1st Edition

Origami 5 Fifth International Meeting of Origami Science, Mathematics, and Education

Edited By Patsy Wang-Iverson, Robert J. Lang, Mark YIM Copyright 2011
    668 Pages
    by A K Peters/CRC Press

    676 Pages
    by A K Peters/CRC Press

    Origami5 continues in the excellent tradition of its four previous incarnations, documenting work presented at an extraordinary series of meetings that explored the connections between origami, mathematics, science, technology, education, and other academic fields.

    The fifth such meeting, 5OSME (July 13–17, 2010, Singapore Management University) followed the precedent previous meetings to explore the interdisciplinary connections between origami and the real world. This book begins with a section on origami history, art, and design. It is followed by sections on origami in education and origami science, engineering, and technology, and culminates with a section on origami mathematics—the pairing that inspired the original meeting.

    Within this one volume, you will find a broad selection of historical information, artists’ descriptions of their processes, various perspectives and approaches to the use of origami in education, mathematical tools for origami design, applications of folding in engineering and technology, as well as original and cutting-edge research on the mathematical underpinnings of origami.

    Origami History, Art, and Design

    History of Origami in the East and the West before Interfusion, Koshiro Hatori
    Introduction
    Origin of Origami: Many Misunderstandings and Some Suppositions
    The East and the West: Different Styles, Different Traditions
    Conclusion
    Bibliography

    Deictic Properties of Origami Technical Terms and Translatability, Koichi Tateishi
    Introduction
    Previous Studies on Origami Terms
    Theoretical Backgrounds
    Maze of (Un)Translatability
    Referentiality/Deictic versus Similarity: The Role of a Japanese Verbal Noun Ori
    Why Don’t the Japanese Use Verbs?
    Further Considerations
    Conclusion
    Bibliography

    Betsy Ross Revisited: General Fold and One-Cut Regular and Star Polygons, Arnold Tubis and Crystal Elaine Mills
    Introduction
    Historical Sources for the Story
    Folding and One-Cut of the Pattern for Stars Artifact
    Generalizing the Betsy Ross Method to Fold and One-Cut Any Regular and Star Polygons
    Discussion
    Bibliography

    Reconstructing David Huffman’s Legacy in Curved-Crease Folding, Erik D. Demaine, Martin L. Demaine, and Duks Koschitz
    Introduction
    Approach
    Reconstructions
    Conclusion
    Bibliography

    Simulation of Nonzero Gaussian Curvature in Origami by Curved-Crease Couplets, Cheng Chit Leong
    Introduction
    Geometry of a Curved Surface
    Representation of Curved Surfaces by Paper Folding
    Ruled Surfaces
    Radial Formed Rotational Symmetric Models
    Cylindrical Formed Rotational Symmetric Models
    Applications of Couplets to Folding Animal Models
    Conclusion
    Bibliography

    Compression and Rotational Limitations of Curved Corrugations, Christine E. Edison
    Introduction
    Method
    Discussion of Results
    Conclusions
    Bibliography

    Polygon Symmetry Systems, Andrew Hudson
    Introduction
    Polygon Construction
    Unified Algorithm for Polygon Construction
    Polygonal Grids and Their Properties
    Symmetrical Twist-Fold Constructions
    Generalizing Folding Algorithms to Different
    Polygons
    Conclusion
    Bibliography

    New Collaboration on Modular Origami and LED, Miyuki Kawamura and Hiroyuki Moriwaki
    Introduction
    AKARI-ORIGAMI: Some Modular Works
    Workshop
    Future of AKARI-ORIGAMI
    Bibliography

    Using the Snapology Technique to Teach Convex Polyhedra, Faye Goldman
    Introduction
    Polyhedra Review
    Snapology Technique
    Brief Introduction to General Snapology
    Observations
    Conclusions
    Bibliography

    A Systematic Approach to Twirl Design, Krystyna Burczyk and Wojciech Burczyk
    Introduction
    The First Attempt: Small (Change) Is Beautiful
    A Square Is a Rectangle: Metamorphosis
    Planar Decorative Area: From Octahedron to Cube, Mosaic Twirls
    Shapes and Lengths: Different Spirals
    Minimal Folding: No Crease Origami
    Consensus Building: Spirals Work Together with Flaps-and-Pockets, Macro-modules
    Conclusion
    Bibliography

    Oribotics: The Future Unfolds, Matthew Gardiner
    Introduction
    Industrial Evolution of Oribotics
    Crease Pattern and Mechanical Design
    Paper versus Fabric
    Interaction Design
    Conclusion and Further Work
    Bibliography

    Origami in Education

    Origametria and the van Hiele Theory of Teaching Geometry, Miri Golan
    Introduction
    The van Hiele Theory of Geometric Teaching
    Origami and the Van Hiele Theory
    Time of Learning
    Gradually Building Knowledge and Concepts
    Using Origametria to Eliminate Misconceptions
    Origametria and van Hiele: An Example from the Classroom
    Conclusion: The Benefits of Using Origametria in the van Hiele System
    Bibliography

    Student-Teachers Introduce Origami in Kindergarten and Primary Schools, Maria Lluïsa Fiol, Neus Dasquens, and Montserrat Prat
    Introduction
    Objectives
    Review of the Literature
    Problem Statements
    Data Collection, Analysis, and Discussion
    Conclusion and Next Steps
    Bibliography

    Narratives of Success: Teaching Origami in Low-Income Urban Communities, Christine Edison
    Introduction
    Origami in Mathematics
    Origami in Art Class
    Origami in Elementary School
    Discussion
    Bibliography

    Origami and Spatial Thinking of College-Age Students, Norma Boakes
    Introduction
    Design and Purpose
    Results
    Conclusion
    Bibliography

    Close Observation and Reverse Engineering of Origami Models, James Morrow and Charlene Morrow
    Introduction
    Rationale/Goals
    Origins and Context
    An Approach to Close Observation and Reverse Engineering
    An Example
    Close Observation and Reverse Engineering in Learning and Problem Solving
    Our Observations of CORE
    Challenges, a Question, and Next Steps
    Conclusion
    Bibliography

    Origami and Learning Mathematics, Sue Pope and Tung Ken Lam
    Introduction
    Using Origami in Mathematics Lessons
    Using Origami to Teach Conceptually Demanding Mathematics
    Conclusions
    Bibliography

    Hands-On Geometry with Origami, Michael J. Winckler, Kathrin D. Wolf, and Hans-Georg Bock
    Introduction
    New Trends in Teaching
    Design of the Coursework
    Evaluation and Lessons Learned

    My Favorite Origamics Lessons on the Volume of Solids, Shi-Pui Kwan
    Introduction
    Exploration: Origami Masu Cubic Box
    Exploration: Origami CK-Octahedron
    Exploration: The Building Block of CK and KC
    Exploration: Origami KC-Heptahedron
    Exploration: Tessellating Solids
    Origami, Science, Mathematics, and Education
    Conclusion
    Bibliography

    Origami Science, Engineering, and Technology

    Rigid-Foldable Thick Origami, Tomohiro Tachi
    Introduction
    Problem Description
    Proposed Method
    Application for Designs
    Conclusion
    Bibliography

    Folding a Patterned Cylinder by Rigid Origami, Kunfeng Wang and Yan Chen
    Introduction
    Kinematics of SphericalR Linkage and Its One DoF Assembly
    Rigid Origami Patterns to Form Cylindrical Structures
    Conclusions and Discussion
    Bibliography

    The Origami Crash Box, Jiayao Ma and Zhong You
    Introduction
    Description of the Origami Pattern Design
    Finite Element Modeling
    Results and Discussion
    Conclusion
    Bibliography

    Origami Folding: A Structural Engineering Approach, Mark Schenk and Simon D. Guest
    Introduction
    Folded Textured Sheets
    Mechanical Modeling Method
    Conclusion
    Bibliography

    Designing Technical Tessellations, Yves Klett and Klaus Drechsler
    Introduction: Paper as Technical Material
    Lightweight Construction with Paper
    Modular Isometric Origami
    Design Strategies
    Not So Serious—Some Fun
    Conclusion
    Bibliography

    A Simulator for Origami-Inspired Self-Reconfigurable Robots, Steven Gray, Nathan J. Zeichner, Mark Yim, and Vijay Kumar
    Introduction
    Foldable Programmable Matter
    Representing Programmable Matter
    Editor Implementation and Usage
    PhysX Simulator and Integration
    Conclusion
    Bibliography

    A CAD System for Diagramming Origami with Prediction of Folding Processes, Naoya Tsuruta, Jun Mitani, Yoshihiro Kanamori, and Yukio Fukui
    Introduction
    Related Work
    Our Proposed System
    Results and Discussion
    Conclusion and Future Work
    Bibliography

    Development of an Intuitive Algorithm for Diagramming and 3D Animated…, Hugo Akitaya, Matheus Ribeiro, Carla Koike, and Jose Ralha
    Introduction
    Computational Origami
    Basic Definitions on Technical Origami
    Turning a CP into a Folding Sequence
    Applicability Scope
    Example
    Software Implementation
    Conclusion
    Bibliography

    Hands-Free Microscale Origami, Noy Bassik, George M. Stern, Alla Brafman, Nana Y. Atuobi, and David H. Gracias
    Introduction
    Stress-Based Microscale Folding
    Miniaturized Microscale Origami Structures:
    Unidirectional Folding
    Bidirectional Microscale Folding Using
    Thin Metal Films
    Conclusion and Future Possibilities
    Bibliography

    Foldable Parylene Origami Sheets Covered with Cells: Toward Applications…, Kaori Kuribayashi-Shigetomi and Shoji Takeuchi
    Introduction
    Materials and Methods
    Results and Discussion
    Conclusions
    Bibliography

    Mathematics of Origami

    Introduction to the Study of Tape Knots, Jun Maekawa
    Introduction
    Why a Knot in a Tape Forms a Regular Pentagon
    Regular Odd-Sided Polygonal Knots
    Regular Even-Sided Polygonal Knots
    Regular Decagonal Knots
    Stability of Knots
    Nonregular Polygonal Knots
    Conclusion and Further Research
    Bibliography

    Universal Hinge Patterns for Folding Orthogonal Shapes, Nadia M. Benbernou, Erik D. Demaine, Martin L. Demaine, and Aviv Ovadya
    Introduction
    Definitions
    Cube Gadgets
    Folding Polycubes
    Implementation
    Rigid Foldability and Self-Folding Sheets
    Bibliography

    A General Method of Drawing Biplanar Crease Patterns, Cheng Herng Yi
    Introduction
    Mathematical Conventions
    Defining the Biplanar
    Drawing the Net
    Drawing the Crease Pattern
    Conclusion
    Bibliography

    A Design Method for Axisymmetric Curved Origami with Triangular Prism Protrusions, Jun Mitani
    Introduction
    Shape of the TargetD Origami
    Designing the Crease Pattern
    Examples and Discussions
    Conclusion
    Bibliography

    Folding Any Orthogonal Maze, Erik D. Demaine, Martin L. Demaine, and Jason Ku
    Introduction
    Algorithm
    Bibliography

    Every Spider Web Has a Simple Flat Twist Tessellation, Robert J. Lang and Alex Bateman
    Introduction
    Shrink and Rotate
    Nonregular Polygons
    Maxwell’s Reciprocal Figures
    Bibliography

    Flat-Unfoldability and Woven Origami Tessellations, Robert J. Lang
    Introduction
    Woven Tessellations
    Simple Woven Patterns
    Flat-Unfoldability
    Parameterizing the Woven Tessellation
    Conclusion
    Bibliography

    Degenerative Coordinates in 22.5° Grid System, Tomohiro Tachi and Erik D. Demaine
    Introduction
    Model
    Construction
    Degeneracy
    Conclusion
    Bibliography

    Two Folding Constructions, Robert Orndorff
    Introduction
    Method A: Descartes’ Construction
    Method B: My Construction
    Conclusion
    Bibliography

    Variations on a Theorem of Haga, Emma Frigerio
    Introduction
    Haga’s Theorem
    Variation
    Variation
    A Comparison
    Variations on the Variations
    Conclusion
    Bibliography

    Precise Division of Rectangular Paper into an Odd Number of Equal Parts. . ., Kazuo Haga
    Introduction
    Preparation
    Trisection
    Five-Section
    Seven-Section
    Pendulum Symmetry
    Nine-section
    Individual Line for Higher-Number Sections
    Conclusions
    Bibliography

    The Speed of Origami Constructions Versus Other Construction Tools, Eulália Tramuns
    Introduction
    Geometric Tools
    Constructions and Measures
    Optimal Constructions
    Conclusions
    Bibliography

    A Note on Operations of Spherical Origami Construction, Toshikazu Kawasaki
    Introduction
    The Eight Operations of Planar Origami
    Construction
    Notation
    Spherical Origami
    Operations of Spherical Origami Construction
    Conclusion
    Bibliography

    Origami Alignments and Constructions in the Hyperbolic Plane, Roger C. Alperin
    Introduction
    Basic Alignments and Folds
    Relations between the Alignment Axioms
    Trigonometry and More Folding in H2
    The Non-Euclidean Parabola
    H6
    Bibliography

    A Combinatorial Definition of 1D Flat-Folding, Hidefumi Kawasaki
    Introduction
    Flat-Foldable 3D Origami
    Mingling
    Proof of the Flat-Foldability Theorem
    Bibliography85

    Stamp Foldings with a Given Mountain-Valley Assignment, Ryuhei Uehara
    Introduction
    Preliminaries
    Universality of the Simple Folding Model
    The Number of Folded States
    Concluding Remarks
    Bibliography

    Flat Vertex Fold Sequences, Thomas C. Hull and Eric Chang
    Introduction
    The Basics of Flat Vertex Folds
    Flat Vertex Fold Sequences
    Conclusion
    Bibliography

    Circle Packing for Origami Design Is Hard, Erik D. Demaine, Sándor P. Fekete, and Robert J. Lang
    Introduction
    Circle-River Design
    Packing and Complexity
    Symmetric-Pockets
    Triangular Paper
    Rectangular Paper
    Square Paper
    Filling Gaps
    Encoding the Input
    A Positive Result
    Conclusions
    Bibliography
    Contributors

    Biography

    Patsy Wang-Iverson is Vice President for Special Projects at the Gabriella and Paul Rosenbaum Foundation. Introduced to origami as a child by her mother, her personal interest in origami merged in recent years with her work in helping to improve students' interest in and success with mathematics. She co-organized, with Eileen Tan and Benjamin Tan, the 2010 Fifth International Meeting on Origami in Science, Mathematics and Education (5OSME) plus a Folding Convention (PLUS!) at the Singapore Management University in Singapore. Robert J. Lang has been an avid student of origami for some forty years and is now recognized as one of the world's leading masters of the art. He is one of the pioneers of the cross-disciplinary marriage of origami with mathematics and organized the 2006 Fourth International Meeting on Origami in Science, Mathematics, and Education at Caltech. He has consulted on applications of origami to medical devices, air-bag design, and space telescopes, is the author or co-author of twelve books and numerous articles on origami and lectures widely on the connections between origami, mathematics, science, and technology.Mark Yim is a professor of mechanical engineering at the University of Pennsylvania. Previously, he was a Principal Scientist at the Palo Alto Research Center (formerly Xerox PARC). His group studies modular self-reconfigurable robots and has demonstrated robots that can transform into different shapes, jump, ride tricycles, climb stairs, poles and fences, manipulate objects and reassemble themselves. Collaborative work with researchers at Harvard, MIT and Berkeley include robotic self-folding origami. He has authored over 100 journal and conference papers and over 40 patents on topics ranging from robotics and videogame feedback devices to education and robotic performance art.

    I will readily admit I became engaged with, even engrossed in, this seminal volume on the multifaceted aspects of the art, history, science, and engineering of origami in almost four dozen short papers astutely organized, really threaded through, four main sections. … There is plenty of mathematics learning and teaching engagingly embedded within the covers. … There’s something for virtually everyone, students and teachers alike, those mathematically or artistically inclined or interested, and those not so much. … novices and the origami-ignorant like me can find starting points to commence their own personalized explorations of this fascinating field. So, pick up this volume and begin unfolding it to reveal all its intricacies. May you find your own treasure ship or sacred crane among its pages!
    —John E. Hammett III, Professor of Mathematics, Saint Peter’s University, and President, New Jersey Association of Mathematics Teacher Educators

    When I first visited Japan in 1983 I was curious to learn how origami was taught and developed in Japan. … My search for a connection of origami with mathematics was not successful at that time. I wished that I could have seen a book or even some articles as they are published now in this book … Until recently, the scientific view of origami was rather underdeveloped and [these] proceedings might change this.
    —Wolfgang Polasek, International Statistical Review, 2012

    Origami 5 continues in the excellent tradition of its four previous incarnations, documenting work presented at an extraordinary series of international meetings that explored the connections between origami, mathematics, science, technology, education, and other academic fields. … Within this one volume, you will find a broad selection of historical information, artists’ descriptions of their processes, various perspectives and approaches to the use of origami in education, mathematical tools for origami design, applications of folding in engineering and technology, as well as original and cutting-edge research on the underpinnings of origami.
    L’Enseignement Mathematique (2) 57, 2011

    … fascinating … With Origami 5 you can make everything from an origami angel to a full-sized baseball cap and learn all the associated mathematics along the way.
    —Kay Gilliland, NCSM Newsletter, Winter 2011

    Praise for the Previous Volume:
    Wow! Fantastic book! It will create new folds in your brain whether you are an artist, scientist, inventor, educator, or simply like to be amazed. The balance between mathematical theory and manipulative practice, and between artistic and educational applications makes this a book for everyone. I look forward to using this book personally and professionally.
    —Robert Root-Bernstein, Ph.D., co-author of Sparks of Genius: The Thirteen Thinking Tools of the World's Most Creative People