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

Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education
	Origami5: Fifth International Meeting of Origami Science, Mathematics, and Education
	Contents
	Preface
	Acknowledgments
	Part I Origami History, Art, and Design
		History of Origami in the East and the West before Interfusion
			1 Introduction
			2 Origin of Origami: Many Misunderstandings and Some Suppositions
			3 The East and the West: Different Styles, Different Traditions
			4 Conclusion
			Bibliography
		Deictic Properties of Origami Technical Terms and Translatability: Cross-Linguistic Differences between English and Japanese
			1 Introduction
			2 Previous Studies on Origami Terms
			3 Theoretical Backgrounds
			4 Maze of (Un)Translatability
				4.1 Translatable but Too Generic
				4.2 Translatable but with Significant Differences
				4.3 Directly Untranslatable Cases
				4.4 Lost in Translation
			5 Referentiality/Deictic versus Similarity: The Roleof a Japanese Verbal Noun Ori
			6 Why Don’t the Japanese Use Verbs?
			7 Further Considerations
			8 Conclusion
			Bibliography
		Betsy Ross Revisited: General Fold and One-Cut Regular and Star Polygons
			1 Introduction
			2 Historical Sources for the Story
			3 Replicating the Pattern for Stars Artifact
			4 Generalizing the Betsy Ross Method to Fold and One- Cut Any Regular or Star Polygon
			5 Discussion
			Bibliography
		Reconstructing David Huffman’s Legacy in Curved-Crease Folding
			1 Introduction
			2 Approach
				2.1 Folding Methods
				2.2 Reconstruction Methods
			3 Reconstructions
				3.1 Degree- 1 and - 2 Vertices
				3.2 Inflated Vertices
				3.3 Tessellations
				3.4 Cones
				3.5 Complex Shapes
			4 Conclusion
			Bibliography
		Simulation of Nonzero Gaussian Curvature in Origami by Curved-Crease Couplets
			1 Introduction
			2 Geometry of a Curved Surface
			3 Representation of Curved Surfaces by Paper Folding
			4 Ruled Surfaces
			5 Radial Formed Rotational Symmetric Models
				5.1 Folding Radial Formed Rotational Symmetric Models from a Crease Pattern
				5.2 Folding Radial Formed Rotational Symmetric Models without Crease Patterns
			6 Cylindrical Formed Rotational Symmetric Models
			7 Applications of Couplets to Folding Animal Models
			8 Conclusion
			Bibliography
		Compression and Rotational Limitations of Curved Corrugations
			1 Introduction
			2 Method
			3 Discussion of Results
			4 Conclusions
			Bibliography
		Polygon Symmetry Systems
			1 Introduction
			2 Polygon Construction
			3 Unified Algorithm for Polygon Construction
			4 Polygonal Grids and Their Properties
			5 Symmetrical Twist-Fold Constructions
			6 Generalizing Folding Algorithms to DifferentPolygons
			7 Conclusion
			Bibliography
		New Collaboration on Modular Origami and LED
			1 Introduction
			2 AKARI- ORIGAMI: Some Modular Works
			3 Workshop
			4 Future of AKARI- ORIGAMI
			Bibliography
		Using the Snapology Technique to Teach Convex Polyhedra
			1 Introduction
			2 Polyhedra Review
				2.1 Platonic Solids
				2.2 Archimedean Solids
			3 Snapology Technique
				3.1 Terminology
				3.2 Assembling the Basic Icosahedron
				3.3 Other Platonic Solids
			4 Brief Introduction to General Snapology
			5 Observations
			6 Conclusions
			Bibliography
		A Systematic Approach to Twirl Design
			1 Introduction
			2 The First Attempt: Small ( Change) Is Beautiful
			3 A Square Is a Rectangle: Metamorphosis
			4 Planar Decorative Area: From Octahedron to Cube, Mosaic Twirls
			5 Shapes and Lengths: Different Spirals
			6 Minimal Folding: No Crease Origami
			7 Consensus Building: Spirals Work Together with Flaps- and- Pockets, Macro- modules
			8 Conclusion
			Bibliography
		Oribotics: The Future Unfolds
			1 Introduction
			2 Industrial Evolution of Oribotics
			3 Crease Pattern and Mechanical Design
			4 Paper versus Fabric
				4.1 Folding Polyester Fabric
				4.2 Paper Choice
				4.3 Crease Pattern Design
			5 Interaction Design
			6 Conclusion and Further Work
			Bibliography
	Part II Origami in Education
		Origametria and the van Hiele Theory of Teaching Geometry
			1 Introduction
			2 The van Hiele Theory of Geometric Teaching
			3 Origami and the Van Hiele Theory
			4 Time of Learning
			5 Gradually Building Knowledge and Concepts
			6 Using Origametria to Eliminate Misconceptions
			7 Origametria and van Hiele: An Example from the Classroom
			8 Conclusion: The Benefits of Using Origametria in the van Hiele System
			Bibliography
		Student Teachers Introduce Origami in Kindergarten and Primary Schools: Froebel Revisited
			1 Introduction
			2 Objectives
			3 Review of the Literature
			4 Problem Statements
			5 Data Collection, Analysis, and Discussion
			6 Conclusion and Next Steps
			Bibliography
		Narratives of Success:Teaching Origami in Low-Income Urban Communities
			1 Introduction
			2 Origami in Mathematics
				2.1 Englewood, Illinois
				2.2 Achievement Academy in Englewood
				2.3 Origami and Student Self- Confidence
			3 Origami in Art Class
				3.1 Learning to Follow Instructions with Origami
			4 Origami in Elementary School
				4.1 Learning English through Origami
				4.2 Origami in a Summer Program
			5 Discussion
			Bibliography
		Origami and Spatial Thinking of College-Age Students
			1 Introduction
			2 Design and Purpose
				2.1 Methodology
				2.2 Sample
				2.3 Instruments
				2.4 Treatment
			3 Results
				3.1 Comparing Pretest and Posttest Spatial Skills
				3.2 Influence of Spatial Experiences on Results
				3.3 Influence of Fields of Study on Results
			4 Conclusion
			Appendix: Excerpt from the Student Survey
			Bibliography
		Close Observation and Reverse Engineering of Origami Models
			1 Introduction
			2 Rationale/ Goals
			3 Origins and Context
			4 An Approach to Close Observation and Reverse Engineering
				4.1 Close Observation of the Modular Origami Object
				4.2 Deconstruction of the Piece
				4.3 Close Observation of the Unit
				4.4 Unfolding the Unit
				4.5 Re- assembly
				4.6 Reflections and Revisions: Re- engineering
			5 An Example
			6 Close Observation and Reverse Engineering in Learning and Problem Solving
			7 Our Observations of CORE
			8 Challenges, a Question, and Next Steps
			9 Conclusion
			Bibliography
		Origami and Learning Mathematics
			1 Introduction
			2 Using Origami in Mathematics Lessons
				2.1 Learning to Fold
				2.2 Common Folding Problems
			3 Using Origami to Teach Conceptually Demanding Mathematics
				3.1 Angles
				3.2 Polygons
				3.3 Symmetry
				3.4 Fractions
				3.5 Reasoning and Proof
			4 Conclusions
			Bibliography
		Hands-On Geometry with Origami
			1 Introduction
			2 New Trends in Teaching
				2.1 Teacher Education in Germany
				2.2 The MINTmachen! Project
				2.3 Research- Driven Projects
				2.4 Teacher Exam Thesis
				2.5 Putting It All Together
			3 Design of the Coursework
				3.1 Overall Goal
				3.2 Example 1: Haga's Theorem
				3.3 Example 2: Trisecting the Angle
				3.4 Example 3: Axiomatics in Geometry and Origami
				3.5 Remarks on the Educational Standards
			4 Evaluation and Lessons Learned
				Future Plans
			Bibliography
		My Favorite Origamics Lessons on the Volume of Solids
			1 Introduction
			2 Exploration 1: Origami Masu Cubic Box
			3 Exploration 2: Origami CK- Octahedron
			4 Exploration 3: The Building Block of CK and KC
			5 Exploration 4: Origami KC- Heptahedron
			6 Exploration 5: Tessellating Solids
			7 Origami, Science, Mathematics, and Education
			8 Conclusion
			Appendix: My Teaching Notes for Exploration 1
			Bibliography
	Part III Origami Science, Engineering, and Technology
		Rigid-Foldable Thick Origami
			1 Introduction
			2 Problem Description
				2.1 Rigid Origami without Thickness
				2.2 Existing Methods
			3 Proposed Method
				3.1 Tapered Panels
				3.2 Limiting the Unfolded State
				3.3 Constant Thickness Panels
				3.4 Global Collision
				3.5 Characteristics
			4 Application for Designs
			5 Conclusion
			Bibliography
		Folding a Patterned Cylinder by Rigid Origami
			1 Introduction
			2 Kinematics of Spherical 4R Linkage and Its One DoF Assembly
			3 Rigid Origami Patterns to Form Cylindrical Structures
			4 Conclusions and Discussion
			Bibliography
		The Origami Crash Box
			1 Introduction
			2 Description of the Origami Pattern Design
			3 Finite Element Modeling
			4 Results and Discussion
				4.1 Axial Crushing of the Conventional Square Tube
				4.2 Collapse Mode and Energy Absorption Properties of the Origami Tube
				4.3 Effects of the Ratio
				4.4 Effects of the Ratio
			5 Conclusion
			Bibliography
		Origami Folding: A Structural Engineering Approach
			1 Introduction
			2 Folded Textured Sheets
				2.1 Engineering Applications
			3 Mechanical Modeling Method
				3.1 Governing Equations
				3.2 Kinematic Analysis
				3.3 Stiffness Analysis
				3.4 Coordinate Transformation
			4 Conclusion
			Bibliography
		Designing Technical Tessellations
			1 Introduction: Paper as Technical Material
			2 Lightweight Construction with Paper
			3 Modular Isometric Origami
				3.1 Modularity
				3.2 Rigid and Isometric Origami
			4 Design Strategies
				4.1 Bottom- Up
				4.2 Top- Down
			5 Not So Serious— Some Fun
			6 Conclusion
			Bibliography
		A Simulator for Origami-Inspired Self-Reconfigurable Robots
			1 Introduction
			2 Foldable Programmable Matter
			3 Representing Programmable Matter
			4 Editor Implementation and Usage
				4.1 Drawing and Creating Crease Patterns
				4.2 Runtime Analysis
			5 PhysX Simulator and Integration
			6 Conclusion
			Bibliography
		A CAD System for Diagramming Origami with Prediction of Folding Processes
			1 Introduction
			2 Related Work
			3 Our Proposed System
				3.1 Listing Candidates by Applying Possible Foldings
				3.2 Removing Duplicate Candidates
				3.3 Ranking of Candidates
			4 Results and Discussion
				4.1 Efficiency of the Prediction Function
				4.2 Enumeration of Simple Origami Shapes
			5 Conclusion and Future Work
			Bibliography
		Development of an Intuitive Algorithm for Diagramming and 3D Animated Tutorial for Folding Crease Patterns
			1 Introduction
			2 Computational Origami
			3 Basic Definitions on Technical Origami
			4 Turning a CP into a Folding Sequence
				4.1 Main Algorithm
				4.2 Choose First Node
				4.3 Maneuver 10
				4.4 Choose and Fold Next Internal Node
				4.5 Maneuvers 1, 2, 6, 7, and 8
				4.6 Choose and Fold Next Molecule
				4.7 Maneuvers 3, 4, 5, and 9
				4.8 Organize Flaps
				4.9 Propagate Folds
			5 Applicability Scope
			6 Example
			7 Software Implementation
			8 Conclusion
			Bibliography
		Hands-Free Microscale Origami
			1 Introduction
			2 Stress- Based Microscale Folding
				2.1 Modeling of Multilayer Thin- Film Curvatures
				2.2 Methods and Materials of Construction
				2.3 Triggers of Folding and Response to Environmental Stimuli
			3 Miniaturized Microscale Origami Structures: Unidirectional Folding
			4 Bidirectional Microscale Folding Using Thin Metal Films
				4.1 Self- Folding of Micropatterned Cubic Cores
				4.2 Self- Folding of Micropatterned Cylindrical Stents with Changing Radii
				4.3 Self- Folding of Micropatterned Paper Airplanes
			5 Conclusion and Future Possibilities
			Bibliography
		Foldable Parylene Origami Sheets Covered with Cells: Toward Applications in Bio-Implantable Devices
			1 Introduction
			2 Materials and Methods
				2.1 Preparation of a Parylene Sheet with Micro- sized Origami Folds
				2.2 Preparation of Cells onto the Parylene Sheet, Folding, and Deploying the Sheet
			3 Results and Discussion
			4 Conclusions
			Bibliography
	Part IV Mathematics of Origami
		Introduction to the Study of Tape Knots
			1 Introduction
			2 Why a Knot in a Tape Forms a Regular Pentagon
			3 Regular Odd- Sided Polygonal Knots
			4 Regular Even- Sided Polygonal Knots
			5 Regular Decagonal Knots
			6 Stability of Knots
			7 Nonregular Polygonal Knots
			8 Conclusion and Further Research
			Bibliography
		Universal Hinge Patterns for Folding Orthogonal Shapes
			1 Introduction
			2 Definitions
			3 Cube Gadgets
			4 Folding Polycubes
				4.1 Hinge Pattern Completeness
				4.2 Paper Dimensions
				4.3 Number of Layers
			5 Implementation
			6 Rigid Foldability and Self- Folding Sheets
			Bibliography
		A General Method of Drawing Biplanar Crease Patterns
			1 Introduction
			2 Mathematical Conventions
			3 Defining the Biplanar
			4 Drawing the Net
			5 Drawing the Crease Pattern
				5.1 The Gadget
				5.2 Overlapping Inside the Biplanar
				5.3 Flat Surrounding Paper
				5.4 Positioning Triangular Walls
			6 Conclusion
			Bibliography
		A Design Method for Axisymmetric Curved Origami with Triangular Prism Protrusions
			1 Introduction
			2 Shape of the Target 3D Origami
			3 Designing the Crease Pattern
				3.1 Conical Type
				3.2 Cylindrical Type
			4 Examples and Discussions
			5 Conclusion
			Bibliography
		Folding Any Orthogonal Maze
			1 Introduction
			2 Algorithm
			Bibliography
		Every Spider Web Has a Simple Flat Twist Tessellation
			1 Introduction
			2 Shrink and Rotate
				2.1 Twist and Aspect Ratio
				2.2 Crease Pattern/ Folded Form Duality
			3 Nonregular Polygons
				3.1 A Broken Tessellation
				3.2 A Valid Rhombus Tessellation
			4 Maxwell's Reciprocal Figures
				4.1 Indeterminateness and Impossibility
				4.2 Positive and Negative Edge Lengths
			Bibliography
		Flat-Unfoldability and Woven Origami Tessellations
			1 Introduction
			2 Woven Tessellations
			3 Simple Woven Patterns
			4 Flat- Unfoldability
			5 Parameterizing the Woven Tessellation
			6 Conclusion
			Bibliography
		Degenerative Coordinates in 22.5 Grid System
			1 Introduction
			2 Model
			3 Construction
			4 Degeneracy
			5 Conclusion
			Bibliography
		Two Folding Constructions
			1 Introduction
			2 Method A: Descartes' Construction
			3 MethodB: MyConstruction
			4 Conclusion
			Bibliography
		Variations on a Theorem of Haga
			1 Introduction
			2 Haga's Theorem
			3 Variation1
			4 Variation2
			5 A Comparison
			6 Variations on the Variations
				6.1 Rectangles
				6.2 From the Strip to the Square
			7 Conclusion
			Bibliography
		Precise Division of Rectangular Paper into an Odd Number of Equal Parts without Tools: An Origamics Exercise
			1 Introduction
			2 Preparation
			3 Trisection
			4 Five- Section
			5 Seven- Section
			6 Pendulum Symmetry
			7 Nine- Section
			8 Individual Line for Higher- Number Sections
			9 Conclusions
			Bibliography
		The Speed of Origami Constructions Versus Other Construction Tools
			1 Introduction
			2 Geometric Tools
				2.1 Origami
				2.2 Ruler and Compass
				2.3 Conics
			3 Constructions and Measures
				3.1 Measures Associated with Constructions
				3.2 Comparison of Constructions
			4 Optimal Constructions
			5 Conclusions
			Bibliography
		A Note on Operations of Spherical Origami Construction
			1 Introduction
			2 The Eight Operations of Planar Origami Construction
			3 Notation
			4 Spherical Origami
			5 Operations of Spherical Origami Construction
			6 Conclusion
			Bibliography
		Origami Alignments and Constructions in the Hyperbolic Plane
			1 Introduction
			2 Basic Alignments and Folds
				2.1 Alignments
				2.2 Folds
				2.3 Unique Alignment Folds
				2.4 Quadratic Folds
				2.5 Quartic Folds
			3 Relations between the Alignment Axioms
				3.1 Totally Real Constructions
				3.2 Radical Axis and Ruler- Compass Constructions
				3.3 Saccheri Quadrilaterals
				3.4 Ruler- Compass Constructions
				3.5 Simulation of Constructions with
			4 Trigonometry and More Folding in
				4.1 Hyperbolic Coordinates, Distances, Angles
				4.2 Parallel Ruler and Its Simulation
				4.3 Theorem of Mordukhai- Boltovskoi
				4.4 Construction of Regular Tessellations
			5 The Non- Euclidean Parabola
			6 H6
				6.1 Higher Origami Constructions
			Bibliography
		A Combinatorial Definition of 1D Flat-Folding
			1 Introduction
			2 Flat- Foldable 1D Origami
			3 Mingling
			4 Proof of the Flat- Foldability Theorem
			Bibliography
		Stamp Foldings with a Given Mountain-Valley Assignment
			1 Introduction
			2 Preliminaries
			3 Universality of the Simple Folding Model
			4 The Number of Folded States
			5 Concluding Remarks
			Bibliography
		Flat Vertex Fold Sequences
			1 Introduction
			2 The Basics of Flat Vertex Folds
			3 Flat Vertex Fold Sequences
			4 Conclusion
			Bibliography
		Circle Packing for Origami Design Is Hard
			1 Introduction
			2 Circle- River Design
			3 Packing and Complexity
			4 Symmetric 3- Pockets
			5 Triangular Paper
			6 Rectangular Paper
			7 Square Paper
			8 Filling Gaps
			9 Encoding the Input
			10 A Positive Result
			11 Conclusions
			Bibliography
		Contributors
		Index
		Color Insert