In dieser Arbeit wird mathematisches Papierfalten und speziell 1-fach-Origami im universitären Kontext untersucht. Die Arbeit besteht aus drei Teilen. Der erste Teil ist im Wesentlichen der Sachanalyse des 1-fach-Origami gewidmet. Im ersten Kapitel gehen wir auf die geschichtliche Einordnung des 1-fach-Origami, betrachten axiomatische Grundlagen und diskutieren, wie das Axiomatisieren von 1-fach-Origami zum Verständnis des Axiomenbegriffs beitragen könnte. Im zweiten Kapitel schildern wir das Design der zugehörigen explorativen Studie, beschreiben unsere Forschungsziele und -fragen. Im dritten Kapitel wird 1-fach-Origami mathematisiert, definiert und eingehend untersucht. Der zweite Teil beschäftigt sich mit den von uns gestalteten und durchgeführten Kursen »Axiomatisieren lernen mit Papierfalten«. Im vierten Kapitel beschreiben wir die Lehrmethodik und die Gestaltung der Kurse, das fünfte Kapitel enthält ein Exzerpt der Kurse. Im dritten Teil werden die zugehörigen Tests beschrieben. Im sechsten Kapitel erläutern wir das Design der Tests sowie die Testmethodik. Im siebten Kapitel findet die Auswertung ebendieser Tests statt. / In this manuscript, mathematical paper folding and specifically 1-fold origami is studied in a university context. The thesis consists of three parts. The first part is mainly devoted to the factual analysis of 1-fold origami. In the first chapter, we elaborate on the historical development of 1-fold origami, consider axiomatic foundations, and discuss how axiomatizing 1-fold origami could contribute to the understanding of the concept of an axiom. In the second chapter, we describe the design of the related exploratory study, describe our research objectives and questions. In the third chapter, 1-fold origami is mathematized, defined, and explored in depth. The second part focuses on the courses with the title "Learning how to axiomatize through paperfolding" which we designed and conducted. In the fourth chapter we describe the teaching methodology and the design of the courses, and the fifth chapter contains an excerpt of the courses. In the third part we describe the associated tests. In the sixth chapter we explain the design of the tests as well as the testing methodology. In the seventh chapter, the evaluation of these tests is carried out.

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The purpose of this research is to develop fundamentals of compliant mechanisms in deployable space systems. The scope was limited to creating methods for thick origami, developing compliant deployable solar arrays, and developing methods for stowing and deploying the arrays. The research on actuation methods was focused on a one-time deployment of the array. Concepts for both passive and active actuation were considered. The primary objective of this work was to develop approaches to accommodate thickness in origami-based deployable arrays with a high ratio of deployed-to-stowed diameter. The HanaFlex design was derived from the origami flasher model and is developed as a deployable solar array for large arrays (150 kW or greater) and CubeSat arrays (60 W). The origami folding concept enables compact stowage of the array, which would be deployed from a hexagonal prism into a flat array with about a 10-times increase in deployed diameter as compared to stowed diameter. The work on the origami pattern for the solar array was also applied to the folding of 80-100 m2 solar sails for two NASA CubeSat missions, NEA-Scout and Lunar Flashlight. The CubeSat program is a promising avenue to put the solar array or solar sails into space for testing and proving their functionality. The deployable array concept is easily scalable, although application to CubeSats changes some of the design constraints. The thickness-to-diameter ratio is larger, making the issues of thickness more pronounced. Methods of actuation are also limited on CubeSats because of the rigorous size and weight constraints. This dissertation also includes the development of a compact, self-deploying array based on a tapered map fold design. The tapered map fold was modified by applying an elastic membrane to one side of the array and adequately spacing the panels adjacent to valley folds. Through this approach, the array can be folded into a fully dense stowed volume. Potential applications for the array include a collapsible solar array for military or backpacking applications. Additional compliant mechanism design was done in support of the HanaFlex array. This included a serpentine flexure to attach the array to the perimeter truss for deployment, and a bistable mechanism that may be used in the deployment of the array or sail.

Lamina emergent mechanisms (LEMs) are a subset of compliant mechanisms which are fabricated from planar materials; use compliance, or flexibility of the material, to transfer energy; and have motion that emerges out of the fabrication plane. LEMs provide potential design advantages by reducing the number of parts, reducing cost, reducing weight, improving recyclability, increasing precision, and eliminating assembly, to name a few. However, there are inherent design and modeling challenges including complexities in large, non-linear deflections, singularities that exist when leaving the planar state, and the coupling of material properties and geometry in predicting mechanism behavior. This thesis examines the planar and spherical LEMs and their relation to origami. Origami, the art of paper folding, is used to better understand spherical LEMs and flat-folding mechanisms in general. All single-layer planar four-bar LEMs are given with their respective layouts. These are all change-point pinned mechanisms (i.e. no slider cranks). Graph representations are used to show the similarities between action origami and mechanisms. Origami principles of flat-folding are shown to be analogous to principles of mechanisms including rules for assembly and motion.

Origami, the Japanese art of paper folding, has been used previously to inspire engineering solutions for compact, deployable designs. Action origami, the subset of origami dealing with models designed to move, is a previously unexplored area for engineering design solutions that are deployable and have additional motion in the deployed state. A literature review of origami in engineering is performed, resulting in seven key areas of technical origami literature from a wide variety of disciplines. Spherical mechanisms are identified as the method by which most action origami models achieve complicated motion while remaining flat-foldable. The subset of action origami whose motion originates from spherical mechanisms is termed "kinematic origami''. Action origami is found to contain large coupled systems of spherical mechanisms. All possible action origami models are classified by their spherical mechanism structure, resulting in eight possible categories. Viewing action origami as spherical mechanisms allows the use of established equations for kinematic analysis. Several kinematic origami categories are used to demonstrate a method for the position analysis of coupled systems of spherical mechanisms. Input-output angle relationships and coupler link motions are obtained for a single spherical mechanism, two spherical mechanisms coupled together, and four spherical mechanisms coupled in a loop arrangement. This lays a groundwork from which it is possible to create compact, deployable mechanisms with motion in the deployed state.

Designers are constantly searching for new sourcing of inspiration for innovative design. Recently, origami has gained interest as one of these potential sources. Origami literally translated from Japanese means “paper folding” where “oru” means “to fold” and “kami” means “paper”. Since paper is insufficient to solve many engineering design problems, designers must turn to other materials. These materials will inevitably be thicker than paper and will often require different folding techniques and considerations. This thesis provides background information describing previous methods to accommodate thickness in origami-inspired design, presents a newly developed technique to address limitations of other methods, and explores the application of the technique. The newly developed technique allows designers to identify a desired motion behavior in an origami model and implement it into a thick mechanism. Many previous methods were incapable of preserving the kinematics and/or restricted usable range of motion. Understanding the capabilities and limitations of thickness accommodation methods empowers designers to better implement inspiration from origami into engineering design. The offset panel technique is further extended to include arbitrary thickness and arbitrary folding plane locations. The technique is verified through creation and testing of hardware, showcasing capabilities and limitations. Demonstration of these capabilities will serve as inspiration for furthering application of thick origami in engineering design. Preliminary work in thick origami led to the design of a thick origami-inspired medical gripper. These origami-inspired forceps, Oriceps, were designed by starting with an origami model exhibiting desired motion, grasping. The Oriceps show some challenges faced with accommodating thickness in adapting an origami model for application.

This work evaluates a specific origami device known as the kaleidocycle and the broad classof rigidly foldable origami. Both of these have potential for application in the design of deployableand foldable arrays of spatial mechanisms.Origami is considered a compliant mechanisms because it achieves its motion through thedeflection of paper creases. Compliant mechanisms generally do not allow for continuous rotation;however, the compliant kaleidocycle represents an exception to this generality. Along with theirability to rotate continuously, kaleidocycles may also be designed to exhibit multistable behaviorduring this rotation. These two characteristics make the kaleidocycle an interesting device withpotential for applications in engineering. This work presents the multistable characteristics ofkaleidocycles, showing that devices can be made which exhibit one, two, three, or four distinctstable equilibrium positions. Kaleiocycles may also be designed to exhibit a range over which thedevice is neutrally stable.The second type of origami presented in this work is rigidly foldable origami, a special classof origami in which all deflection occurs at the creases, allowing the panels to remain rigid. Thistype of origami is of particular interest because of its ability to be constructed from materials muchstiffer than paper while retaining its mobility. This property allows rigidly foldable origami to beapplied to fields such as architecture and deployable mechanisms. This work presents a method forevaluating rigid foldability in origami tessellations. This method is used to define seven theoremsfor the rigid foldability of origami twists and to develop new rigidly foldable origami “gadgets”and tessellations.

<p>  </p> <p>Architectured metamaterials are artificial systems with unique structural characteristics. They show distinct deformation behaviors and improved mechanical properties compared to regular materials. For example, mechanical metamaterials demonstrate negative Poisson's ratios, whereas regular materials have positive values. In theory, the auxetic behaviors arise from periodic cellular architectures regardless of the materials utilized. While this premise is mostly true for macroscopic metamaterials, it may not work well at a very small lengthscale since chemistry may play a critical role in nanostructures. However, this fundamental idea has not been addressed due to the lack of powerful manufacturing strategies at the nanoscale. The majority of architectured metamaterials are manufactured from top down with their unit size of microns or larger. On the other hand, there are also molecular auxetics which are natural crystals and thus are not designable. Therefore, there is a significant gap in lengthscale from 10 nm to 1 µm. DNA self-assembly is a bottom-up approach that can construct complex nanostructures based on sequence complementarity. Examples include DNA origami structures and DNA tile assemblies. This dissertation bridges the gap in the lengthscale by introducing nanoscale auxetic units from DNA and investigates relevant structural properties and mechanical behaviors. This study addresses the premise of metamaterials and elucidates the structure-property relation. The findings from this work formulate design principles for DNA based auxetic metastructures. </p> <p>In this work, we built several two-dimensional (2D) auxetic nanostructures from wireframe DNA origami. They serve as the model systems to demonstrate the feasibility of constructing nanoscale auxetics via DNA self-assembly. DNA origami structures are commonly constructed by a long ‘scaffold’ strand with many ‘staple’ oligonucleotides. Since the DNA metastructures are too small to directly apply external forces, we implemented chemical deformation by inserting ‘jack’ edges. Like a car jack, the length of the jack edges can be modulated via two-step DNA reactions: toehold-mediated strand displacement and annealing with a new set of jack staples. The DNA nanostructures reconfigure accordingly. To complement the experiment, we performed molecular dynamics (MD) simulations based on coarse-grained models using an open-source oxDNA platform. In the numerical computation, external loads were directly applied to deform the metastructures, providing details of structural deformation. We discovered that the auxetic behaviors of DNA metamaterials can be estimated by architectural designs, however the material properties are also crucial in the structures and deformations. Our mechanistic study provided general design guidelines for 2D auxetic DNA metamaterials. We also designed and constructed a Hoberman flight ring from DNA, a simplified planar version of Hoberman sphere. This structure consists of six equilateral triangles that are topologically organized into two layers, resembling a trefoil knot. The DNA flight ring deploys upon external forces, expanding (open state) or contracting (closed state) by sliding the two layers of triangles. This is the first synthetic deployable nanostructure and offers a versatile platform for topological research.</p> <p>This thesis also investigates 3D effects in DNA assemblies and related mechanics. We used a DNA origami tile designed with an intrinsic twist as a model system and explored its cyclization process using MD simulations. The numerical computation revealed the detailed process where the structure untwists and curves for cyclization simultaneously under external forces. The force and energy required to overcome the initial curvature and cause the 3D deformation were also calculated. The results agree well with the previous experiment and theory, further verifying the simulation method. Direct mechanical forces and DNA responses were realized experimentally with 3D DNA crystals built from triangular DNA tiles. Nanoindentation was performed on macroscopic ligated crystals using atomic force microscopy (AFM). MD simulations were performed in parallel, which revealed the full spectrum of several distinct deformation modes from linear elasticity to structural failure. The combined experiment, computation, and theoretical calculation showed that the complex behaviors can only be understood fully by considering the structure and its components. </p> <p>The scientific findings from this thesis should contribute to the construction of auxetic metastructures, the design methods for DNA based metamaterials as well as the prediction of their structural properties and mechanical behaviors. This thesis will pave the way for building architectured materials from DNA with tailored properties and functionalities, opening the door for new opportunities and unique applications.</p>

The ancient art of ‘origami’ has recently become the source of inspiration for engineers to create structures that can unfold from a compact state to a fully deployed one. For instance, researchers have currently adopted origami designs in various engineering disciplines, including aerospace engineering, robotics, biomedical engineering, and architecture. In particular, architects have been interested in designing origami-inspired rigid walled structures that can be deployed as disaster-relief shelters. This type of design has three main advantages: transportability, constructability, and rigidity. Although there has been increased interests in deployable structures, limited research has been conducted on evaluating their structural performance, specifically the mechanical performance of the hinges that allow for the rotation of the rigid panels. To address the limitation, this thesis proposes a novel design of hinge connections for rigid origami structures. The hinges utilize snap fit connections to allow for the structure to achieve and maintain a locked state once unfolded without the need for any additional connections. Prototypes of the hinge design were fabricated using a 3D printer and their flexural strength was experimentally and computationally studied. It was concluded that the design could resist typical flexural loads for residential structures, and future research should be performed to minimize deflection.

We work with two different mechanisms to generate geometric frustration on thin elastic sheets; isotropic differential growth and folding. We describe how controlled growth and prescribing folding patterns are useful tools for designing three-dimensional objects from information printed in two dimensions. The first mechanism is inspired by the possibility to control shapes by swelling polymer films, where we propose a solution for the problem of shape formation by asking the question, ``what 2D metric should be prescribed to achieve a given 3D shape?'', namely the reverse problem. We choose two different types of initial configurations of sheets, disk-like with one boundary and annular with two boundaries. We demonstrate our technique by choosing four examples of 3D axisymmetric shapes and finding the respective swelling factors to achieve the desired shape. Second, we present a mechanical model for a single curved fold that explains both the buckled shape of a closed fold and its mechanical stiffness. The buckling arises from the geometrical frustration between the prescribed crease angle and the bending energy of the sheet away from the crease. This frustration increases as the sheet's area increases. Stiff folds result in creases with constant space curvature while softer folds inherit the broken symmetry of the buckled shape. We extend the application of our numerical model to show the potential to study multiple fold structures.