Background: Mental Rotation

Shepard, who studied physics at Stanford University, earned his Ph.D. in psychology from Yale in 1955. He had a keen interest in human perception — the ways we see, hear, and interpret the world around us. During the 1950s, psychology largely focused on behaviorism, the study of observable actions, as demonstrated by Ivan Pavlov and B.F. Skinner.

In contrast, Shepard was captivated by the inner workings of the mind. He joined Bell Labs in 1958, where he began to investigate the notion that perception is an active process rather than a passive one. He recognized that the information we take in is transformed and interpreted based on our prior experiences. One of his notable contributions to cognitive science was his research on mental rotation.

In 1968, Shepard became a professor at Stanford, where he designed experiments aimed at measuring our ability to mentally rotate objects. Inspired by a dream involving rotating three-dimensional forms, he published a significant paper in Science in 1971 titled “Mental Rotation of Three-Dimensional Objects,” co-authored with Jacqueline Metzler. This study assessed the time it took participants to mentally evaluate the equivalence of pairs of cuboid shapes displayed on a plane.

The research established a direct correlation between the perceived rotation of an object (in relation to a baseline position) and the time required for subjects to determine whether two objects were identical. This was an early stride toward quantifying a fundamentally invisible psychological process. Interestingly, participants rotated the objects at an average speed of about 60 degrees per second.

The Illusion

By the time he published Mental Images and Their Transformation (1982) and later Mind Sights (1990), Shepard had accumulated extensive expertise regarding how the mind processes representations of three-dimensional objects. One of the most recognized illusions from Mind Sights is his Figure A2, titled “Turning the Tables.” Below is a recreation of that image:

The illustration showcases two tables viewed from a perspective. The table on the left appears longer and narrower than the one on the right, although the truth is that the quadrilaterals forming the tabletops are identical.

“How can this be?” your mind questions as you shift your gaze between the tables, attempting to reconcile this baffling information. As Shepard elaborates in his book, this is not merely a visual trick; the illusion reveals significant insights into “fundamental perceptual principles” that have emerged through natural selection over millennia.

(Note: The tabletops are frequently described as parallelograms, but they are not. My analysis using ImageJ confirms that, at best, they are trapezoids. Shepard himself simply refers to them as “quadrilaterals.”)

How This Could Be

Throughout much of human history, our survival has hinged on our ability to accurately reconstruct the three-dimensional environment from the two-dimensional images projected onto our retinas. This interpretative mechanism has evolved to function autonomously and largely outside our conscious control, regardless of whether the input is from the natural world or a flat image.

In contrast, the human skill of translating a three-dimensional object into a two-dimensional drawing seems to be a relatively recent development. The oldest known example is a 45,500-year-old cave painting of a pig found in Indonesia:

World's oldest known figurative artwork found in Indonesian cave

Archaeologists have uncovered what they assert to be the oldest instance of figurative art created by humans. An ochre…

It took another 44,000 years before the concept of perspective in painting was introduced, with one of the earliest examples being Masaccio’s The Tribute Money (c. 1426–27):

I believe that since the Renaissance, the emergence of perspective in art, followed by the proliferation of computer graphics and virtual reality, has reinforced our innate tendency for automatic image processing, conditioning us to perceive that in certain instances, 2D can be indistinguishable from 3D. For images that adhere to the principles of linear perspective, the cues of depth and lighting become overwhelmingly persuasive.

“Turning the Tables” utilizes depth perception along with our capacity to comprehend three-dimensional rotation. When observing a square rotating away from us, it can be interpreted as such; however, it can also easily be perceived as a melting shape with varying side lengths or a distorted quadrilateral tilting into the page:

Adding perspective, context, and lighting makes it challenging not to interpret the square as maintaining its dimensions while pivoting backward. Intriguingly, we recognize this version as a square, even though none of the projections are genuinely rectangular:

Similarly, when observing a “tabletop” positioned in the x-y plane and rotating around the z-axis within a bounding box, we instinctively perceive that its dimensions remain constant, although their projected lengths on the screen are continuously fluctuating:

In particular, Shepard’s illusion capitalizes on a visual cue known as retinal size — the actual size of an object's image on the retina. Depth cues inform us that the long axis of the left table extends back and away from us, while the long axis of the right table appears to be almost perpendicular to our line of sight.

Nevertheless, the retinal size of both tabletops is the same. If the left table extends backward, its retinal size is perceived as foreshortened, leading our brains to conclude that its “actual” length must be greater than that of the right table. Conversely, the width of the right table is assumed to be larger than that of its counterpart.

What renders this illusion particularly compelling is that, in contrast to many striking visual tricks (such as Akiyoshi Kitaoka’s “Out of Focus”), it is easy to grasp precisely what is happening here — yet this understanding does not alter our fundamental belief in the accuracy of our perception.

Shepard’s Legacy

Roger Shepard was not just a cognitive scientist; he was an artist, musician, photographer, and poet — a truly creative individual. His inspirations extended beyond Newton and Einstein to include figures like Escher and Magritte. His creativity infused his research across various domains, from multi-dimensional scaling to musical cognition and a universal law of generalization explaining how animals can generalize from familiar situations to new ones.

In 1995, Shepard received the esteemed National Medal of Science. The citation for his award states:

> “For his theoretical and experimental work elucidating the human mind’s perception of the physical world and why the human mind has evolved to represent objects as it does; and for giving purpose to the field of cognitive science and demonstrating the value of integrating insights from multiple scientific disciplines in addressing scientific challenges.”

From his early school days, Shepard often sketched the dreams he experienced, occasionally facing reprimands for doodling in class. Some of these sketches later served as illustrations in Mind Sights. Throughout his life, he navigated effortlessly between art and science, dreams and reality, seeking to comprehend the intricacies of our experiences. We are all enriched by his delightful explorations of the mind.

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Further Resources

• Book by Roger Shepard: Mind Sights: Original Visual Illusions, Ambiguities, and Other Anomalies, With a Commentary on the Play of Mind in Perception and Art
• Research paper: Paradoxical perception of surfaces in the Shepard tabletop illusion
• An engaging YouTube illustration of the illusion

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