Triaxial weaving, a craft technique that enables the generation of surfaces with tri-directional arrays of initially straight elastic strips, has long been loved by basket makers and artists seeking a combination of practical and aesthetically-pleasing structures. The design principles of traditional weaving are based on the observation that the non-hexagonal topology of unit cells imparts out-of-plane shapes. In the realm of differential geometry, the weaving tradition is rooted in the concept of Euler characteristics through the Gauss-Bonnet theorem, with discrete topological defects being used as building blocks. Taking an alternative point of departure, we introduce a novel approach for triaxial weaving that enables us to continuously span a variety of 3D shapes of the weave by tuning the natural in-plane curvature of the strips. We systematically explore the validity of the new strategy by quantifying the shape of experimental specimens with X-ray tomography in combination with continuum-based simulations. To demonstrate the potential of our design scheme, and as a canonical example, we present a fullerene-like weave that is perfectly spherical, which cannot be readily achieved using straight strips. Ellipsoidal and toroidal structures are also explored.