Weaving curved surfaces from flat ribbons

Basket weaving has always forced curvature in by inserting defects, leaving surfaces faceted. We show instead that giving each flat ribbon a prescribed in-plane curvature tunes a weave's Gaussian curvature continuously, a largely geometric effect rooted in the Gauss-Bonnet theorem. A computational inverse-design pipeline then solves, for any target surface, the flat ribbon shapes that relax into it once woven and laser-cut, and we characterized the resulting domes, which stiffen, snap through, and can be tuned from monostable to bistable. The work turns an empirical craft into a predictive framework for smooth, self-supporting curved shells.