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Twists, Tilings, and Tessellation describes the underlying principles and mathematics of the broad and exciting field of abstract and mathematical origami, most notably the field of origami tessellations. It contains folding instructions, underlying principles, mathematical concepts, and many beautiful photos of the latest work in this fast-expanding field.
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Table of Contents

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

About the Author

Robert J. Lang has been an avid student of origami for over fifty years and is now recognized as one of the world's leading masters of the art. He is noted for designs of great detail and realism, and his repertoire includes some of the most complex origami designs ever created. His work combines aspects of the Western school of mathematical origami design with the Eastern emphasis upon line and form to yield designs that are at once distinctive, elegant, and challenging to fold. They have been shown in exhibitions in New York (Museum of Modern Art), Paris (Carrousel du Louvre), Salem (Peabody Essex Museum), San Diego (Mingei Museum of World Folk Art), and Kaga, Japan (Nippon Museum of Origami), among others. He is one of the pioneers of computational origami techniques, and has published widely on the theory and mathematics of folding. Dr. Lang was born in Ohio and raised in Atlanta, Georgia. Along the way to his current career as a full-time origami artist and consultant, he worked as a physicist, engineer, and R&D manager, during which time he authored or co-authored over 80 technical publications and 50 patents on semiconductor lasers, optics, and integrated optoelectronics. He was elected a Fellow of the Optical Society of America and served as Editor-in-Chief of the IEEE Journal of Quantum Electronics from 2007-2010. After switching his primary focus to origami, he authored or co-authored numerous technical articles on the mathematics of folding and on design techniques for folding in technological applications. In 2009, he received Caltech's Distinguished Alumni Award for his origami work, and in 2013 was elected a Fellow of the American Mathematical Society. Dr. Lang resides in Alamo, California.

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