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M.C. Escher’s creations serve as an entrance into a realm of depth-defying visual deceptions, showcasing “impossible entities” that defy the principles of physics with intricate geometries. Your interpretation of his artworks is contingent on your vantage point — for instance, an individual appearing to ascend may actually be descending if you tilt your head horizontally.
Computer graphics researchers and designers can replicate these illusions in three dimensions, but only by distorting or slicing a real object and angling it in a specific manner. However, this workaround comes with drawbacks: Modifying the smoothness or illumination of the form will reveal that it isn’t genuinely an optical illusion, which implies you cannot accurately resolve geometric problems on it.
Scholars at MIT’s Computer Science and Artificial Intelligence Laboratory (CSAIL) have crafted an innovative method to portray “impossible” objects in a more adaptable manner. Their “Meschers” tool transforms images and 3D models into 2.5-dimensional representations, producing Escher-like images of entities such as windows, buildings, and even donuts. This approach allows users to adjust lighting, smoothen surfaces, and analyze unique geometries while maintaining their optical illusion.
This instrument could help geometric researchers calculate the distance between two points on a curved impossible surface (“geodesics”) and simulate how heat disperses across it (“heat diffusion”). It may also assist artists and computer graphics experts in developing physics-defying designs in several dimensions.
Lead author and MIT PhD candidate Ana Dodik intends to establish computer graphic resources that aren’t confined to mimicking reality, empowering creators to convey their vision irrespective of whether a shape can be materialized in the tangible world. “Utilizing Meschers, we’ve unveiled a fresh category of shapes for creators to explore on the computer,” she states. “They could also aid perception scientists in discerning the juncture at which an object genuinely becomes impossible.”
Dodik and her team will share their research at the SIGGRAPH conference this August.
Transforming impossible objects into possibilities
Impossible objects can’t be completely replicated in three dimensions. Their individual components often appear credible, yet these sections fail to bond correctly when assembled in 3D. Nonetheless, what can be computationally simulated, as the CSAIL researchers discovered, is the manner in which we perceive these forms.
Consider the Penrose Triangle, for example. The object in its entirety is physically unattainable because the depths do not “add up,” yet we can identify real-world 3D forms (like its three L-shaped edges) within it. These smaller segments can be accurately created in three dimensions — a characteristic termed “local consistency” — but when we attempt to assemble them, they do not yield a globally coherent shape.
The Meschers method models locally consistent areas without necessitating global coherence, assembling an Escher-style structure. Behind the curtain, Meschers interprets impossible objects as if we know their x and y positions in the image, alongside variations in z coordinates (depth) among adjacent pixels; the tool employs these depth variations to conjecture about impossible objects in an indirect manner.
The myriad applications of Meschers
Aside from rendering impossible objects, Meschers can subdivide their forms into smaller fragments for more exact geometric calculations and smoothing processes. This methodology allowed the researchers to minimize visual flaws of impossible shapes, such as a red heart silhouette they refined.
The team also examined their tool on an “impossibagel,” where a bagel is shadowed in a physically unattainable manner. Meschers enabled Dodik and her colleagues to simulate heat diffusion and compute geodesic distances between various points of the model.
“Envision you’re an ant navigating this bagel, and you wish to determine how long it’ll take to traverse, for instance,” Dodik explains. “Similarly, our tool could assist mathematicians in analyzing the underlying geometry of impossible shapes closely, much like we scrutinize real-world ones.”
Much like an illusionist, the tool can generate optical illusions from otherwise functional objects, simplifying the task for computer graphics artists to design impossible structures. It can also employ “inverse rendering” techniques to convert sketches and images of impossible objects into high-dimensional models.
“Meschers illustrates how computer graphics tools need not be confined by the parameters of physical reality,” asserts senior author Justin Solomon, associate professor of electrical engineering and computer science and head of the CSAIL Geometric Data Processing Group. “Remarkably, artists utilizing Meschers can conceptualize shapes that we will never encounter in the physical realm.”
Meschers can further assist computer graphics artists in adjusting the shading of their creations while maintaining an optical illusion. This adaptability would enable creatives to modify the illumination of their artwork to portray a broader range of scenes (like dawn or dusk) — as Meschers demonstrated by relighting a model of a dog on a skateboard.
Despite its adaptability, Meschers is merely the beginning for Dodik and her collaborators. The group is contemplating designing an interface to simplify the tool’s use while constructing more intricate scenarios. They are also collaborating with perception scientists to explore how the computer graphics tool might be employed more extensively.
Dodik and Solomon co-authored the paper with CSAIL associates Isabella Yu ’24, SM ’25; PhD candidate Kartik Chandra SM ’23; MIT professors Jonathan Ragan-Kelley and Joshua Tenenbaum; and MIT Assistant Professor Vincent Sitzmann.
Their research was partially funded by the MIT Presidential Fellowship, the Mathworks Fellowship, the Hertz Foundation, the U.S. National Science Foundation, the Schmidt Sciences AI2050 Fellowship, MIT Quest for Intelligence, the U.S. Army Research Office, U.S. Air Force Office of Scientific Research, the SystemsThatLearn@CSAIL initiative, Google, the MIT–IBM Watson AI Laboratory, the Toyota–CSAIL Joint Research Center, Adobe Systems, the Singapore Defence Science and Technology Agency, and the U.S. Intelligence Advanced Research Projects Activity.
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