In this research, we study a method to produce families of origami tessellations from given polyhedral surfaces. The resulting tessellated surfaces generalize the patterns proposed by Ron Resch and allow the construction of an origami tessellation that approximates a given surface. We will achieve these patterns by first constructing an initial configuration of the tessellated surfaces by separating each facets and inserting folded parts between them based on the local configuration. The initial configuration is then modified by solving the vertex coordinates to satisfy geometric constraints of developability, folding angle limitation, and local nonintersection. We propose a novel robust method for avoiding intersections between facets sharing vertices. Such generated polyhedral surfaces are not only applied to folding paper but also sheets of metal that does not allow 180 deg folding.

We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the model can be folded with additional creases, suggesting that real paper “folds” into this model via small such creases. We conjecture that the circular version of this model, consisting simply of concentric circular creases, also folds without extra creases. At the heart of our results is a new structural theorem characterizing uncreased intrinsically flat surfaces—the portions of paper between the creases. Differential geometry has much to say about the local behavior of such surfaces when they are sufficiently smooth, e.g., that they are torsal ruled. But this classic result is simply false in the context of the whole surface. Our structural characterization tells the whole story, and even applies to surfaces with discontinuities in the second derivative. We use our theorem to prove fundamental properties about how paper folds, for example, that straight creases on the piece of paper must remain piecewise-straight (polygonal) by folding.

When actuating a rigid origami mechanism by applying moments at the crease lines, we often confront the bifurcation problem where it is not possible to predict the way the model will fold when it is in a flat state. In this paper, we develop a mathematical model of self-folding and propose the concept of self-foldability of rigid origami when a set of moments, which we call a driving force, are applied. In particular, we desire to design a driving force such that a given crease pattern can uniquely self-fold to a desired mode without getting caught in a bifurcation. We provide necessary conditions for self-foldability that serve as tools to analyze and design self-foldable crease patterns. Using these tools, we analyze the unique self-foldability of several fundamental patterns and demonstrate the usefulness of the proposed model for mechanical design.

We propose Inkjet 4D Print, a self-folding fabrication method of 3D origami tessellations by printing 2D patterns on both sides of a heat-shrinkable base sheet, using a commercialized inkjet ultraviolet (UV) printer. Compared to the previous folding-based 4D printing approach using fused deposition modeling (FDM) 3D printers [An et al. 2018], our method has merits in (1) more than 1200 times higher resolution in terms of the number of self-foldable facets, (2) 2.8 times faster printing speed, and (3) optional full-color decoration. This paper describes the material selection, the folding mechanism, the heating condition, and the printing patterns to self-fold both known and freeform tessellations. We also evaluated the self-folding resolution, the printing and transformation speed, and the shape accuracy of our method. Finally, we demonstrated applications enabled by our self-foldable tessellated objects.

Despite the recent trend of computational origami for human-computer interaction (HCI) and digital fabrication, it is still difficult for designers to complete a series of design, simulation, and fabrication of objects leveraging computational origami theory. In this paper, we propose Crane, an integrated origami design platform implemented with Grasshopper. With this platform, users can seamlessly (1) design the 2D and 3D crease pattern, (2) simulate 3D folding transformation from the given crease pattern, (3) inversely find a new pattern under design constraints, (4) thicken the 2D pattern into a 3D volume along with the appropriate hinge structures for different fabrication methods, and (5) optionally connect the resulting design to other Rhinoceros or Grasshopper plugins for post-processes. To help understand how to use our system and demonstrate its feasibility, we showed three examples of origami products designed using our system. We also reported user feedback from the workshop as an evaluation.

Structural instability, once a catastrophic phenomenon to be avoided in engineering applications, is being harnessed to improve functionality of structures and materials, and has been a catalyst of substantial research in the field. One important application is to create functional metamaterials that deform their internal structure to adjust performance, resembling phase transformations in natural materials. In this paper, we propose a novel origami pattern, named the Shrimp pattern, with application to multi-phase architected metamaterials whose phase transition is achieved mechanically by snap-through. The Shrimp pattern consists of units that can be easily tessellated in two dimensions, either periodically with homogeneous local geometry or non-periodically with heterogeneous local geometries. We can use a few design parameters to program the unit cell to become either monostable or tune the energy barrier between the bistable states. By tessellating these unit cells into an architected metamaterial, we can create complex yet navigable energy landscapes, leading to multiple metastable phases of the material. As each phase has different geometries, the metamaterial can switch between different mechanical properties and shapes. The geometric origin of the multi-stable behavior implies that, conceptually, our designs are scale-independent, making them candidates for a variety of innovative applications, including reprogramable materials, reconfigurable acoustic waveguides, and microelectronic mechanical systems and energy storage systems.