We present a design method that prioritises in-context design for origami surfaces with periodic tessellations in a parametric CAD workflow using Grasshopper 3D. The key design criteria are: target geometry surface, user-defined folding patterns as periodic tessellations, and fold resolution. Using an error minimisation solver, we generate developable crease patterns from non-developable meshes. We evaluate our method through a study of a target geometry , Fold Mapped with various fold molecules at variable resolutions, and present a visual analysis as proof of form-fit to the target. This method affords rapid development of origami surfaces, bypassing significant trial and error in by-hand design processes.
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... An origami structure is built by a tessellation that is a repetition of a unit cell covering the entire sheet. Different unit cells generate different tessellations, with specific properties related to motion, configuration, and actuation . One of the challenges of the origami design is to deal with the large number of variables and degrees of freedom (DoFs) associated with such complex structures. ...
Origamis are becoming the inspiration of new adaptive structures applied for several purposes. One of the challenges of the design of the origami inspired structures is to deal with the large number of variables and degrees of freedom (DoFs) associated with such complex structures. Closed tessellations have a reduced number of DoF when compared to the opened ones. Besides, the coupling due to the closure of the tessellation promotes some periodicity along the structure. Symmetric behaviors allow the description of the structure from a unit cell behavior, establishing reduced-order models. This paper investigates the origami waterbomb pattern, exploring the unit cell behavior and its symmetries. Initially, kinematics analysis based on an equivalent mechanism approach establishes a reduced-order model associated with symmetry hypotheses. Afterward, mechanical analysis is investigated using a nonlinear finite element analysis through bar-and-hinge formulation. A comparison between both formulations is performed showing the range of validity of the reduced-order model description. The general conclusions are applied to a cylindrical tessellation under symmetric actuation showing the capability of the reduced-order model for the origami description. Results show that the rigid foldability hypothesis is the essential point for the equivalence between the two descriptions.
... The Flasher model is not directly related to buckling. However, the Flasher model and the Kresling model, both have tilted flaps (Fig. 2H-J) and it is possible that both models can be created by applying similar twist forces on basic curvilinear geometrical shapes (cylinder for the Kresling model and hemisphere for the Flasher model) . Therefore, we tested whether a thin hemisphere that twist buckles is similar to the TRiC protein (and the analog Kresling based origami model). ...
Protein structure is an important field of research, with particular significance in its potential applications in biomedicine and nanotechnology. In a recent study, we presented a general approach for comparing protein structures and origami models and demonstrated it with one-domain proteins. For example, the analysis of the α-helical barrel of the outer membrane protein A (OmpA) suggests that there are similar patterns between its structure and the Kresling origami model, providing insight into structure-activity relationships. Here we demonstrate that our approach can be expanded beyond single-domain proteins to also include multi-domain proteins, and to study dynamic processes of biomolecules. Two examples are given: (1) The eukaryotic chaperonin (TRiC) protein is compared with a newly generated origami model, and with an origami model that is constructed from two copies of the Flasher origami model, and (2) the CorA Magnesium transport system is compared with a newly generated origami model and with an origami model that combines the Kresling and Flasher origami models. Based on the analysis of the analog origami models, it is indicated that it is possible to identify building blocks for constructing assembled origami models that are analogous to protein structures. In addition, it is identified that the expansion/collapse mechanisms of the TRiC and CorA are auxetic. Namely, these proteins require a single motion for synchronized folding along two or three axes.
... The survey paper [Demaine et al. 2015b] reviews masterpieces of curved-crease folding and the corresponding design methods. The parametric design of origami surfaces with periodic tessellations is the topic of [Gardiner et al. 2018]. Analysis together with design has been done by Demaine et al. [2015a; for 'lens' crease patterns, and crease patterns formed by conics. ...
In this paper we study pleated structures generated by folding paper along curved creases. We discuss their properties and the special case of principal pleated structures. A discrete version of pleated structures is particularly interesting because of the rich geometric properties of the principal case, where we are able to establish a series of analogies between the smooth and discrete situations, as well as several equivalent characterizations of the principal property. These include being a conical mesh, and being flat-foldable. This structure-preserving discretization is the basis of computation and design. We propose a new method for designing pleated structures and reconstructing reference shapes as pleated structures: we first gain an overview of possible crease patterns by establishing a connection to pseudogeodesics, and then initialize and optimize a quad mesh so as to become a discrete pleated structure. We conclude by showing applications in design and reconstruction, including cases with combinatorial singularities. Our work is relevant to fabrication in so far as the offset properties of principal pleated structures allow us to construct curved sculptures of finite thickness.
Kirigami is a Japanese art of paper cutting. It is used to obtain three-dimensional shapes via cutting and folding the paper. Origami, however, is based on a series of precise geometric folding without any other changes to the paper. With the characteristics of dimensional change and form transformation as well as the advantages of being able to expand and zoom, both Kirigami and Origami have great potential in cross-domain applications. These include biomedical materials, deformable robots, adaptable building cortex, and aerospace science. These applications show different requirements for folding by Kirigami or Origami. This study considered that the knowledge of design must include the operational process used to solve design problems.
With the arrival of the internet of things and the rise of wearable computing, electronics are playing an increasingly important role in our everyday lives. Until recently, however, the rigid angular nature of traditional electronics has prevented them from being integrated into many of the organic, curved shapes that interface with our bodies (such as ergonomic equipment or medical devices) or the natural world (such as aerodynamic or optical components). In the past few years, many groups working in advanced manufacturing and soft robotics have endeavored to develop strategies for fabricating electronics on these curved surfaces. This is their story. In this work, we describe the motivations, challenges, methodologies, and applications of curved electronics, and provide a outlook for this promising field.
In the past decade, touch surfaces revolutionized our interactions with digital systems and accelerated the conception of flexible user interfaces depending on their use cases. By now, we can already think of a future in which static and inflexible materials can also be programmed in three-dimensional space.
ORI*Sense is a design for a mobile shape display that not only carries information with its transformation, but also introduces new interactions with digital devices through flexible features - by exposing haptic interface elements such as buttons, potis and sliders on top of both flat and curved surfaces, depending on the stage of the application.
The prototype was designed in the course of the diploma thesis of Christoph Kirmaier within the ORI* project at the Ars Electronica Futurelab under Dr. Matthew Gardiner’s direction. It consists of a 3D-printed resch pattern enhanced with sensoric components, to illustrate the idea of a shape pixel that can be lifted out of the surface and sense deformation, touch and proximity of a user.
By implementing soft robotic actuators mounted below the 3D-printed surface, the shape pixel can also be deformed programmatically. The surface could theoretically be of any material - for this concept it is designed as a user interface element that appears in any color, through even patterning of red, green and blue lighting elements.
ORI*sense meets four core qualities that were identified during the design process - multi-materiality, 4-dimensionality, modularity and mobility. It is conceptualized as a system of stackable elements, that can be produced out of any material, move in any direction and be used in any mobile scenario. Finally, the project outlines the simplest form of user interaction with computer systems: a button which grows out of a display when needed.
The art of oribotics involves a process of imagining and creating geometry for static and kinetic origami structures to fit a desired form. It is a two-fold difficult task. Firstly, tools for calculating the geometry are few, and those existing lack key aesthetic and functional criteria specific to my artistic practice. Secondly, material issues, such as durability and complex foldability, compound the issues for fabrication. Existing methods, even those applying digital fabrication, pose complex folding problems that confound origami experts.
Case studies are provided of leading origamists working with software, fabrication and materials to analyse and summarise processes. Analysis of these led to the synthesis and identification of differentiating criteria that inspired the invention of two key methods: Fold-Mapping and Fold-Printing. Fold-Mapping abstracts naturally occurring origami patterns into fold-molecules for tessellation across target geometric surfaces. It allows an artist to prioritise the sculptural shape of the result while seeking a kinetic solution through experimentation with different fold-molecules. A developability algorithm then flattens the crease pattern into geometry for fabrication. Fold-Printing allows the fabrication of Fold-Mapping results. It includes results of high-order complex-foldability by 3D printing whereby polymers are deposited onto textiles forming a durable polymer plate-structure separated by perfect textile hinges.
Evaluation of the methods is two-fold. Firstly, the methods were continuously evaluated and refined through invention based trial-and-error. Secondly, the artefacts were evaluated according to a set of aesthetic criteria which in this thesis are collectively called ORI* theory. ORI* theory includes origami and oribotic criteria, and a metric for complex-foldability.
For proof of concept, the methods are presented as an evaluated set of successful traits and developed solutions. These produce developable crease patterns from target geometries and afford the fabrication and foldability of complex ORI* objects. Fold-Printing allows near impossible-to-fold patterns to become foldable objects. The methods do not succeed in all circumstances, and that success is additionally dependent on the author’s experience of origami structures. The aesthetic qualities and material properties of the artistic results have distinct qualities that qualify them to meet the particular criteria required for ORI* objects.
The thesis concludes with the proposing of future work: 1) in the direction of soft-robotic applications using both Fold-Mapping and Fold-Printing methods, and 2) the creation of enhanced aesthetic and technical ORI* objects across many disciplinary domains.
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 ε 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. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems 1 Introduction The ultimate challenge in computational origami design is to devise an algorithm that tells you the best way to fold anything you want. Several results tackle this problem for various notions of " best " and " anything ". We highlight two key such results, from SoCG'96 and SoCG'99 respectively. The tree method [6, 7, 4] finds an efficient folding of a given square of paper into a shape with an orthogonal projection equal to a scaled copy of a given metric tree. We use the term " efficient " because the method works well in practice, being the foundation for most modern origami design, but exact optimization of the scale factor (the usual measure of efficiency) is a difficult computational problem, recently shown NP-hard , but one that can be handled reasonably well by heuristics. The strip method  finds a *
Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures-we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.
Surface developability is required in a variety of applications in product design, such as clothing, ship hulls, automobile parts, etc. However, most current geometric modeling systems using polygonal surfaces ignore this important intrinsic geometric property. This paper investigates the problem of how to minimally deform a polygonal surface to attain developability, or the so-called developability-by-deformation problem. In our study, this problem is first formulated as a global constrained optimization problem and a penalty-function-based numerical solution is proposed for solving this global optimization problem. Next, as an alternative to the global optimization approach, which usually requires lengthy computing time, we present an iterative solution based on a local optimization criterion that achieves near real-time computing speed.
The generation of arbitrary patterns and shapes at very small scales is at the heart of our effort to miniaturize circuits and is fundamental to the development of nanotechnology. Here I review a recently developed method for folding long single strands of DNA into arbitrary two-dimensional shapes using a raster fill technique - 'scaffolded DNA origami'. Shapes up to 100 nanometers in diameter can be approximated with a resolution of 6 nanometers and decorated with patterns of roughly 200 binary pixels at the same resolution. Experimentally verified by the creation of a dozen shapes and patterns, the method is easy, high yield, and lends itself well to automated design and manufacture. So far, CAD tools for scaffolded DNA origami are simple, require hand-design of the folding path, and are restricted to two dimensional designs. If the method gains wide acceptance, better CAD tools will be required.
Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.