is sometimes assumed to be a fundamental law of visual perception. But it is this relationship that makes the Moon Illusion such a puzzle, because applying the SDIH requires that the moon appear farther away if it appears large, and close if it appears small. This is also why the illusion has sometimes been called "paradoxical."
Since observation contradicts the hypothesis, modern explanations of the Moon Illusion have proposed a variety of changes to the SDIH. One modification allows perceived distance to be evoked simultaneously by different behavioral responses. Thus, the moon may appear more distant to people who say: "The moon looks close."
Other explanations exclude perceived distance. Verbal statements about the perceived distance of the moon are described as inferences based on perceived size rather than descriptions of experience: "The Moon looks big so it must be close."
Other researchers have gone beyond the SDIH altogether, substituting a perceptual outcome - perceived visual angle - for the stimulus input. Unlike the SDIH, this implies a purely psychological relationship.
But none of these theories answers the age-old question: what accounts for the simultaneous perception of a large and near moon?
I have proposed a solution that approaches the illusion from a different starting point. I begin with the nature of the perceptual system: What were the circumstances in which it evolved?
Clearly, motion was an essential feature. Our visual systems evolved in an environment that contained (mostly) rigid moving objects. Objects that move radially, i.e., directly toward or away from a viewer, produce a stimulus on the eye that is continuously increasing or decreasing in size. Our visual system automatically transforms such changing stimulus inputs into objects that appear to be rigid - i.e., unchanging in perceived size - but moving radially in three-dimensional space. Call this mechanism a type of kinetic form of SDIH.
Now, how would such a perceptual system respond to the stimulus of the Moon Illusion - a stimulus that changes only in the elevation of the Moon over the ground? The perceived distance of the moon would be determined by contextual stimulus information from the ground and the horizon - when the moon is low, it would appear close (that is, at or near the apparent distance of the horizon) and when the moon is high in the sky, it would appear to be at a far greater distance.
The perceived size of the moon would be determined by the kinetic SDIH, which produces the perception of rigid objects moving radially when stimulus size changes continuously. Accordingly, when the object appears to be at different distances, the perceived size of the object must change. This makes the moon that appears closer (i.e., the moon at the horizon) also appear larger than the moon that appears farther away.
Understanding the perceptual system in evolutionary terms makes it possible to uncover the processes that determine our awareness of objects moving in space. Presented with an anomalous stimulus such as the moon, this system has graced humanity with a sublime illusion about both the object's size and its distance.