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Mc Escher’s work of art is a portal to a world with deep optic illusions, featuring “impossible objects” that destroy the laws of physics with complex geometry. You think his illustration depends on your point of view – for example, if you tilt your head sideways, it seems that someone walking upstairs might go towards the steps.
Computer graphics scientists and designers can recreate these fantasies in 3D, but can only bend or cut real shapes and position them at specific angles. However, this solution has a negative effect: changing the stationarity of the structure or lighting will indicate that it is not actually an optical illusion, which also means you cannot accurately solve the geometric problem above.
Researchers at the MIT Computer Science and Artificial Intelligence Laboratory (CSAIL) have developed a unique approach to representing “impossible” objects in a more general way. Their “Meschers” tool converts images and 3D models into 2.5-dimensional structures, resulting in an Escher-like description like Windows, buildings and even donuts. This method can help users re-preserve and study unique geometry while retaining their illusion.
The tool can help geometry researchers calculate the distance between two points on a curved impossible surface (“gear measurement”) and simulate how heat dissipates over it (“heat diffusion”). It can also help artists and computer graphics scientists create physically damaging designs in multiple dimensions.
Lead author and MIT PhD student ANA DODIK aims to design computer graphics tools that are not limited to copying reality, allowing artists to express their intentions independently of the shapes in the physical world. “Using Meschers, we unlock a new class of shapes for artists to use on computers,” she said. “They can also help perception scientists understand where objects are truly impossible.”
Dodik and her colleagues will present their papers at the Siggraph conference in August.
Make impossible objects possible
Impossible objects cannot be completely copied in 3D. Their components usually look reasonable, but when assembled in 3D, these parts won’t glue together correctly. But, as Csail researchers have found, what can be computed to mimic the process of how we view these shapes.
Take the Penrose triangle as an example. The entire object is physically impossible because depth does not “accumulate”, but we can identify 3D shapes in the real world (such as its three L-shaped angles). These smaller areas can be implemented in 3D (a property called “local consistency”), but when we try to assemble them together, they don’t form a globally consistent shape.
Meschers approach the locally consistent regions of the model without forcing them to be consistent globally, pieced together the Escher-style structure. Behind the scenes, Meschers represent impossible objects, as if we know their X and Y coordinates in the image, and the differences in Z coordinates (depth) between adjacent pixels; the tool uses these differences in depth to indirectly reason about impossible objects.
Many uses of Mechez
In addition to rendering impossible objects, the cheque can subdivide its structure into smaller shapes for more precise geometric calculations and smoothing operations. This process allows researchers to reduce visual defects in impossible shapes, such as their sparse red heart profile.
The researchers also tested their tools on “The Impossible Book”, where bagels are obscured in physically impossible ways. Meschers helped Dodik and her colleagues simulate heat diffusion and calculated geodetic distances between different points in the model.
“Imagine you’re an ant traveling through this bagel, for example, you want to know how long it will take you to go through it,” Dodik said. “In the same way, our tools can help mathematicians analyze basic geometric shapes of impossible shapes, just like how we study real-world geometry.”
Just like a magician, the tool creates illusions with otherwise practical objects, making it easier for computer graphics artists to create impossible objects. It can also use the Inverse Rendering tool to convert images of drawings and impossible objects into high-dimensional designs.
“Meschers show how computer graphics tools don’t have to be constrained by the rules of physical reality,” said senior writer Justin Solomon. “It’s incredible that artists using Meschers can reason about shapes that we will never find in the real world.”
Meschers can also help computer graphics artists by adjusting the shadows of their creations while still retaining the optical illusion. This versatility will allow the creatives to change the lighting of their artwork to portray a wider range of scenes (such as sunrise or sunset) – as Meschers demonstrated by reconfirming the dog’s model on the skateboard.
Despite its versatility, Meschers is just the beginning of Dodik and his colleagues. The team is considering designing an interface that makes the tool easy to use when building more carefully designed scenarios. They also work with perceptual scientists to understand how computer graphics tools can be used more widely.
Dodik and Solomon wrote the paper with Csail affiliate Isabella Yu ’24, SM ’25; PhD student Kartik Chandra SM ’23; MIT professors Jonathan Ragan-Kelley and Joshua Tenenbaum; and MIT assistant professor Vincent Sitzmann.
MIT Presidential Scholarship, Mathematics Scholarship, Hertz Foundation, National Science Foundation, Schmidt Science AI2050 Scholarship, MIT to Intelligence Agency, U.S. Army Research Office, U.S. Air Force Office of Scientific Research, Systems thatatelen@csail interative to toyot to toyot – watsibm witto, Adobe Systems, Singapore Defense Science and Technology Agency, and U.S. Intelligence Advanced Research Projects Activities.