Twists, tilings, and tessellations : mathematical methods for geometric origami

Introduction 
Genesis 
What to Expect, What You Need
Vertices 
Modeling Origami 
Crease Patterns 
Creases and Folds 
Vertices 
Kawasaki-Justin Theorem 
Justin Ordering Conditions 
Three-Facet Theorem 
Big-Little-Big Angle Theorem 
Maekawa-Justin Theorem 
Vertex Type 
Vertex Validity 
Degree- Vertices 
Degree- Vertices 
Unique Smallest Sector 
Two Consecutive Smallest Sectors 
Four Equal Sectors 
Constructing Degree- Vertices 
Half-Plane Properties 
Multivertex Flat-Foldability 
Isometry Conditions and Semifoldability 
Injectivity Conditions and Non-Twist Relation 
Local Flat-Foldability Graph 
Vector Formulations of Vertices 
Vector Notation: Points 
Vector Notation: Lines 
Translation 
Rotation 
Reection 
Line Intersection 
2D Developability 
2D Flat-Foldability 
Analytic vs Numerical 
Terms 
Periodicity 
Repeating Vertices 
1D Periodicity 
Periodicity and Symmetry 
Tiles 
Linear Chains 
2D Periodicity 
Hu_man Grid 
Yoshimura Pattern 
Miura-ori 
Miura-ori Variations 
Barreto`s Mars 
Generalized Mars 
Partial Periodicity 
Yoshimura-Miura Hybrids 
Semigeneralized Miura-ori 
Predistortion 
Tachi-Miura Mechanisms 
Triangulated Cylinders 
Triangulated Cylinder Geometry 
Waterbomb Tessellation 
Troublewit and Pleats 
Corrugations and More 
Terms 
Simple Twists 
Twist-Based Tessellations 
Folding a Twist 
Diagrams Versus Crease Patterns 
A Square Twist Tessellation 
Elements of a Twist 
Regular Polygonal Twists 
Cyclic Regular Twists 
Open and Closed Back Twists 
Rotation Angle of the Central Polygon 
Iso-Area Twists 
Twist Flat-Foldability 
Local Flat-Foldability 
Pleat Crease Parity 
Pleat Assignments 
mm=vv Condition 
mv=vm Condition 
MM=V V Condition 
MV=VM Condition 
Cyclic Overlap Conditions 
Summary of Limits 
General Polygonal Twists 
Triangle Twists 
Higher Order Irregular Twists 
Cyclic Overlaps in Irregular Twists 
Closed Back Irregular Twists 
Joining Twists 
Terms 
Twist Tiles 
Introduction to Twist Tiles 
What is a Tile? 
Ways of Mating 
Centered Twist Tiles 
O_set Twist Tiles 
Vertex Types 
Vertices and Angles 
Unit Polygons 
Centered Twist Tiles 
O_set Twist Tiles 
Folded Form Tiles 
Centered Twist Folded Form Tiles 
O_set Twist Folded Form Tiles 
Triangle Tiles 
Centered Twist Triangle Tiles 
O_set Twist Triangle Tiles 
Higher-Order Polygon Tiles 
Centered Twist Cyclic Polygon Tiles 
Cyclic Polygon Construction 
Quadrilateral O_set Twist Polygon Tiles 
O_set Twist Higher-Order Polygon Tiles 
Pathological Twist Tiles 
Split-Twist Quadrilateral Tiles 
Terms 
Tilings 
Introduction to Tilings 
Archimedean Tilings 
Uniform Tilings 
Constructing Archimedean Tilings 
Lattice Patches and Vectors 
Edge-Oriented Tilings 
Centered Twist Tiles 
O_set Twist Tiles 
k-Uniform Tilings 
-Uniform Tilings 
Two-Colorable -Uniform Tilings 
Higher-Order Uniform Tilings 
Periodic Tilings with Other Shapes 
Gridded Tessellations 
Non-Periodic Tilings 
Goldberg Tiling 
Self-Similar Tilings 
Terms 
Primal-Dual Tessellations 
Introduction 
Shrink and Rotate 
Twist and Aspect Ratio 
Crease Pattern/Folded Form Duality 
Nonregular Polygons 
A Broken Tessellation 
Dual Graphs and Interior Duals 
A Valid Rhombus Tessellation 
Relation Between Primal and Dual Graphs 
Maxwell`s Reciprocal Figures 
Indeterminateness and Impossibility 
Positive and Negative Edge Lengths 
Crease Assignment 
Triangle Graphs 
Voronoi and Delaunay 
Flagstone Tessellations 
Spiderwebs Revisited 
The Flagstone Geometry 
Flagstone Vertex Construction 
Examples 
Woven Tessellations 
Woven Concepts 
Simple Woven Patterns 
Woven Algorithm 
Flat Unfoldability 
Woven Algorithm, Continued 
Woven Examples 
Terms 
Rigid Foldability 
Half-Open Vertices 
Spherical Geometry 
A Degree- Vertex in Spherical Geometry 
Opposite Fold Angles 
Adjacent Fold Angles 
Conditions on Rigid Foldability 
The Weighted Fold Angle Graph 
Distinctness of Fold Angle 
Matching Fold Angle 
General Twists 
Triangle Twists 
Mechanical Advantage 
Non-Twist Folds 
Quadrilateral Meshes 
Non-Flat-Foldable Vertices 
Terms 
Spherical Vertices 
The Gaussian Sphere
Plane Orientation
The Trace 
Polyhedral Vertices 
A Degree- Vertex 
Sector and Fold Angles 
Osculating Plane 
Binding Conditions 
Ruling Plane 
Real Space Solid Angle 
Ruling Angle 
Osculating Angle 
Adjacent Fold Angles 
Flat-Foldable and Straight-Major/Minor Vertices 
Sector Angle/Fold Angle Relations 
More Angles and Planes 
Sector Elevation Angles 
Sector Angles 
Bend Angle 
Edge Torsion Angle 
Midfold Angles and Planes 
In_nitesimal Trace 
What Speci_es a Vertex? 
Grids of Vertices 
Hu_man Grid 
Gauss Map 
Miura-ori and Mars 
Terms 
3D Vectors 
3D Analysis 
3D Vectors 
3D Vertices 
Direction Vectors 
Vertex from Crease Directions 
Degree- Vertex from Sector Elevation Angles 
Discrete Space Curve 
Plate Model 
Folding a Crease Pattern 
Fold Angle Consistency 
Solving for Fold Angles 
Truss Model 
3D Isometry and Planarity 
Explicit Stress/Strain 
3D Developability 
Time E_ciency 
Terms 
Rotational Solids 
Three-Dimensional Twists 
Pu_y Twists 
Folding a Sphere 
Thin-Flange Algorithm 
Solid-Flanged Structures 
Mosely`s \\Bud\" 
Solid-Flange Algorithm 
Speci_ed Gores 
Generalized Flanges 
Cylindrical Unfoldings 
Unwrapping 
Artists of Revolution 
Variations on the Theme 
Twist Lateral Shifts 
Triangulated Gores 
Terms 
Afterword
Acknowledgements