Smooth triaxial weaving with naturally curved ribbons

Triaxial weaving is a handcrafting technique to create shell-like structures using initially straight and flat ribbons in tri-directional arrays. To produce nonzero Gaussian curvature, traditional weavers intentionally introduce discrete topological defects, leading to faceted surfaces in the overall structure. Here, we demonstrate smooth gradation of curvature in the woven structures by prescribing in-plane curvatures to the ribbons. We first investigate a representative unit in triaxial weaves consisting of curved ribbons using rapid prototyping, X-ray micro-computed tomography, and finite element analysis. By controlling the in-plane curvature of the ribbons in the representative unit, we demonstrate that a continuous range of integrated Gaussian curvatures can be achieved, which is not feasible using straight ribbons. We further reveal that the shape of the representative units is dictated by the geometry of ribbons, not elasticity. Finally, we design a set of ribbon profiles to achieve various canonical shapes - sphere, ellipsoid, torus - by leveraging the geometry-driven nature of triaxial weaving. Our strategy can be applied to expand the design space of general discrete combinatorial structures for achieving smooth shapes.