From Folds and Cuts to Linkages

Over the past decade, Professor L. Mahadevan’s Soft Math Lab at the Harvard John A. Paulson School of Engineering and Applied Sciences (SEAS) has helped establish how the ancient Japanese paper arts of folding or cutting can be used to inverse design   structures that transform dramatically in shape and function. 

Now, the researchers have created a new class of shape-changing matter, based not on folds or cuts, but linkages — networks of interconnected scissor mechanisms that collapse into lines and deploy into curved surfaces. 

A study published in Proceedings of the National Academy of Sciences, led by physics graduate student Noah Toyonaga, establishes a mathematical and physical framework for what the authors call collapsible scissored surfaces — deployable lattices of two-bar linkages that can transform from a one-dimensional collapsed state into two-dimensional structures with prescribed geometry.

“Origami showed how folds can encode shape,” said senior author Mahadevan, the Lola England de Valpine Professor of Applied Mathematics, of Organismic and Evolutionary Biology, and of Physics. “Kirigami showed how cuts can unlock motion and functionality. This work asks a complementary question: What can be achieved when the basic building block is not a fold or a cut, but a linkage?" 

These approaches could be used to design deployable aerospace structures, adaptive architecture, robotic systems, medical devices, and programmable materials whose shape can be encoded directly into their geometry.

“Each begins with simple discrete elements and ends with complex, programmable behavior,” Toyonaga  said. “Together they suggest that the future of mechanical design may lie not in the materials alone, but in the geometry of how those materials are assembled.”

A unified framework

The new work uses the scissor mechanism, the elementary linkage found in both everyday objects such as scissors or expandable gates as well as more exotic uses in deployable space structures. While individual scissor chains have been studied for centuries, extending them into fully collapsible surfaces has remained unexplored.

The researchers show that these linkage-based architectures occupy a distinct region of the design landscape. If origami is the geometry of folds and kirigami is the geometry of cuts, pantograph lattices — what the researchers call their linked structures — are the geometry of articulated connections.

This completes a trilogy for the geometric design of mechanical metamaterials: Origami, or shape through folds; kirigami, or shape through cuts; and pantograph lattices— shape through linkages, allowing for the emergence of global behaviors from local geometric rules.

Growing surfaces one linkage at a time

The central advance of the work is an analytical design algorithm that allows complex scissored surfaces to be built incrementally. Rather than solving a large optimization problem, the researchers discovered that deployable scissor lattices can be generated through a local additive construction process. Starting from a boundary, new linkages are added one at a time while preserving compatibility, deployability, and collapsibility.

This approach reveals that the geometry of an entire deployable surface can be encoded in a small set of design parameters. The resulting structures can collapse into a compact one-dimensional bundle and then deploy smoothly to a target shape. 

“These lattices are a nice example of how global form can be understood through purely local rules,” Toyonaga said. “That is, we can design the shape of an entire structure from a sequence of simple geometric decisions.”

The team validated the theory through computational design and physical prototypes fabricated using multimaterial 3D printing, in collaboration with Colter Decker from the Robert Wood group in SEAS. The resulting structures deploy reliably into a wide range of shapes including helical, toroidal, and doubly curved “eggbox” geometries.

Beyond the specific structures demonstrated, the research advances a vision of geometry serving as a universal language for designing matter.

The work as also coauthored with Seri Nishimoto and Tomohiro Tachi from the University of Tokyo. The research was supported by the Harvard NSF MRSEC 20-11754, the Simons Foundation, the Henri Seydoux Foundation, and the NSF Graduate Research Fellowship under Grant Nos. DGE2140743, JSPS 24KJ0637, JST FOREST JPMJFR232T, and JST AdCORP JPMJKB2302.