The miniaturization of shell structures presents a versatile and complex challenge, bridging geometry with diverse practical applications. In this paper, we introduce a novel approach for computing origami crease patterns to compress arbitrary 3D shell objects. First, we employ the adapted Material Point Method (MPM) to simulate the compression of a target surface and obtain an initial folded configuration. Since MPM produces overly smooth curved surfaces, their crease patterns are unsuitable for practical origami fabrication. We then propose a novel Folding Line Extraction (FLE) method that optimizes these smoothed surfaces to extract folding lines that achieve the target compression with minimal deformation and stretching outside the crease lines. This method produces smooth curved folding lines. Fabrication and experimental validation of the extracted patterns demonstrate their effectiveness and applicability in real-world scenarios.

We introduce a novel class of polyhedral tori (PQ-toroids) that snap between two stable configurations - a flat state and a deployed one separated by an energy barrier. Being able to create PQ-toroids from any set of given planar bottom and side faces opens the possibility to assemble the bistable blocks into a thick freeform curved shell structure to follow a planar quadrilateral (PQ) net with coplanar adjacent offset directions. A design pipeline is developed and presented for inversely computing PQ-toroid modules using conjugate net decompositions of a given surface. We analyze the snapping behavior and energy barriers through simulation and build physical prototypes to validate the feasibility of the proposed system. This work expands the geometric design space of multistable origami for lightweight modular structures and offers practical applications in architectural and deployable systems.

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.

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.

It was established at SoCG'99 that every polyhedral complex can be folded from a sufficiently large square of paper, but the known algorithms are extremely impractical, wasting most of the material and making folds through many layers of paper. At a deeper level, these foldings get the topology wrong, introducing many gaps (boundaries) in the surface, which results in flimsy foldings in practice. We develop a new algorithm designed specifically for the practical folding of real paper into complicated polyhedral models. We prove that the algorithm correctly folds any oriented polyhedral manifold, plus an arbitrarily small amount of additional structure on one side of the surface (so for closed manifolds, inside the model). This algorithm is the first to attain the watertight property: for a specified cutting of the manifold into a topological disk with boundary, the folding maps the boundary of the paper to within epsilon of the specified boundary of the surface (in Fréchet distance). Our foldings also have the geometric feature that every convex face is folded seamlessly, i.e., as one unfolded convex polygon of the piece of paper. This work provides the theoretical underpinnings for Origamizer, freely available software written by the second author, which has enabled practical folding of many complex polyhedral models such as the Stanford bunny.

This paper presents the first practical method for "origamizing" or obtaining the folding pattern that folds a single sheet of material into a given polyhedral surface without any cut. The basic idea is to tuck fold a planar paper to form a three-dimensional shape. The main contribution is to solve the inverse problem; the input is an arbitrary polyhedral surface and the output is the folding pattern. Our approach is to convert this problem into a problem of laying out the polygons of the surface on a planar paper by introducing the concept of tucking molecules. We investigate the equality and inequality conditions required for constructing a valid crease pattern. We propose an algorithm based on two-step mapping and edge splitting to solve these conditions. The two-step mapping precalculates linear equalities and separates them from other conditions. This allows an interactive manipulation of the crease pattern in the system implementation. We present the first system for designing three-dimensional origami, enabling a user can interactively design complex spatial origami models that have not been realizable thus far.