Folding Your Way to Understanding*
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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. -
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. -
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. -
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. -
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. -
Modular Origami by Trisha Kodama
ETHNOMATHEMATICS MODULAR ORIGAMI BY TRISHA KODAMA How does the appearance and form of three-dimensional shapes relate to math? How do shipping companies use volume to transport cargo in the most cost efficient way? ELEMENTARY FIFTH GRADE TIMEFRAME FIVE CLASS PERIODS (45 MIN.) STANDARD BENCHMARKS AND VALUES MATHEMATICAL PRACTICE • ‘Ike Piko‘u-Personal Connection Pathway: Students CCSS.MATH.CONTENT.5.MD.C.3 will develop a sense of pride and self-worth while contributing to the learning of our class and the other • Recognize volume as an attribute of solid figures and 5th grade class. understand concepts of volume measurement. • ‘Ike Na‘auao-Intellectual Pathway: Students nurture CCSS.MATH.CONTENT.5.MD.C.4 their curiosity of origami by learning how to fold • Measure volumes by counting unit cubes, using cubic other objects. cm, cubic in, cubic ft, and improvised units. NA HOPENA A‘O CCSS.MATH.CONTENT.5.MD.C.5 • Strengthened Sense of Responsibility: Students will • Relate volume to the operations of multiplication demonstrate commitment and concern for others and addition and solve real world and mathematical when working with their partner. They will give each problems involving volume. other helpful feedback in order to complete tasks. General Learner Outcome #2 – Community Contributor • Strengthened Sense of Excellence: Students will learn General Learner Outcome #5 – Complex Thinker that precise paper folding will lead to producing quality General Learner Outcome #6-Effective and Ethical User work, a perfect Sonobe cube. of Technology • Strengthened Sense of Aloha: Students will show NA HONUA MAULI OLA PATHWAYS Aloha to students in the other 5th grade class when presenting their slideshow and teaching them how to • ‘Ike Pilina-Relationship Pathway: Students share in the fold a Sonobe unit and put together a Sonobe cube. -
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. -
Making an Origami Sonobe Octahedron
Making an Origami Sonobe Octahedron Now that you have made a Sonobe unit and a Sonobe cube, you are ready to constuct a more complicated polyhedron. (We highly recommend building a cube before attempting to construct other polyhedra.) This handout will show you how to assemble twelve units into a small triakis octahedron. You can find instructions for making some other polyhedra online. To make the triakis octahedron, you can use any colour combination you like. In this example we use twelve units in three colours (four units of each colour). 1. Fold twelve Sonobe units (see the Making an Origami Sonobe Unit handout for instructions). Make an extra fold in each Sonobe unit by placing the unit with the side containing the X facing up and folding along the dotted line shown in the leftmost image below. 2. Assemble three units (one of each colour) as described in Step 2 of the Making an Origami Sonobe Cube handout to create a single corner, or pyramid, with three flaps (below, left). We are going to make more pyramids using these three flaps. Each new pyramid will also contain all three colours. 3. To make a second pyramid of three colours, choose a flap. We chose the purple flap, so we will add one new pink unit and one new blue unit to our construction. To add the new pink unit, insert a flap of the pink unit into the pocket of the chosen purple unit (above, centre). To add the new blue unit, insert a flap of the blue unit into the pocket of the new pink unit (shown by the white arrow above). -
Sonobe Variations Ashley Shimabuku
Sonobe Variations Ashley Shimabuku 1 Introduction This origami activity is a variation on the Sonobe unit activity. The students loved the original origami activity and they like how these variations are just as easy to build but look more complicated. The same kinds of questions can be asked about these Sonobe units. There is a worksheet included in the Paper Folding and Polyhedra paper. The polyhedra are built in the same fashion. So the stellated octahedron and stellated icosahedron can also be made using these variations. Additionally, the original Sonobe unit and the variations can be mixed and matched. There are two variations presented below the \bar" Sonobe and the \zig zag" Sonobe. There is a video with instructions [1] however it can be hard to follow. Instructions with pictures are given below. 2 Materials Required 1. Origami paper (two sided) 2. Diagram on how to make a Sonobe unit or variations on the Sonobe unit 3. Example of a constructed cube 3 Lesson Plan This activity can be done in a class period or less. If the students are familiar with the Sonobe unit it may take less time as the polyhedra are constructed in the same way. Students seem to like the variations slightly better than the original Sonobe as the variations look more complicated. The same counting questions can be asked and discussed with the students. Two sided origami paper is recommended as the color of backside of the paper is what makes the \bars" or \zig zag" pattern. 1 4 Instructions for Bar Variation (a) Step 1 Fold paper down the (b) Step 2 Fold edges to center (c) Step 3 Fold edges outward middle and unfold line and unfold (d) Step 4 Fold top left corner (e) Step 5 Tuck upper left corner (f) Step 6 Turn over and fold down to right edge, then fold top under flap on opposite side and bottom point straight up right corner down to left edge repeat for lower right corner (g) Finished 2 5 References 1. -
Marvelous Modular Origami
www.ATIBOOK.ir Marvelous Modular Origami www.ATIBOOK.ir Mukerji_book.indd 1 8/13/2010 4:44:46 PM Jasmine Dodecahedron 1 (top) and 3 (bottom). (See pages 50 and 54.) www.ATIBOOK.ir Mukerji_book.indd 2 8/13/2010 4:44:49 PM Marvelous Modular Origami Meenakshi Mukerji A K Peters, Ltd. Natick, Massachusetts www.ATIBOOK.ir Mukerji_book.indd 3 8/13/2010 4:44:49 PM Editorial, Sales, and Customer Service Office A K Peters, Ltd. 5 Commonwealth Road, Suite 2C Natick, MA 01760 www.akpeters.com Copyright © 2007 by A K Peters, Ltd. All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photo- copying, recording, or by any information storage and retrieval system, without written permission from the copyright owner. Library of Congress Cataloging-in-Publication Data Mukerji, Meenakshi, 1962– Marvelous modular origami / Meenakshi Mukerji. p. cm. Includes bibliographical references. ISBN 978-1-56881-316-5 (alk. paper) 1. Origami. I. Title. TT870.M82 2007 736΄.982--dc22 2006052457 ISBN-10 1-56881-316-3 Cover Photographs Front cover: Poinsettia Floral Ball. Back cover: Poinsettia Floral Ball (top) and Cosmos Ball Variation (bottom). Printed in India 14 13 12 11 10 10 9 8 7 6 5 4 3 2 www.ATIBOOK.ir Mukerji_book.indd 4 8/13/2010 4:44:50 PM To all who inspired me and to my parents www.ATIBOOK.ir Mukerji_book.indd 5 8/13/2010 4:44:50 PM www.ATIBOOK.ir Contents Preface ix Acknowledgments x Photo Credits x Platonic & Archimedean Solids xi Origami Basics xii