Twists, Tilings, and Tessellations: Mathematical Methods for Geometric Origami

Cover
Title Page
Copyright Page
Dedication Page
Table of Contents
Introduction
1. Genesis
2. What to expect and What You Need
1 Vertices
1.1 Modeling Origami
1.1.1 Crease Patterns
1.1.2 Creases and Folds
1.2 Vertices
1.2.1 Kawasaki-Justin Theorem
1.2.2 Justin Ordering Conditions
1.2.3 Three Facet Theorem
1.2.4 Big-Little-Big Angle Theorem
1.2.5 Maekawa-Justin Theorem
1.2.6 Vertex Type
1.2.7 Vertex Validity
1.3 Degree-2 Vertices
1.4 Degree-4 Vertices
1.4.1 Unique Smallest Sector
1.4.2 Two Consecutive Smallest Sectors
1.4.3 Four Equal Sectors
1.4.4 Constructing Degree-4 Vertices
1.4.5 Half-Plane Properties
1.5 Multivertex Flat-Foldability
1.5.1 Isometry Conditions and Semifoldability
1.5.2 Injectivity Conditions and Non-Twist Relation
1.5.3 Local Flat-Foldability Graph
1.6 Vector Formulations of Vertices
1.6.1 Vector Notation: Points
1.6.2 Vector Notation: Lines
1.6.3 Translation
1.6.4 Rotation
1.6.5 Reflection
1.6.6 Line Intersection
1.6.7 2D Developability
1.6.8 2D Flat-Foldability
1.6.9 Analytic versus Numerical
1.7 Terms
2 Periodicity
2.1 Repeating Vertices
2.2 1D Periodicity
2.2.1 Periodicity and Symmetry
2.2.2 Tiles
2.2.3 Linear Chains
2.3 2D Periodicity
2.3.1 Huffman Grid
2.3.2 Yoshimura Pattern
2.3.3 Miura-ori
2.3.4 Miura-ori Variations
2.3.5 Barreto’s Mars
2.3.6 Generalized Mars
2.4 Partial Periodicity , ,
2.4.1 Yoshimura-Miura Hybrids
2.4.2 Semigeneralized Miura-ori
2.4.3 Predistortion
2.4.4 Tachi-Miura Mechanisms
2.4.5 Triangulated Cylinders
2.4.6 Triangulated Cylinder Geometry
2.4.7 Waterbomb Tessellation
2.4.8 Troublewit and Pleats
2.4.9 Corrugations and More
2.5 Terms
3 Simple Twists
3.1 Twist-Based Tessellations
3.2 Folding a Twist
3.2.1 Diagrams versus Crease Patterns
3.2.2 A Square Twist Tessellation
3.3 Elements of a Twist
3.4 Regular Polygonal Twists ,
3.4.1 Cyclic Regular Twists
3.4.2 Open- and Closed-Back Twists
3.4.3 Rotation Angle of the Central Polygon
3.4.4 Iso-Area Twists
3.5 Twist Flat-Foldability
3.5.1 Local Flat-Foldability
3.5.2 Pleat Crease Parity
3.5.3 Pleat Assignments
3.5.4 mm/vv Condition
3.5.5 mv/vm Condition
3.5.6 MM/VV Condition
3.5.7 MV/VM Condition
3.5.8 Cyclic Overlap Conditions
3.5.9 Summary of Limits
3.6 General Polygonal Twists ,
3.6.1 Triangle Twists
3.6.2 Higher-Order Irregular Twists
3.6.3 Cyclic Overlaps in Irregular Twists
3.6.4 Closed-Back Irregular Twists
3.6.5 Open-Back Brocard Polygon Twists
3.7 Joining Twists
3.8 Terms
4 Twist Tiles
4.1 Introduction to Twist Tiles
4.1.1 What is a Tile
4.1.2 Ways of Mating
4.1.3 Centered Twist Tiles
4.1.4 Offset Twist Tiles
4.2 Vertex Figures
4.3 Vertices and Angles
4.3.1 Unit Polygons
4.3.2 Centered Twist Tiles
4.3.3 Offset Twist Tiles
4.4 Folded Form Tiles ,
4.4.1 Centered Twist Folded Form Tiles
4.4.2 Offset Twist Folded Form Tiles
4.5 Triangle Tiles
4.5.1 Centered Twist Triangle Tiles
4.5.2 Offset Twist Triangle Tiles
4.6 Higher-Order Polygon Tiles , ,
4.6.1 Centered Twist Cyclic Polygon Tiles
4.6.2 Cyclic Polygon Construction
4.6.3 Quadrilateral Offset Twist Polygon Tiles
4.6.4 Offset Twist Higher-Order Polygon Tiles
4.6.5 Pathological Twist Tiles
4.6.6 Split-Twist Quadrilateral Tiles
4.7 Terms
5 Tilings
5.1 Introduction to Tilings
5.2 Archimedean Tilings ,
5.2.1 Uniform Tilings
5.2.2 Constructing Archimedean Tilings
5.2.3 Lattice Patches and Vectors
5.3 Edge-Oriented Tilings
5.3.1 Centered Twist Tiles
5.3.2 Offset Twist Tiles
5.4 k-Uniform Tilings
5.4.1 2-Uniform Tilings
5.4.2 Two-Colorable 2-Uniform Tilings
5.4.3 Higher-Order Uniform Tilings
5.4.4 Periodic Tilings with Other Shapes
5.4.5 Grid Tessellations
5.5 Non-Periodic Tilings ,
5.5.1 Goldberg Tiling
5.5.2 Self-Similar Tilings
5.6 Terms
6 Primal-Dual Tessellations
6.1 Shrink and Rotate
6.2 Properties
6.2.1 Twist and Aspect Ratio
6.2.2 Crease Pattern/Folded Form Duality
6.3 Nonregular Polygons
6.3.1 A Broken Tessellation
6.3.2 Dual Graphs and Interior Duals
6.3.3 A Valid Rhombus Tessellation
6.3.4 Relation Between Primal and Dual Graphs
6.4 Maxwell’s Reciprocal Figures ,
6.4.1 Indeterminateness and Impossibility
6.4.2 Positive and Negative Edge Lengths
6.4.3 Crease Assignment
6.4.4 Triangle Graphs
6.4.5 Voronoi and Delaunay
6.5 Flagstone Tessellations
6.5.1 Spiderwebs Revisited
6.5.2 The Flagstone Geometry
6.5.3 Flagstone Vertex Construction
6.5.4 Examples
6.6 Woven Tessellations ,
6.6.1 Woven Concepts
6.6.2 Simple Woven Patterns
6.6.3 Woven Algorithm
6.6.4 Flat Unfoldability
6.6.5 Woven Algorithm, Continued
6.6.6 Woven Examples
6.7 Terms
7 Rigid Foldability
7.1 The Easy Way or the Hard Way
7.2 Half-Open Vertices
7.3 Spherical Geometry
7.4 A Degree-4 Vertex in Spherical Geometry
7.4.1 Opposite Fold Angles
7.4.2 Adjacent Fold Angles
7.5 Conditions on Rigid Foldability
7.5.1 The Weighted Fold Angle Graph
7.5.2 Distinctness of Fold Angle
7.5.3 Matching Fold Angle
7.6 General Twists
7.6.1 Triangle Twists
7.6.2 Mechanical Advantage
7.7 Non-Twist Folds
7.7.1 General Meshes
7.7.2 Quadrilateral Meshes
7.8 Non-Quadrilateral Meshes
7.8.1 Forced Rigid Foldability
7.8.2 Non-Flat-Foldable Vertices
7.9 Terms
8 Spherical Vertices
8.1 Generalizing Vertices
8.2 The Gaussian Sphere
8.2.1 Plane Orientation
8.2.2 The Trace
8.2.3 Polyhedral Vertices
8.2.4 A Degree-4 Vertex
8.3 Sector and Fold Angles
8.3.1 Osculating Plane
8.3.2 Binding Condition
8.3.3 Ruling Plane
8.3.4 Real Space Solid Angle
8.3.5 Ruling Angle
8.3.6 Osculating Angle
8.3.7 Adjacent Fold Angles
8.3.8 Flat-Foldable and Straight-Major/Minor Vertices
8.3.9 Sector Angle/Fold Angle Relations
8.4 More Angles and Planes
8.4.1 Sector Elevation Angles
8.4.2 Sector Angles
8.4.3 Bend Angle
8.4.4 Edge Torsion Angle
8.4.5 Midfold Angles and Planes
8.4.6 Infinitesimal Trace
8.4.7 What Specifies a Vertex
8.5 Networks of Vertices
8.5.1 Huffman Grid
8.5.2 Gauss Map
8.5.3 Miura-ori and Mars
8.6 Terms
9 3D Analysis
9.1 3D Vectors
9.2 3D Vertices
9.2.1 Fold Direction Vectors
9.2.2 Vertex from Fold Directions
9.2.3 Degree-4 Vertex from Sector Elevation Angles
9.3 Discrete Space Curve
9.4 Plate Model
9.4.1 Folding a Crease Pattern
9.4.2 Fold Angle Consistency
9.4.3 Solving for Fold Angles
9.5 Truss Model
9.5.1 3D Isometry and Planarity
9.5.2 Explicit Stress/Strain
9.5.3 3D Developability
9.6 Time Efficiency
9.7 Terms
10 Rotational Solids
10.1 Three-Dimensional Twists ,
10.1.1 Puffy Twists
10.1.2 Folding a Sphere
10.2 Thin-Flange Algorithm
10.3 Thick-Flange Structures ,
10.3.1 Mosely’s “Bud”
10.3.2 Thick-Flange Algorithm
10.3.3 Specified Gores
10.3.4 Generalized Flanges
10.4 Axial Unfoldings
10.5 Variations on the Theme
10.5.1 Twist Lateral Shifts
10.5.2 Triangulated Gores
10.6 Artists of Revolution
10.7 Terms
Afterword
Acknowledgements
Bibliography
Index