On rigid origami I: Piecewise-planar paper with straight-line creases

We develop a theoretical framework for rigid origami, and show how this framework can be used to connect rigid origami and results from cognate areas, such as the rigidity theory, graph theory, linkage folding and computer science. First, we give definitions on important concepts in rigid origami, then focus on how to describe the configuration space of a creased paper. The shape and 0-connectedness of the configuration space are analyzed using algebraic, geometric and numeric methods, where the key results from each method are gathered and reviewed

The grammar of developable double corrugations (for formal architectural applications)

This paper investigates the geometrical basis of regular corrugations, with specific emphasis on Developable Double Corrugations (DDCs), which form a unique sub-branch of Origami Folding and Creasing Algorithms. The aim of the exercise is three fold – (1) To define and isolate a ‘single smallest starting block’ for a given set of distinct and divergent DDC patterns, such that this starting block becomes the generator of all DDCs when different generative rules are applied to it. (2) To delineate those generic parameters and generative rules which would apply to the starting block, such that different DDCs are created as a result (3) To use the knowledge from points (1) and (2) to create a complete family of architectural forms and shapes using DDCs. For this purpose, a matrix of 12 underlying geometry types are identified and used as archetypes. The objective is to mathematically explore DDCs for architectural form finding, using physical folding as a primary algorithmic tool. Some DDCs have more degrees of freedom than others and can fit varied geometries, while others cannot. The discussion and conclusions involve - (a) identifying why certain DDCs are ideal for certain forms and not others, when all of them are generated using the same/or similar starting block(s), (b) discussing the critical significance of flat-foldability in this specific context and (c) what we can do with this knowledge of DDCs in the field of architectural research and practice in the future

Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes.Comment: 54 pages, 33 figure

Geometry Synthesis and Multi-Configuration Rigidity of Reconfigurable Structures

Reconfigurable structures are structures that can change their shapes to change their functionalities. Origami-inspired folding offers a path to achieving shape changes that enables multi-functional structures in electronics, robotics, architecture and beyond. Folding structures with many kinematic degrees of freedom are appealing because they are capable of achieving drastic shape changes, but are consequently highly flexible and therefore challenging to implement as load-bearing engineering structures. This thesis presents two contributions with the aim of enabling folding structures with many degrees of freedom to be load-bearing engineering structures. The first contribution is the synthesis of kirigami patterns capable of achieving multiple target surfaces. The inverse design problem of generating origami or kirigami patterns to achieve a single target shape has been extensively studied. However, the problem of designing a single fold pattern capable of achieving multiple target surfaces has received little attention. In this work, a constrained optimization framework is presented to generate kirigami fold patterns that can transform between several target surfaces with varying Gaussian curvature. The resulting fold patterns have many kinematic degrees of freedom to achieve these drastic geometric changes, complicating their use in the design of practical load-bearing structures. To address this challenge, the second part of this thesis introduces the concept of multi-configuration rigidity as a means of achieving load-bearing capabilities in structures with multiple degrees of freedom. By embedding springs and unilateral constraints, multiple configurations are rigidly held due to the prestress between the springs and unilateral constraints. This results in a structure capable of rigidly supporting finite loads in multiple configurations so long as the loads do not exceed some threshold magnitude. A theoretical framework for rigidity due to embedded springs and unilateral constraints is developed, followed by a systematic method for designing springs to maximize the load-bearing capacity in a set of target configurations. An experimental study then validates theoretical predictions for a linkage structure. Together, the application of geometry synthesis and multi-configuration rigidity constitute a path towards engineering reconfigurable load-bearing structures.</p

EXTENDING ORIGAMI TECHNIQUE TO FOLD FORMING OF SHEET METAL PRODUCTS

This dissertation presents a scientific based approach for the analysis of folded sheet metal products. Such analysis initializes the examination in terms of topological exploration using set of graph modeling and traversal algorithms. The geometrical validity and optimization are followed by utilizing boundary representation and overlapping detection during a geometrical analysis stage, in this phase the optimization metrics are established to evaluate the unfolded sheet metal design in terms of its manufacturability and cost parameters, such as nesting efficiency, total welding cost, bend lines orientation, and maximum part extent, which aides in handling purposes. The proposed approach evaluates the design in terms of the stressed-based behavior to indicate initial stress performance by utilizing a structural matrix analysis while developing modification factors for the stiffness matrix to cope with the stress-based differences of the diverse flat pattern designs. The outcome from the stressed-based ranking study is mainly the axial stresses as exerted on each element of folded geometry; this knowledge leads to initial optimizing the flat pattern in terms of its stress-based behavior. Furthermore, the sheet folding can also find application in composites manufacturing. Thus, this dissertation optimizes fiber orientation based on the elasticity theory principles, and the best fiber alignment for a flat pattern is determined under certain stresses along with the peel shear on adhesively bonded edges. This study also explores the implementation of the fold forming process within the automotive production lines. This is done using a tool that adopts Quality Function Deployment (QFD) principle and Analytical Hierarchy Process (AHP) methodology to structure the reasoning logic for design decisions. Moreover, the proposed tool accumulates all the knowledge for specific production line and parts design inside an interactive knowledge base. Thus, the system is knowledge-based oriented and exhibits the ability to address design problems as changes occur to the product or the manufacturing process options. Additionally, this technique offers two knowledge bases; the first holds the production requirements and their correlations to essential process attributes, while the second contains available manufacturing processes options and their characteristics to satisfy the needs to fabricate Body in White (BiW) panels. Lastly, the dissertation showcases the developed tools and mathematics using several case studies to verify the developed system\u27s functionality and merits. The results demonstrate the feasibility of the developed methodology in designing sheet metal products via folding

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

Efficient Enumeration of Flat-Foldable Single Vertex Crease Patterns

We investigate enumeration of distinct flat-foldable crease patterns under the following assumptions: positive integer n is given; every pattern is composed of n lines incident to the center of a sheet of paper; every angle between adjacent lines is equal to 2π/n; every line is assigned one of “mountain,” “valley,” and “flat (or consequently unfolded)”; crease patterns are considered to be equivalent if they are equal up to rotation and reflection. In this natural problem, we can use two well-known theorems for flat-foldability: the Kawasaki Theorem and the Maekawa Theorem in computational origami. Unfortunately, however, they are not enough to characterize all flat-foldable crease patterns. Therefore, so far, we have to enumerate and check flat-foldability one by one using computer. In this study, we develop the first algorithm for the above stated problem by combining these results in a nontrivial way and show its analysis of efficiency

Efficient Enumeration of Flat-Foldable Single Vertex Crease Patterns

We investigate the following computational origami problem; the input is a positive integer n. We then draw n lines in a radial pattern. They are incident to the central point of a sheet of paper, and every angle between two consecutive lines is equal to 2π/n. Each line is assigned one of “mountain,” “valley,” and “flat” (or consequently unfolded), and only flat-foldable patterns will be output. We consider two crease patterns are the same if they can be equal with rotations and reflections. We propose an efficient enumeration algorithm for flat-foldable single vertex crease patterns for given n. In computational origami, there are well-known theorems for flat-foldability; Kawasaki Theorem and Maekawa Theorem. However, they give us necessary conditions, and sufficient condition is not known. Therefore, we have to enumerate and check flat-foldability one by one using the other algorithm. In this paper, we develop the first algorithm for the above stated problem by combining these results in nontrivial way, and show its analysis of efficiency.11th International Conference and Workshops, WALCOM 2017, Hsinchu, Taiwan, March 29–31, 2017, Proceeding