Noah Toyonaga was sketching with scissor mechanisms on a whiteboard in Cambridge when he realized the lines weren’t just moving—they were encoding shape. At Harvard’s Soft Math Lab, led by Professor L. Mahadevan, a decade of work decoding the geometry of origami and kirigami had already transformed how engineers think about shape-shifting materials. Now, Toyonaga and his team have completed the trilogy with a third principle: collapsible scissored surfaces—lattices of interconnected two-bar linkages that collapse into a line and deploy into complex curved forms. Published in the Proceedings of the National Academy of Sciences, this breakthrough introduces a new class of mechanical metamaterials rooted not in folds or cuts, but in articulated connections, opening doors to adaptive architecture, deployable space habitats, and medical devices that transform on demand.
The significance lies in control and simplicity. While origami uses folds and kirigami uses cuts to sculpt 3D forms from flat sheets, this new approach leverages the pantograph lattice—a network inspired by the humble scissor mechanism found in expandable gates and drafting tools. For centuries, engineers have used scissor chains in linear applications, but extending them into fully collapsible 2D surfaces remained unexplored—until now. The team’s mathematical framework shows that these linkage-based structures occupy a unique space in mechanical design, where global shape emerges from local rules. That means instead of solving complex optimization problems, designers can build deployable surfaces one linkage at a time, guided by a small set of geometric parameters.
The core innovation is an analytical design algorithm that allows structures to grow incrementally while preserving deployability and collapsibility. Starting from a boundary, each new linkage is added in a way that maintains compatibility with the whole system. This local construction process enables the creation of surfaces that can transform from a tightly packed, one-dimensional bundle into prescribed 2D shapes—helices, toroids, even doubly curved "eggbox" geometries. To validate their theory, the team collaborated with Colter Decker from the Robert Wood group at Harvard’s John A. Paulson School of Engineering and Applied Sciences, using multimaterial 3D printing to fabricate physical prototypes that reliably deploy into target forms.
"Each begins with simple discrete elements and ends with complex, programmable behavior," Toyonaga said. This philosophy points to a future where materials aren’t just strong or lightweight, but intelligent in their geometry. The work suggests that shape can be encoded directly into a material’s architecture, making it programmable without electronics or external controls. As Mahadevan put it, if origami is the geometry of folds and kirigami the geometry of cuts, then pantograph lattices are the geometry of connections—completing a unified framework for mechanical metamaterials. With this trilogy now complete, the path forward is clear: the next generation of adaptive structures won’t just be built—they’ll be geometrically designed.