PARADOXES—in which the same information may lead to two contradictory conclusions—give us pleasure and torment at the same time. They are a source of endless fascination and frustration, whether they involve philosophy (consider Russell's paradox, “This statement is false”), science—or perception. The Nobel Prize winner Peter Medawar once said that such puzzles have the same effect on a scientist or philosopher as the smell of burning rubber on an engineer: they create an irresistible urge to find the cause. As neuroscientists who study perception, we feel compelled to study the nature of visual paradoxes.

Let us take the simplest case. If different sources of information are not consistent with one another, what happens? Typically the brain will heed the one that is statistically more reliable and simply ignore the other source. For example, if you view the inside of a hollow mask from a distance, you will see the face as normal—that is, convex—even though your stereovision correctly signals that the mask is actually a hollow, concave face. In this case, your brain's cumulative experience with convex faces overrides and vetoes perception of the unusual occurrence of a hollow face.

Most tantalizing are the situations in which perception contradicts logic, leading to “impossible figures.” British painter and printmaker William Hogarth created perhaps the earliest such figure in the 18th century (a). A brief view of this image suggests nothing abnormal. Yet closer inspection reveals that it is logically impossible. Another example is the “devil's pitchfork,” or Schuster's conundrum (b). Such impossible figures raise profound questions about the relation between perception and rationality.

In modern times, interest in such effects was partly revived by Swedish artist Oscar Reutersvrd. Known as the father of impossible figures, he devised numerous geometric paradoxes, including the “endless staircase” and the “impossible triangle.” These two were also independently developed by Lionel and Roger Penrose, the famous father-and-son scientists, and c shows their version of what is now commonly called the Penrose triangle.

Dutch artist M. C. Escher playfully embedded such figures in his engravings exploring space and geometry. Consider Escher's staircase (d): no single part of the staircase is impossible or ambiguous, but the entire ensemble is logically impossible. You could be climbing the staircase upward for-ever and yet keep going in circles, never reaching the top. It epitomizes the human condition: we perpetually reach for perfection, never quite getting there!

Is this staircase truly a perceptual paradox? That is, is the brain unable to construct a coherent percept (or token of perception) because it has to simultaneously entertain two contradictory perceptions? We think not. Perception, almost by definition, has to be unified and stable at any given instant because its whole purpose is to lead to an appropriate goal-directed action on our part. Indeed, some philosophers have referred to perception as “conditional readiness to act,” which may seem like a bit of an overstatement.

Despite the common view that “we see what we believe,” the perceptual mechanisms are really on autopilot as they compute and signal various aspects of the visual environment. You cannot choose to see what you want to see. (If I show you a blue lion, you see it as blue. You cannot say, “I will choose to see it as gold because it ought to be.”) On the contrary, the paradox in d arises precisely because the perceptual mechanism performs a strictly local computation signaling “ascending stairs,” whereas your conceptual/intellectual mechanism deduces that it is impossible logically for such an ascending staircase to form a closed loop. The goal of perception is to compute rapidly the approximate answers that are good enough for immediate survival; you cannot ruminate over whether the lion is near or far. The goal of rational conception—of logic—is to take time to produce a more accurate appraisal.

Genuine or Not?

Are impossible figures (aside from the triangle, to which we will return) genuine paradoxes within the domain of perception itself? One could argue that the perception itself remains, or appears to remain, internally consistent, coherent and stable and that a genuinely paradoxical percept is an oxymoron. The staircase is no more a paradox than our seeing a visual illusion such as the Mueller-Lyer (e)—in which two lines of equal length appear to differ—but then measuring the two lines with a ruler and convincing ourselves at an intellectual level that the two lines are of identical length. The clash is between perception and intellect, not a genuine paradox within perception itself. On the other hand, “This statement is false” is a paradox entirely in the conceptual/linguistic realm.

Another compelling perception is the motion aftereffect. If you stare for a minute at stripes moving in one direction and then transfer your gaze to a stationary object, the object appears to move in the opposite direction that the stripes moved. This effect arises because your visual system has motion-detecting neurons signaling different directions, and the stripes constantly moving in one direction “fatigue” the neurons that would normally signal that direction [see “Stability of the Visual World,” by Vilayanur S. Ramachandran and Diane Rogers-Ramachandran; Scientific American Mind, February/March 2006]. The result is a “rebound” that makes even stationary objects appear to move in the opposite direction.

Yet curiously, when you look at the object it seems to be moving in one direction, but it does not seem to get anywhere; it does not progress to a goal. This effect is often touted as a perceptual paradox: How can something appear to move but not change location? But once again, the percept itself is not paradoxical; rather it is signaling with certainty that the object is moving. It is your intellect that deduces it is not moving and infers a paradox.

Consider the much more familiar converse situation. You know (deduce) that the hour hand of your clock is moving, even though it looks stationary. It is not moving fast enough to excite motion-detecting neurons. Yet no one would call a clock hand's movement a paradox.

Perception-Cognition Boundary

There are borderline cases, as exemplified by the devil's pitchfork. In this display, some people can “see” the whole in a single glance. The local and global perceptual cues themselves are perceived as a single gestalt with internal contradictions. That is, one can apprehend the whole in a single glance and appreciate its paradoxical nature without thinking about it. Such displays remind us that despite the modular quasi-autonomous nature of perception and its apparent immunity from the intellect, the boundary between perception and cognition can blur.

The impossible triangle is similar. As shown by Richard L. Gregory, emeritus professor of neuropsychology at the University of Bristol in England, you can construct a complicated 3-D object (f) that would produce the image in g only when viewed from one particular vantage point. From that specific angle, the object appears to be a triangle confined to a single plane. But your perception rejects such highly improbable events, even when your intellect is convinced of their possibility (after being shown the view at g). Thus, even when you understand conceptually the unusual shape of object f, you continue to see a closed triangle when viewing g, rather than the object (f) that actually gives rise to it.

How would one test these notions empirically? With the Escher staircase, one could exploit the fact that perception is virtually instantaneous, whereas cogitation takes time. One could present the display briefly—a short enough time to prevent cognition from kicking in—say, a tenth of a second followed by a masking stimulus (which prevents continued visual processing after removal of the test figure). The prediction would be that the picture should no longer look paradoxical unless the stimulus duration were lengthened adequately. The same could be tried for the devil's pitchfork, which is more likely to be a genuine perceptual paradox. In this case, the mask may not be able to “dissect” it into two distinct (perception or cognition) stages. It may boil down to a matter of scale or complexity.

Whatever paradoxes' origins, no one can fail to be intrigued by these enigmatic displays. They perpetually titillate our senses and challenge all our notions of reality and illusion. Human life, it would seem, is delightfully bedeviled by paradox.