Rolling a sheet of paper around a cylinder is easy. In contrast, giving it the shape of a dome or a saddle is much more complex and leads to folds or tearing. Gauss, through his Theorema Egregium, showed that the intrinsic geometry of a surface specifically the product of its principal curvatures limits transformations that avoid stretching or compression. Transforming a developable surface into a non-developable one therefore requires internal deformation.To overcome this constraint, various systems have been developed, particularly in soft robotics. Some draw inspiration from differential growth in biological tissues, such as the wavy edges of iris petals. However, this biomimetic strategy requires highly stretchable materials like elastomers, which excludes rigid materials such as paper.Other approaches rely on the geometry of folding (origami) or cutting (kirigami), which we study in the first part. Kirigami enables an apparent extension of inextensible materials through cutting, creating zones that behave like flexible beams. We investigated axisymmetric patterns made of circular arcs, where pulling the structure from its center generates step-like formations. The local slope of these steps depends on the cutting pattern and the stiffness of the beams, allowing us to program shapes such as cones, domes, or trumpet-like forms.We also considered distributed loadings, such as self-weight or drag forces due to fluid flow, and irreversible shaping using plastic materials (metals, polyester films). We developed an algorithm that determines a cutting pattern suited to a target shape, accounting for moment balance in non-symmetric geometries.If paper is replaced by fabric, it is also possible to bypass Gauss’s constraint thanks to the orthotropy of the material. Though barely stretchable along warp and weft, a fabric can stretch along the bias. This property is exploited in architectural gridshells.In the second part, we studied structures made of crossed ribbons, assembled via slots similar to lap joints in carpentry. These slots, with controlled clearance, act as rotational hinges up to a certain angle. The ribbons can bend and twist, enabling in-plane and out-of-plane deformations. One system, consisting of perpendicularly crossed ribbons, forms a grid of square cells that can deform through shear while maintaining overall planarity. By stiffening certain cells with 3D-printed blocks, local rigidity can be increased without altering the overall geometric properties.By reducing the number of longitudinal ribbons, we created original beams, whose sagging shape differs from a simple ribbon. The torsion of the ribbons enables deformations with uniformly negative Gaussian curvature, characteristic of asymptotic grids: the ribbons remain perpendicular to the surface and cannot bend laterally. As a result, only zero or negative curvatures are achievable.However, by modifying the intersection angles through the slot geometry, this constraint can be relaxed. A gradual variation of these angles along the ribbons enables the creation of positive curvatures and target 3D shapes.The two systems we explored reveal a strong coupling between geometry and mechanics. They enable the formation of lightweight, low-cost, and easily manufacturable structures, suitable for a wide range of applications from microsystems and deformable sensors for human-machine interaction to deployable architectural structures.
Joo-Won Hong (Fri,) studied this question.