Mathematician Discovers Most Efficient Way to Fold Paper into a Doughnut Shape

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A mathematical discovery regarding the most efficient way to fold a flat sheet of paper into a torus, or doughnut-like shape, offers new potential for the development of minimally invasive medical devices. According to reporting from Science News on May 29, 2026, a mathematician has identified a method to achieve this complex geometry using the fewest number of folds possible.

While the discovery originates in the field of geometry, the ability to reduce the number of folds required to create a three-dimensional structure from a two-dimensional plane has direct implications for biomedical engineering. In healthcare, the capacity to compress complex shapes into small volumes is a fundamental requirement for devices that must be delivered through the human body via catheters or small incisions.

Applications in Minimally Invasive Surgery

The primary health application of efficient folding patterns lies in the design of deployable implants. Many modern medical interventions rely on stents, heart valves, and vascular grafts that are folded or compressed for insertion and then expanded once they reach the target site in the artery or organ.

Reducing the number of folds in these structures is critical for several reasons. Every crease in a material represents a point of concentrated stress, which can lead to material fatigue or structural failure during the deployment process. By minimizing the number of folds, engineers can create devices that are more durable and less likely to fracture when they transition from a compressed state to their final functional shape.

The torus shape specifically is relevant for certain types of specialized implants and soft robotic tools used in surgery. A torus-shaped device can provide uniform radial support or act as a seal within a biological lumen, and the ability to fold such a shape efficiently allows for a smaller delivery profile, which typically reduces patient trauma and shortens recovery times.

The Role of Origami Engineering in Biotechnology

This mathematical breakthrough contributes to a broader field known as origami engineering, which applies the principles of paper folding to synthetic materials and biological molecules. This intersection of mathematics and medicine is already being explored in the development of targeted drug delivery systems.

In molecular biology, researchers use a technique called DNA origami to fold genetic material into specific shapes that can encapsulate therapeutic payloads. These molecular containers are designed to remain closed during transport through the bloodstream and open only when they encounter a specific chemical trigger at a disease site, such as a tumor.

The efficiency of the fold determines how stably a container can be closed and how reliably it can be opened. Mathematical models that minimize folds help scientists design more stable molecular structures that protect medication from premature degradation by the immune system.

Material Stress and Device Longevity

The focus on the fewest folds possible addresses a significant challenge in materials science: the degradation of structural integrity. In medical-grade polymers and nitinol—a shape-memory alloy commonly used in stents—repeated folding or extreme compression can create micro-cracks.

When a device is deployed in a high-pressure environment, such as a coronary artery, any structural weakness caused by excessive folding can lead to a collapse of the device or the triggering of an inflammatory response from the surrounding tissue. An optimized folding pattern ensures that the material is distributed more evenly, reducing the risk of mechanical failure.

the mathematical precision of these folds allows for more predictable deployment. In surgical settings, the predictability of how a device unfolds is essential for ensuring that the implant sits correctly against the vessel wall, preventing complications such as migration or thrombosis.

Future Directions in Deployable Health Tech

As the mathematical understanding of torus folding evolves, researchers are expected to explore the use of biodegradable materials that follow these optimized patterns. This would allow for the creation of temporary scaffolds that support healing tissue and then dissolve, leaving no permanent foreign object in the body.

The integration of these folding techniques with 3D printing and smart materials may also lead to the development of soft robots capable of navigating the complex pathways of the human anatomy to perform biopsies or deliver localized treatments with higher precision.

While the current discovery is a theoretical mathematical achievement, its transition into clinical application will require further testing in biomaterials to ensure that the efficiency of the fold translates to safety and efficacy within the human biological environment.