Paper-Fold Physics: Newton’S Three Laws of Motion Lisa (Yuk Kuen) Yau Revised Date: Monday, July 30, 2018
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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. -
3.2.1 Crimpable Sequences [6]
JAIST Repository https://dspace.jaist.ac.jp/ Title 折り畳み可能な単頂点展開図に関する研究 Author(s) 大内, 康治 Citation Issue Date 2020-03-25 Type Thesis or Dissertation Text version ETD URL http://hdl.handle.net/10119/16649 Rights Description Supervisor:上原 隆平, 先端科学技術研究科, 博士 Japan Advanced Institute of Science and Technology Doctoral thesis Research on Flat-Foldable Single-Vertex Crease Patterns by Koji Ouchi Supervisor: Ryuhei Uehara Graduate School of Advanced Science and Technology Japan Advanced Institute of Science and Technology [Information Science] March, 2020 Abstract This paper aims to help origami designers by providing methods and knowledge related to a simple origami structure called flat-foldable single-vertex crease pattern.A crease pattern is the set of all given creases. A crease is a line on a sheet of paper, which can be labeled as “mountain” or “valley”. Such labeling is called mountain-valley assignment, or MV assignment. MV-assigned crease pattern denotes a crease pattern with an MV assignment. A sheet of paper with an MV-assigned crease pattern is flat-foldable if it can be transformed from the completely unfolded state into the flat state that all creases are completely folded without penetration. In applications, a material is often desired to be flat-foldable in order to store the material in a compact room. A single-vertex crease pattern (SVCP for short) is a crease pattern whose all creases are incident to the center of the sheet of paper. A deep insight of SVCP must contribute to development of both basics and applications of origami because SVCPs are basic units that form an origami structure. -
Pleat Folding, 6.849 Fall 2010
Demaine, Demaine, Lubiw Courtesy of Erik D. Demaine, Martin L. Demaine, and Anna Lubiw. Used with permission. 1999 1 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Albers at Bauhaus, 1927–1928] 2 Circular Variation from Bauhaus [Albers at Bauhaus, 1927–1928] 3 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 4 Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. 5 “Black Hexagon” Demaine, Demaine, Fizel 2006 Courtesy of Erik Demaine, Martin Demaine, and Jenna Fizel. Used with permission. 6 Hyparhedra: Platonic Solids [Demaine, Demaine, Lubiw 1999] 7 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 8 “Computational Origami” Erik & Martin Demaine MoMA, 2008– Elephant hide paper ~9”x15”x7” Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Computational/. 9 Peel Gallery, Houston Nov. 2009 Demaine & Demaine 2009 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Limit/. 10 “Natural Cycles” Erik & Martin Demaine JMM Exhibition of Mathematical Art, San Francisco, 2010 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/NaturalCycles/. 11 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/BlindGlass/. Demaine & Demaine 2010 12 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 13 θ = 30° n = 16 Courtesy of Erik D. -
How to Make an Origami Octahedron Ball
How to make an Origami Octahedron Ball Introduction: These instructions will teach you how to make an origami octahedron. The ball is constructed by assembling 12 folded units together. A unit This is called a Sonobe unit, named after the person who invented them. The process of using identical units and assembling them in a pattern to create large, complex, beautiful structures is considered modular origami. This type of origami requires multiple sheets of paper to make a model. By varying the number of units, you can make several other shapes. A cube requires 6 units and an icosahedron requires 30. Time Frame: 30 minutes Materials: 2- 8.5”x11” sheets of paper (Multi-colored ball: use 3 different colored sheets of paper) Scissors Ruler (measures inches) Pencil Preparation: Make sure you have enough space and a flat surface to fold the origami. Procedure: Step 1. Cut 12 – 3”x3” square pieces of paper Note: The size of the squares depends on your preference of how large you want the ball to be. These instructions create a ball that fits in your hand. To save time, origami squares (4”x4”) can be used and step 1 skipped. 1.1. Draw 6 squares that are 3”x3” on a sheet of paper with the pencil and ruler. Note: For a multi-color effect, as depicted in these instructions, use 3 different colors of paper and cut 4 squares from each. 1.2.Stack the second sheet of paper behind the first. 1.3. Cut out 12 squares of paper with the scissors. -
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. -
Constructing Points Through Folding and Intersection
CONSTRUCTING POINTS THROUGH FOLDING AND INTERSECTION The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Butler, Steve, Erik Demaine, Ron Graham and Tomohiro Tachi. "Constructing points through folding and intersection." International Journal of Computational Geometry & Applications, Vol. 23 (01) 2016: 49-64. As Published 10.1142/S0218195913500039 Publisher World Scientific Pub Co Pte Lt Version Author's final manuscript Citable link https://hdl.handle.net/1721.1/121356 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Constructing points through folding and intersection Steve Butler∗ Erik Demaine† Ron Graham‡ Tomohiro Tachi§ Abstract Fix an n 3. Consider the following two operations: given a line with a specified point on the line we≥ can construct a new line through the point which forms an angle with the new line which is a multiple of π/n (folding); and given two lines we can construct the point where they cross (intersection). Starting with the line y = 0 and the points (0, 0) and (1, 0) we determine which points in the plane can be constructed using only these two operations for n =3, 4, 5, 6, 8, 10, 12, 24 and also consider the problem of the minimum number of steps it takes to construct such a point. 1 Introduction If an origami model is laid flat the piece of paper will retain a memory of the folds that went into the construction of the model as creases (or lines) in the paper. -
Sonobe Origami Units
Sonobe Origami Units Crease paper down the middle Fold edges to center line and Fold top right and bottom left Fold triangles down to make and unfold unfold corners | do not let the sharper triangles | do not triangles cross crease lines cross the crease lines Fold along vertical crease lines Fold top left corner down to Fold bottom right corner up to Undo the last two folds right edge left edge Tuck upper left corner under Turn the paper over, crease Corners of one module ¯t into along dashed lines as shown to flap on opposite side and pockets of another A cube requires six modules repeat for lower right corner complete the module Sonobe modules also make many other polyhedra Where's the Math in Origami? Origami may not seem like it involves very much mathematics. Yes, origami involves symmetry. If we build a polyhedron then, sure, we encounter a shape from geometry. Is that as far as it goes? Do any interesting mathematical questions arise from the process of folding paper? Is there any deep mathematics in origami? Is the mathematics behind origami useful for anything other than making pretty decorations? People who spend time folding paper often ask themselves questions that are ultimately mathematical in nature. Is there a simpler procedure for folding a certain ¯gure? Where on the original square paper do the wings of a crane come from? What size paper should I use to make a chair to sit at the origami table I already made? Is it possible to make an origami beetle that has six legs and two antennae from a single square sheet of paper? Is there a precise procedure for folding a paper into ¯ve equal strips? In the last few decades, folders inspired by questions like these have revolutionized origami by bringing mathematical techniques to their art. -
Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi
HEXAFLEXAGONS, PROBABILITY PARADOXES, AND THE TOWER OF HANOI For 25 of his 90 years, Martin Gard- ner wrote “Mathematical Games and Recreations,” a monthly column for Scientific American magazine. These columns have inspired hundreds of thousands of readers to delve more deeply into the large world of math- ematics. He has also made signifi- cant contributions to magic, philos- ophy, debunking pseudoscience, and children’s literature. He has produced more than 60 books, including many best sellers, most of which are still in print. His Annotated Alice has sold more than a million copies. He continues to write a regular column for the Skeptical Inquirer magazine. (The photograph is of the author at the time of the first edition.) THE NEW MARTIN GARDNER MATHEMATICAL LIBRARY Editorial Board Donald J. Albers, Menlo College Gerald L. Alexanderson, Santa Clara University John H. Conway, F.R. S., Princeton University Richard K. Guy, University of Calgary Harold R. Jacobs Donald E. Knuth, Stanford University Peter L. Renz From 1957 through 1986 Martin Gardner wrote the “Mathematical Games” columns for Scientific American that are the basis for these books. Scientific American editor Dennis Flanagan noted that this column contributed substantially to the success of the magazine. The exchanges between Martin Gardner and his readers gave life to these columns and books. These exchanges have continued and the impact of the columns and books has grown. These new editions give Martin Gardner the chance to bring readers up to date on newer twists on old puzzles and games, on new explanations and proofs, and on links to recent developments and discoveries. -
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. -
Origami Design Secrets Reveals the Underlying Concepts of Origami and How to Create Original Origami Designs
SECOND EDITION PRAISE FOR THE FIRST EDITION “Lang chose to strike a balance between a book that describes origami design algorithmically and one that appeals to the origami community … For mathematicians and origamists alike, Lang’s expository approach introduces the reader to technical aspects of folding and the mathematical models with clarity and good humor … highly recommended for mathematicians and students alike who want to view, explore, wrestle with open problems in, or even try their own hand at the complexity of origami model design.” —Thomas C. Hull, The Mathematical Intelligencer “Nothing like this has ever been attempted before; finally, the secrets of an origami master are revealed! It feels like Lang has taken you on as an apprentice as he teaches you his techniques, stepping you through examples of real origami designs and their development.” —Erik D. Demaine, Massachusetts Institute of Technology ORIGAMI “This magisterial work, splendidly produced, covers all aspects of the art and science.” —SIAM Book Review The magnum opus of one of the world’s leading origami artists, the second DESIGN edition of Origami Design Secrets reveals the underlying concepts of origami and how to create original origami designs. Containing step-by-step instructions for 26 models, this book is not just an origami cookbook or list of instructions—it introduces SECRETS the fundamental building blocks of origami, building up to advanced methods such as the combination of uniaxial bases, the circle/river method, and tree theory. With corrections and improved Mathematical Methods illustrations, this new expanded edition also for an Ancient Art covers uniaxial box pleating, introduces the new design technique of hex pleating, and describes methods of generalizing polygon packing to arbitrary angles. -
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. -
The Miura-Ori Opened out Like a Fan INTERNATIONAL SOCIETY for the INTERDISCIPLINARY STUDY of SYMMETRY (ISIS-SYMMETRY)
The Quarterly of the Editors: International Society for the GyiSrgy Darvas and D~nes Nag¥ interdisciplinary Study of Symmetry (ISIS-Symmetry) Volume 5, Number 2, 1994 The Miura-ori opened out like a fan INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) President ASIA D~nes Nagy, lnslltute of Apphed Physics, University of China. t~R. Da-Fu Ding, Shangha~ Institute of Biochemistry. Tsukuba, Tsukuba Soence C~ty 305, Japan Academia Stoma, 320 Yue-Yang Road, (on leave from Eotvos Lot’find Umve~ty, Budapest, Hungary) Shanghai 200031, PR China IGeometry and Crystallography, H~story of Science and [Theoreucal B~ology] Tecbnology, Lmgmsucs] Le~Xiao Yu, Department of Fine Arts. Nanjmg Normal Umvers~ty, Nanjmg 210024, P.R China Honorary Presidents }Free Art, Folk Art, Calhgraphy] Konstantin V. Frolov (Moscow) and lndta. Kirti Trivedi, Industrial Design Cenlre, lndmn Maval Ne’eman (TeI-Avw) Institute of Technology, Powa~, Bombay 400076, India lDes~gn, lndmn Art] Vice-President Israel. Hanan Bruen, School of Education, Arthur L. Loeb, Carpenter Center for the V~sual Arts, Umvers~ty of Hallo, Mount Carmel, Haffa 31999, Israel Harvard Umverslty. Cambridge, MA 02138, [Educanon] U S A. [Crystallography, Chemical Physics, Visual Art~, Jim Rosen, School of Physics and Astronomy, Choreography, Music} TeI-Av~v Umvers~ty, Ramat-Avtv, Tel-Av~v 69978. Israel and [Theoretical Physms] Sergei V Petukhov, Instnut mashmovedemya RAN (Mechamcal Engineering Research Institute, Russian, Japan. Yasushi Kajfl~awa, Synergel~cs Institute. Academy of Scmnces 101830 Moskva, ul Griboedova 4, Russia (also Head of the Russian Branch Office of the Society) 206 Nakammurahara, Odawara 256, Japan }Design, Geometry] }B~omechanlcs, B~ontcs, Informauon Mechamcs] Koichtro Mat~uno, Department of BioEngineering.