Many materials are built from a grid of flexible but nearly inextensible rods that behaves as a shell-like structure. Everyday examples range from fabrics made of 1000s of interwoven yarns; to kitchen strainers made of 100s of plastically deforming wires; to architectural gridshells or medical stents made of 10s of elastically deforming rods. In this talk, I emphasize the geometric constraints common to these different physical systems. We build from a differential geometric model for woven fabric, initially introduced by Pafnuty Chebyshev in 1878, that directly encodes the inextensibility of the two families of rods.

We discuss the theory of Chebyshev nets through a series of applied, collaborative efforts in computational fabrication and inverse design. Theoretical obstructions expose the challenges in finding Chebyshev nets on surfaces with large amounts of curvature, suggesting a limited shape space. However, we show that a careful reformulation of the problem, combined with a discrete analog of Chebyshev nets, leads to computational tools that reveal a vibrant design space.

This talk is an in-depth example of my research goal to provide a foundation for material-aware computational design tools by harnessing differential geometry and actively developing its discrete analogs.