An Overview of Mechanisms and Patterns with Origami David Dureisseix
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Geometry of Transformable Metamaterials Inspired by Modular Origami
Geometry of Transformable Metamaterials Inspired by Yunfang Yang Department of Engineering Science, Modular Origami University of Oxford Magdalen College, Oxford OX1 4AU, UK Modular origami is a type of origami where multiple pieces of paper are folded into mod- e-mail: [email protected] ules, and these modules are then interlocked with each other forming an assembly. Some 1 of them turn out to be capable of large-scale shape transformation, making them ideal to Zhong You create metamaterials with tuned mechanical properties. In this paper, we carry out a fun- Department of Engineering Science, damental research on two-dimensional (2D) transformable assemblies inspired by modu- Downloaded from http://asmedigitalcollection.asme.org/mechanismsrobotics/article-pdf/10/2/021001/6404042/jmr_010_02_021001.pdf by guest on 25 September 2021 University of Oxford, lar origami. Using mathematical tiling and patterns and mechanism analysis, we are Parks Road, able to develop various structures consisting of interconnected quadrilateral modules. Oxford OX1 3PJ, UK Due to the existence of 4R linkages within the assemblies, they become transformable, e-mail: [email protected] and can be compactly packaged. Moreover, by the introduction of paired modules, we are able to adjust the expansion ratio of the pattern. Moreover, we also show that trans- formable patterns with higher mobility exist for other polygonal modules. The design flex- ibility among these structures makes them ideal to be used for creation of truly programmable metamaterials. [DOI: 10.1115/1.4038969] Introduction be made responsive to external loading conditions. These struc- tures and materials based on such structures can find their usage Recently, there has been a surge of interest in creating mechani- in automotive, aerospace structures, and body armors. -
What Lies Between Rigidity and Flexibility of Structures
S A J _ 2011 _ 3 _ original scientific article approval date 03 05 2011 UDK BROJEVI: 514.113.5 ; 614.073.5.042 ID BROJ: 184974860 WHAT LIES BETWEEN RIGIDITY AND FLEXIBILITY OF STRUCTURES A B S T R A C T The borderline between continuous flexibility and rigidity of structures like polyhedra or frameworks is not strict. There can be different levels of infinitesimal flexibility. This article presents the mathematical background and some examples of structures which under particular conditions are flexible or almost flexible and otherwise rigid. Hellmuth Stachel 102 KEY WORDS Vienna University of Technology - Institute of Discrete Mathematics and Geometry RIGIDITY FLEXIBILITY INFINITESIMAL FLEXIBILITY FLEXIBLE POLYHEDRA SNAPPING POLYHEDRA KOKOTSAKIS MESHES ORIGAMI MECHANISMS S A J _ 2011 _ 3 _ INTRODUCTION A framework or a polyhedron will be called “rigid” when the edge lengths determine its planar or spatial shape uniquely; under the term “shape” we mean its spatial form – apart from movements in space. More generally and under inclusion of smooth or piecewise linear surfaces, a structure is called rigid when its intrinsic metric defines its spatial shape uniquely. In this sense, the intrinsic metric of a polyhedron is defined by its net (unfolding), i.e., the coplanar set of faces with identified pairs of edges originating from the same edge of the spatial form. After cutting out this net from paper or cardboard, a paper- or cardboard-model of this polyhedron can be built in the usual way. Does such a net really define the shape of a polyhedron uniquely? Think of a cube where one face is replaced by a four-sided pyramid with a small height. -
Mathematics of Origami
Mathematics of Origami Angela Kohlhaas Loras College February 17, 2012 Introduction Origami ori + kami, “folding paper” Tools: one uncut square of paper, mountain and valley folds Goal: create art with elegance, balance, detail Outline History Applications Foldability Design History of Origami 105 A.D.: Invention of paper in China Paper-folding begins shortly after in China, Korea, Japan 800s: Japanese develop basic models for ceremonial folding 1200s: Origami globalized throughout Japan 1682: Earliest book to describe origami 1797: How to fold 1,000 cranes published 1954: Yoshizawa’s book formalizes a notational system 1940s-1960s: Origami popularized in the U.S. and throughout the world History of Origami Mathematics 1893: Geometric exercises in paper folding by Row 1936: Origami first analyzed according to axioms by Beloch 1989-present: Huzita-Hatori axioms Flat-folding theorems: Maekawa, Kawasaki, Justin, Hull TreeMaker designed by Lang Origami sekkei – “technical origami” Rigid origami Applications from the large to very small Miura-Ori Japanese solar sail “Eyeglass” space telescope Lawrence Livermore National Laboratory Science of the small Heart stents Titanium hydride printing DNA origami Protein-folding Two broad categories Foldability (discrete, computational complexity) Given a pattern of creases, when does the folded model lie flat? Design (geometry, optimization) How much detail can added to an origami model, and how efficiently can this be done? Flat-Foldability of Crease Patterns 훗 Three criteria for 훗: Continuity, Piecewise isometry, Noncrossing 2-Colorable Under the mapping 훗, some faces are flipped while others are only translated and rotated. Maekawa-Justin Theorem At any interior vertex, the number of mountain and valley folds differ by two. -
Jitterbug ★★★✩✩ Use 12 Squares Finished Model Ratio 1.0
Jitterbug ★★★✩✩ Use 12 squares Finished model ratio 1.0 uckminister Fuller gave the name Jitterbug to this transformation. I first came across B the Jitterbug in Amy C. Edmondson’s A Fuller Explanation (1987).As I didn’t have dowels and four-way rubber connectors, I made several cuboctahedra that worked as Jitterbugs but were not very reversible. Some were from paper and others from drinking straws and elastic thread. My first unit,Triangle Unit, was partly successful as a Jitterbug.A better result came from using three units per triangular face (or one unit per vertex) which was published in 2000. This improved version is slightly harder to assemble but has a stronger lock.The sequence is pleasingly rhythmical and all steps have location points.The pleats in step 18 form the spring that make the model return to its cuboctahedral shape. Module ★★★ Make two folds in half. Do Fold the sides to the centre. Fold the bottom half behind. 1 not crease the middle for 2 Unfold the right flap. 3 the vertical fold. Fold the top right corner to This creates a 60 degree Fold the right edge to the 4 lie on the left edge whilst the 5 angle. Unfold. 6 centre. fold starts from the original centre of the square. Reproduced with permission from Tarquin Group publication Action Modular Origami (ISBN 978-1-911093-94-7) 978-1-911093-95-4 (ebook). © 2018 Tung Ken Lam. All rights reserved. 90 ° Fold the bottom half Repeat step 4 and unfold, Fold the left edge to the 7 behind. -
Folding a Paper Strip to Minimize Thickness✩
Folding a Paper Strip to Minimize Thickness✩ Erik D. Demainea, David Eppsteinb, Adam Hesterbergc, Hiro Itod, Anna Lubiwe, Ryuhei Ueharaf, Yushi Unog aComputer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, Cambridge, USA bComputer Science Department, University of California, Irvine, USA cDepartment of Mathematics, Massachusetts Institute of Technology, USA dSchool of Informatics and Engineering, University of Electro-Communications, Tokyo, Japan eDavid R. Cheriton School of Computer Science, University of Waterloo, Ontario, Canada fSchool of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, Japan gGraduate School of Science, Osaka Prefecture University, Osaka, Japan Abstract In this paper, we study how to fold a specified origami crease pattern in order to minimize the impact of paper thickness. Specifically, origami designs are often expressed by a mountain-valley pattern (plane graph of creases with relative fold orientations), but in general this specification is consistent with exponentially many possible folded states. We analyze the complexity of finding the best consistent folded state according to two metrics: minimizing the total number of layers in the folded state (so that a “flat folding” is indeed close to flat), and minimizing the total amount of paper required to execute the folding (where “thicker” creases consume more paper). We prove both problems strongly NP- complete even for 1D folding. On the other hand, we prove both problems fixed-parameter tractable in 1D with respect to the number of layers. Keywords: linkage, NP-complete, optimization problem, rigid origami. 1. Introduction Most results in computational origami design assume an idealized, zero- thickness piece of paper. This approach has been highly successful, revolution- izing artistic origami over the past few decades. -
The Geometry Junkyard: Origami
Table of Contents Table of Contents 1 Origami 2 Origami The Japanese art of paper folding is obviously geometrical in nature. Some origami masters have looked at constructing geometric figures such as regular polyhedra from paper. In the other direction, some people have begun using computers to help fold more traditional origami designs. This idea works best for tree-like structures, which can be formed by laying out the tree onto a paper square so that the vertices are well separated from each other, allowing room to fold up the remaining paper away from the tree. Bern and Hayes (SODA 1996) asked, given a pattern of creases on a square piece of paper, whether one can find a way of folding the paper along those creases to form a flat origami shape; they showed this to be NP-complete. Related theoretical questions include how many different ways a given pattern of creases can be folded, whether folding a flat polygon from a square always decreases the perimeter, and whether it is always possible to fold a square piece of paper so that it forms (a small copy of) a given flat polygon. Krystyna Burczyk's Origami Gallery - regular polyhedra. The business card Menger sponge project. Jeannine Mosely wants to build a fractal cube out of 66048 business cards. The MIT Origami Club has already made a smaller version of the same shape. Cardahedra. Business card polyhedral origami. Cranes, planes, and cuckoo clocks. Announcement for a talk on mathematical origami by Robert Lang. Crumpling paper: states of an inextensible sheet. Cut-the-knot logo. -
Generating Folding Sequences from Crease Patterns of Flat-Foldable Origami
Generating Folding Sequences from Crease Patterns of Flat-Foldable Origami Hugo A. Akitaya∗ Jun Mitaniy Yoshihiro Kanamoriy Yukio Fukuiy University of Tsukuba University of Tsukuba / JST ERATO University of Tsukuba University of Tsukuba (b) (a) Figure 1: (a) Example of a step sequence graph containing several possible ways of simplifying the input crease pattern. (b) Results obtained using the path highlighted in orange from Figure 1(a). to fold the paper into the origami design, since crease patterns show only where each crease must be made and not folding instructions. In fact, folding an origami model by its crease pattern may be a difficult task even for people experienced in origami [Lang 2012]. According to Lang, a linear sequence of small steps, such as usually (a) (b) portrayed in traditional diagrams, might not even exist and all the folds would have to be executed simultaneously. Figure 2: (a) Crease pattern and (b) folded form of the “Penguin”. We introduce a system capable of identifying if a folding sequence Design by Hideo Komatsu. Images redrawn by authors. exists using a predetermined set of folds by modeling the origami steps as graph rewriting steps. It also creates origami diagrams semi-automatically from the input of a crease pattern of a flat 1 Problem and Motivation origami. Our system provides the automatic addition of traditional symbols used in origami diagrams and 3D animations in order to The most common form to convey origami is through origami di- help people who are inexperienced in folding crease patterns as agrams, which are step-by-step sequences as shown in Figure 1b. -
Use of Origami in Mathematics Teaching: an Exemplary Activity
Asian Journal of Education and Training Vol. 6, No. 2, 284-296, 2020 ISSN(E) 2519-5387 DOI: 10.20448/journal.522.2020.62.284.296 © 2020 by the authors; licensee Asian Online Journal Publishing Group Use of Origami in Mathematics Teaching: An Exemplary Activity Davut Köğce Niğde Ömer Halisdemir University, Faculty of Education, Department of Elementary Mathematics Education Turkey. Abstract Students’ attitudes and motivations toward mathematics decrease as they intensively confront cognitive information along the process of mathematics teaching in general. One of the most important reasons for this is the lack of activities for affective and psychomotor domains in the mathematics teaching process. One way to overcome this problem is to include activities through which students can participate effectively in the teaching process. For example, activities performed using origami can be one of them in teaching the attainments covered by the field of geometry learning in mathematics curriculum. Therefore, this study was carried out to present an exemplary activity about how origami can be used when teaching mathematics in secondary schools and to identify preservice teacher opinions on this activity. The activity presented in this study was performed with 32 preservice elementary mathematics teachers attending the faculty of education at a university and taking the Mathematics Teaching with Origami course, and their opinions were taken afterwards. At the end of the application, the preservice teachers stated that such activities would have very positive contributions to students’ mathematics learning. In addition, the preservice teachers communicated and exchanged ideas with each other during the application. It is therefore recommended to use such activities in mathematics teaching. -
Paper Pentasia: an Aperiodic Surface in Modular Origami
Paper Pentasia: An Aperiodic Surface in Modular Origami a b Robert J. Lang ∗ and Barry Hayes † aLangorigami.com, Alamo, California, USA, bStanford University, Stanford, CA 2013-05-26 Origami, the Japanese art of paper-folding, has numerous connections to mathematics, but some of the most direct appear in the genre of modular origami. In modular origami, one folds many sheets into identical units (or a few types of unit), and then fits the units together into larger constructions, most often, some polyhedral form. Modular origami is a diverse and dynamic field, with many practitioners (see, e.g., [12, 3]). While most modular origami is created primarily for its artistic or decorative value, it can be used effectively in mathematics education to provide physical models of geometric forms ranging from the Platonic solids to 900-unit pentagon-hexagon-heptagon torii [5]. As mathematicians have expanded their catalog of interesting solids and surfaces, origami designers have followed not far behind, rendering mathematical forms via folding, a notable recent example being a level-3 Menger Sponge folded from 66,048 business cards by Jeannine Mosely and co-workers [10]. In some cases, the origami explorations themselves can lead to new mathematical structures and/or insights. Mosely’s developments of business-card modulars led to the discovery of a new fractal polyhedron with a novel connection to the famous Snowflake curve [11]. One of the most popular geometric mathematical objects has been the sets of aperiodic tilings developed by Roger Penrose [14, 15], which acquired new significance with the dis- covery of quasi-crystals, their three-dimensional analogs in the physical world, in 1982 by Daniel Schechtman, who was awarded the 2011 Nobel Prize in Chemistry for his discovery. -
GEOMETRIC FOLDING ALGORITHMS I
P1: FYX/FYX P2: FYX 0521857570pre CUNY758/Demaine 0 521 81095 7 February 25, 2007 7:5 GEOMETRIC FOLDING ALGORITHMS Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s but have only recently been studied in the mathemat- ical literature. Over the past decade, there has been a surge of interest in these problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this comprehensive treatment of the geometry of folding and unfolding presents hundreds of results and more than 60 unsolved “open prob- lems” to spur further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Among the results in Part I is that there is a planar linkage that can trace out any algebraic curve, even “sign your name.” Part II features the “fold-and-cut” algorithm, establishing that any straight-line drawing on paper can be folded so that the com- plete drawing can be cut out with one straight scissors cut. In Part III, readers will see that the “Latin cross” unfolding of a cube can be refolded to 23 different convex polyhedra. Aimed primarily at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from high school students to researchers. Erik D. Demaine is the Esther and Harold E. Edgerton Professor of Elec- trical Engineering and Computer Science at the Massachusetts Institute of Technology, where he joined the faculty in 2001. He is the recipient of several awards, including a MacArthur Fellowship, a Sloan Fellowship, the Harold E. -
Industrial Product Design by Using Two-Dimensional Material in the Context of Origamic Structure and Integrity
Industrial Product Design by Using Two-Dimensional Material in the Context of Origamic Structure and Integrity By Nergiz YİĞİT A Dissertation Submitted to the Graduate School in Partial Fulfillment of the Requirements for the Degree of MASTER OF INDUSTRIAL DESIGN Department: Industrial Design Major: Industrial Design İzmir Institute of Technology İzmir, Turkey July, 2004 We approve the thesis of Nergiz YİĞİT Date of Signature .................................................. 28.07.2004 Assist. Prof. Yavuz SEÇKİN Supervisor Department of Industrial Design .................................................. 28.07.2004 Assist.Prof. Dr. Önder ERKARSLAN Department of Industrial Design .................................................. 28.07.2004 Assist. Prof. Dr. A. Can ÖZCAN İzmir University of Economics, Department of Industrial Design .................................................. 28.07.2004 Assist. Prof. Yavuz SEÇKİN Head of Department ACKNOWLEDGEMENTS I would like to thank my advisor Assist. Prof. Yavuz Seçkin for his continual advice, supervision and understanding in the research and writing of this thesis. I would also like to thank Assist. Prof. Dr. A. Can Özcan, and Assist.Prof. Dr. Önder Erkarslan for their advices and supports throughout my master’s studies. I am grateful to my friends Aslı Çetin and Deniz Deniz for their invaluable friendships, and I would like to thank to Yankı Göktepe for his being. I would also like to thank my family for their patience, encouragement, care, and endless support during my whole life. ABSTRACT Throughout the history of industrial product design, there have always been attempts to shape everyday objects from a single piece of semi-finished industrial materials such as plywood, sheet metal, plastic sheet and paper-based sheet. One of the ways to form these two-dimensional materials into three-dimensional products is bending following cutting. -
Folding & Unfolding: Folding Polygons to Convex Polyhedra
FoldingFolding && Unfolding:Unfolding: FoldingFolding PolygonsPolygons toto ConvexConvex PolyhedraPolyhedra JosephJoseph O’RourkeO’Rourke SmithSmith CollegeCollege FoldingFolding andand UnfoldingUnfolding TalksTalks Linkage folding Tuesday Erik Demaine Paper folding Wednesday Erik Demaine Folding polygons into Saturday Joe O’Rourke convex 1 polyhedra Unfolding Saturday Joe O’Rourke polyhedra 2 OutlineOutline Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Folding Polygons Algorithms Examples Questions OutlineOutline11 Reconstruction of Convex Polyhedra Cauchy to Sabitov (to an Open Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm OutlineOutline22 Folding Polygons Algorithms Edge-to-Edge Foldings Gluing Trees; exponential lower bound Gluing Algorithm Examples Foldings of the Latin Cross Foldings of the Square Questions Transforming shapes? ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra graph Steinitz’s Theorem face angles edge lengths face areas Minkowski’s Theorem face normals dihedral angles inscribed/circumscribed Minkowski’sMinkowski’s TheoremTheorem (a) (b) ReconstructionReconstruction ofof ConvexConvex PolyhedraPolyhedra graph face angles Cauchy’s Theorem edge lengths face areas face normals dihedral angles inscribed/circumscribed Cauchy’sCauchy’s RigidityRigidity TheoremTheorem If two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent; in particular,