What can theoretical physics teach us about knitting?
Penn physicist Randall Kamien, visiting scholar Lauren Niu, and collaborator Geneviève Dion of Drexel bring unprecedented levels of predictability to the ancient practice of knitting by developing a mathematical model that could be used to create a new c
University of Pennsylvania
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A close-up of a self-folding knitted fabric, demonstrating how specific stitch patterns—knits and purls—encode geometric rules that dictate the material’s three-dimensional shape. Researchers from Penn and Drexel have developed a mathematical model that predicts these folding behaviors, opening new possibilities for programmable textiles in fields ranging from soft robotics to deployable structures.
view moreCredit: (Image: Courtesy of Lauren Niu)
The practice of purposely looping thread to create intricate knit garments and blankets has existed for millennia. Though its precise origins have been lost to history, artifacts like a pair of wool socks from ancient Egypt suggest it dates back as early as the 3rd to 5th century CE. Yet, for all its long-standing ubiquity, the physics behind knitting remains surprisingly elusive.
“Knitting is one of those weird, seemingly simple but deceptively complex things we take for granted,” says theoretical physicist and visiting scholar at the University of Pennsylvania, Lauren Niu, who recently took up the craft as a means to study how “geometry influences the mechanical properties and behavior of materials.”
Despite centuries of accumulated knowledge, predicting how a particular knit pattern will behave remains difficult—even with modern digital tools and automated knitting machines. “It’s been around for so long, but we don’t really know how it works,” Niu notes. “We rely on intuition and trial and error, but translating that into precise, predictive science is a challenge.”
These experimental knitted structures showcase how stitch patterns can be engineered to create self-folding and shape-morphing textiles. On the left, a variety of stitch motifs demonstrate different programmed curvatures in a scarf-like garment, while the right displays a take on an eerie, mask-like appearance, showcasing how controlled stitch arrangements can create complex three-dimensional forms. (Image: Courtesy of Lauren Niu)
In a paper published in the Proceedings of the National Academy of Science, Niu and her mentors Randall Kamien of Penn’s School of Arts & Sciences and Geneviève Dion of the Center for Functional Fabrics at Drexel University have presented a model that seeks to decode the ancient practice of knitting by ascribing a mathematical language to the stitches in knits and purls.
“The beautiful thing about Lauren’s approach is that it doesn’t just describe what’s happening—it predicts it,” says Kamien. “We’re taking the tools we use to study everything from gravity to soap bubbles and applying them to knitting. And, remarkably, it works.”
Dion, drawing from her experience as a textile researcher and garment designer, views this work as a crucial convergence of scientific theory and practical design applications. “Right now, fabric design relies on experience, experimentation, and intuition,” she says. “If we can apply predictive models to textiles, we open the door to fabrics with precise, engineered properties—whether it’s self-folding medical materials, reconfigurable structures for soft robotics, or garments that adapt to the body in new ways.”
Decoding knitting, one stitch at a time
To build their model, Niu borrowed mathematical techniques from an unexpected source: general relativity, the theory used to describe the warping of space and time. While relativity explains how gravity bends space-time, the researchers applied similar geometric principles to explain how the looping paths of yarn create curvature in knitted fabrics.
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“It started with the simple observation that knit fabrics curl in specific ways,” Niu says. “Think of when you cut the sleeves off a T-shirt, and it curls up; that usually means it’s made with only knit stitches. Purls just curl in the other direction.”
However, when knits and purls are combined, “that’s when the magic happens,” Dion says. “You get these incredible self-folding structures that can be soft and flexible but also structured and resilient.”
Niu explains that knitting, at its core, is a method of transforming a one-dimensional strand of yarn into a structured, flexible two-dimensional sheet, which can then fold itself into complex three-dimensional shapes. The researchers realized that this transformation could be described mathematically using the same principles that govern how surfaces curve in space.
Instead of seeing a knitted fabric as just a collection of interlocking loops, the team treated it as a continuous surface with an intrinsic curvature determined by the arrangement of stitches. By applying the formalism used to describe how materials bend and stretch—known as elasticity theory—they built an energetic model that simulates the forces acting on the loops of yarn and predicts how a piece of fabric will deform in space.
“The key insight was recognizing that knitting operates like a programmable material,” Kamien says. “By controlling the stitch pattern—just knits and purls—you can essentially encode instructions for how the fabric will behave once it comes off the needles. That’s why a scarf, a sock, and a sweater can all come from the same kind of yarn but behave so differently.”
Their simulations revealed that the mechanical properties of knitted fabrics often depend more on stitch geometry than on the material itself. Whether the yarn was wool, cotton, or synthetic, the fabric’s tendency to curl, pleat, or expand followed universal geometric rules. This suggests that knitting is governed by fundamental mathematical principles—ones that could be harnessed to design materials with precise, tunable behaviors.
Where knitting meets origami
Niu explains that the work fits into the larger focus of Kamien’s group, particularly its research on kirigami, the art of cutting paper to create complex, foldable structures.
“Kirigami, much like knitting, is an example of how geometry can be used to encode mechanical properties into a material,” she says.
The team’s previous work has explored how strategically placed cuts in a sheet can cause it to morph into specific three-dimensional shapes when stretched. The insights from knitting take this idea further, showing that a material’s internal structure—not just its cuts—can dictate how it folds and unfolds.
“The parallels between knitting and kirigami are striking,” Kamien says. “In kirigami, you add cuts; in knitting, you add loops. But in both cases, you’re programming geometry directly into the material so that it shapes itself without requiring extra inputs like heat, hinges, or reinforcements.”
Dion coined a new term for this approach: knitogami™—a fusion of knitting and origami that captures the idea of self-folding textiles. “We call it knitogami because it extends the principles of origami into a soft, fabric-based medium,” she explains. “Instead of relying on folds and creases in paper, we’re using the inherent elasticity and structure of knitted loops to create dynamic, shape-shifting materials.”
By mapping out these rules, the team developed a framework that could be used to create programmable textiles—fabrics that shape themselves without requiring external forces like heat or manual pleating.
“If we can predict how a piece of fabric will shape itself just by changing the stitch pattern, we can start designing textiles with built-in functionality,” Dion says. “This could lead to garments that adapt to movement, medical textiles that mold to the body, or even large-scale deployable structures that assemble themselves.”
The next steps
Looking ahead, the team hopes to refine their model to incorporate even more complex stitch patterns and fabric behaviors.
“Right now, we’re focused on fundamental stitches—knits and purls—but the real world of knitting is much richer,” Niu says. “The goal is to take this mathematical approach and expand it to include cables, lace, and other advanced techniques that knitters have developed over centuries.”
Randall Kamien is the Vicki and William Abrams Professor in the Natural Sciences in the Department of Physics and Astronomy in the School of Arts & Sciences at the University of Pennsylvania.
Lauren Niu is senior research scientist at the Center for Functional Fabrics at Drexel University and a visiting postdoctoral researcher at Penn Arts & Sciences.
Geneviève Dion is a professor in the Department of Design at the Westphal College of Media Arts and Design and Founding Director of the Center for Functional Fabrics at Drexel University.
This research was supported by the Simons Foundation, the Kaufman Foundation, the Advanced Functional Fabrics of America, Inc. (Transaction HQ00342190016), the Air Force Research Laboratory (FA8650-20-2-5506), and NextFlex, as well as the U.S. Army DEVCOM–Soldier Center.
These experimental knitted structures showcase how stitch patterns can be engineered to create self-folding and shape-morphing textiles. On the left, a variety of stitch motifs demonstrate different programmed curvatures in a scarf-like garment, while the right displays a take on an eerie, mask-like appearance, showcasing how controlled stitch arrangements can create complex three-dimensional forms.
Credit
(Image: Courtesy of Lauren Niu)
A comparative analysis of stitch patterns, simulated fabric deformations, and corresponding physical samples. The left column shows the programmed knit-purl stitch arrangements, the center column presents computational simulations predicting the fabric’s three-dimensional folding behavior, and the right column displays real-world knitted samples demonstrating the accuracy of these predictions. (Image: Courtesy of Lauren Niu)
Credit
Lauren Niu and Randall Kamien
Journal
Proceedings of the National Academy of Sciences
Method of Research
Computational simulation/modeling
Subject of Research
Not applicable
Article Title
Geometric modeling of knitted fabrics
Fiber computer allows apparel to run apps and “understand” the wearer
MIT researchers developed a fiber computer and networked several of them into a garment that learns to identify physical activities.
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US Army Major Hefner training in Norway wearing Fiber computer base layer.
view moreCredit: US Army Cold Regions Research & Engineering Lab
What if the clothes you wear could care for your health?
MIT researchers have developed an autonomous programmable computer in the form of an elastic fiber, which could monitor health conditions and physical activity, alerting the wearer to potential health risks in real-time. Clothing containing the fiber computer was comfortable and machine washable, and the fibers were nearly imperceptible to the wearer, the researchers report.
Unlike on-body monitoring systems known as “wearables,” which are located at a single point like the chest, wrist, or finger, fabrics and apparel have an advantage of being in contact with large areas of the body close to vital organs. As such, they present a unique opportunity to measure and understand human physiology and health.
The fiber computer contains a series of microdevices, including sensors, a microcontroller, digital memory, bluetooth modules, optical communications, and a battery, making up all the necessary components of a computer in a single elastic fiber.
The researchers added four fiber computers to a top and a pair of leggings, with the fibers running along each limb. In their experiments, each independently programmable fiber computer operated a machine-learning model that was trained to autonomously recognize exercises performed by the wearer, resulting in an average accuracy of about 70 percent.
Surprisingly, once the researchers allowed the individual fiber computers to communicate among themselves, their collective accuracy increased to nearly 95 percent.
“Our bodies broadcast gigabytes of data through the skin every second in the form of heat, sound, biochemicals, electrical potentials, and light, all of which carry information about our activities, emotions, and health. Unfortunately, most if not all of it gets absorbed and then lost in the clothes we wear. Wouldn’t it be great if we could teach clothes to capture, analyze, store, and communicate this important information in the form of valuable health and activity insights?” says Yoel Fink, a professor of materials science and engineering at MIT, a principal investigator in the Research Laboratory of Electronics (RLE) and the Institute for Soldier Nanotechnologies (ISN), and senior author of a paper on the research, which appears today in Nature.
The use of the fiber computer to understand health conditions and help prevent injury will soon undergo a significant real-world test as well. U.S. Army and Navy service members will be conducting a month-long winter research mission to the Arctic, covering 1,000 kilometers in average temperatures of -40 degrees Fahrenheit. Dozens of base layer merino mesh shirts with fiber computers will be providing real-time information on the health and activity of the individuals participating on this mission, called Musk Ox II.
“In the not-too-distant future, fiber computers will allow us to run apps and get valuable health care and safety services from simple everyday apparel. We are excited to see glimpses of this future in the upcoming Arctic mission through our partners in the U.S. Army, Navy, and DARPA. Helping to keep our service members safe in the harshest environments is a honor and privilege,” Fink says.
He is joined on the paper by co-lead authors Nikhil Gupta, a materials science and engineering graduate student; Henry Cheung MEng ’23; and Syamantak Payra ’22, currently a graduate student at Stanford University; John Joannopoulos, the Francis Wright Professor of Physics and director of the Institute for Soldier Nanotechnologies; as well as others at MIT, Rhode Island School of Design, and Brown University.
Fiber focus
The fiber computer builds on more than a decade of work in the Fibers@MIT lab at the RLE and was supported primarily by ISN. In previous papers, the researchers demonstrated methods for incorporating semiconductor devices, optical diodes, memory units, elastic electrical contacts, and sensors into fibers that could be formed into fabrics and garments.
“But we hit a wall in terms of the complexity of the devices we could incorporate into the fiber because of how we were making it. We had to rethink the whole process. At the same time, we wanted to make it elastic and flexible so it would match the properties of traditional fabrics,” says Gupta.
One of the challenges that researchers surmounted is the geometric mismatch between a cylindrical fiber and a planar chip. Connecting wires to small, conductive areas, known as pads, on the outside of each planar microdevice proved to be difficult and prone to failure because complex microdevices have many pads, making it increasingly difficult to find room to attach each wire reliably.
In this new design, the researchers map the 2D pad alignment of each microdevice to a 3D layout using a flexible circuit board called an interposer, which they wrapped into a cylinder. They call this the “maki” design. Then, they attach four separate wires to the sides of the “maki” roll and connected all the components together.
“This advance was crucial for us in terms of being able to incorporate higher functionality computing elements, like the microcontroller and Bluetooth sensor, into the fiber,” says Gupta.
This versatile folding technique could be used with a variety of microelectronic devices, enabling them to incorporate additional functionality.
In addition, the researchers fabricated the new fiber computer using a type of thermoplastic elastomer that is several times more flexible than the thermoplastics they used previously. This material enabled them to form a machine-washable, elastic fiber that can stretch more than 60 percent without failure.
They fabricate the fiber computer using a thermal draw process that the Fibers@MIT group pioneered in the early 2000s. The process involves creating a macroscopic version of the fiber computer, called a preform, that contains each connected microdevice.
This preform is hung in a furnace, melted, and pulled down to form a fiber, which also contains embedded lithium-ion batteries so it can power itself.
“A former group member, Juliette Marion, figured out how to create elastic conductors, so even when you stretch the fiber, the conductors don’t break. We can maintain functionality while stretching it, which is crucial for processes like knitting, but also for clothes in general,” Gupta says.
Bring out the vote
Once the fiber computer is fabricated, the researchers use a braiding technique to cover the fiber with traditional yarns, such as polyester, merino wool, nylon, and even silk.
In addition to gathering data on the human body using sensors, each fiber computer incorporates LEDs and light sensors that enable multiple fibers in one garment to communicate, creating a textile network that can perform computation
Each fiber computer also includes a Bluetooth communication system to send data wirelessly to a device like a smartphone, which can be read by a user.
The researchers leveraged these communication systems to create a textile network by sewing four fiber computers into a garment, one in each sleeve. Each fiber ran an independent neural network that was trained to identify exercises like squats, planks, arm circles, and lunges.
“What we found is that the ability of a fiber computer to identify human activity was only about 70 percent accurate when located on a single limb, the arms or legs. However, when we allowed the fibers sitting on all four limbs to ‘vote,’ they collectively reached nearly 95 percent accuracy, demonstrating the importance of residing on multiple body areas and forming a network between autonomous fiber computers that does not need wires and interconnects,” Fink says.
Moving forward, the researchers want to use the interposer technique to incorporate additional microdevices.
Arctic insights
In February, a multinational team equipped with computing fabrics will travel for 30 days and 1,000 kilometers in the Arctic. The fabrics will help keep the team safe, and set the stage for future physiological “digital twinning” models.
“As a leader with more than a decade of Arctic operational experience, one of my main concerns is how to keep my team safe from debilitating cold weather injuries — a primary threat to operators in the extreme cold. Conventional systems just don’t provide me with a complete picture. We will be wearing the base layer computing fabrics on us 24/7 to help us better understand the body’s response to extreme cold and ultimately predict and prevent injury,” says U.S. Army Major Hefner, the commander of Musk Ox II.
Karl Friedl, U.S. Army senior research scientist of performance physiology, noted that the MIT programmable computing fabric technology may become a “gamechanger for everyday lives.”
“Imagine near-term fiber computers in fabrics and apparel that sense and respond to the environment and to the physiological status of the individual, increasing comfort and performance, providing real-time health monitoring and providing protection against external threats. Soldiers will be the early adopters and beneficiaries of this new technology, integrated with AI systems using predictive physiological models and mission-relevant tools to enhance survivability in austere environments,” Friedl says.
“The convergence of classical fibers and fabrics with computation and machine learning has only begun. We are exploring this exciting future not only through research and field testing, but importantly in an MIT Department of Materials Science and Engineering course ‘Computing Fabrics,’ taught with Professor Anais Missakian from the Rhode Island School of Design,” adds Fink.
This research was supported, in part, by the U.S. Army Research Office Institute for Soldier Nanotechnology (ISN), the U.S. Defense Threat Reduction Agency, the U.S. National Science Foundation, the Fannie and John Hertz Foundation Fellowship, the Paul and Daisy Soros Foundation Fellowship for New Americans, the Stanford-Knight Hennessy Scholars Program, and the Astronaut Scholarship Foundation.
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Written by Adam Zewe, MIT News
Knotted computer fiber.
Credit
Yoel Fink
Braiding a computer fiber with a combination of metal and textile yarns
Credit
Hamilton Osoy, IFM
Journal
Nature
Article Title
"A single-fibre computer enables textile networks and distributed inference"
