1st Edition

Twists, Tilings, and Tessellations Mathematical Methods for Geometric Origami

By Robert J. Lang Copyright 2018
    756 Pages
    by A K Peters/CRC Press

    756 Pages
    by A K Peters/CRC Press

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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.



    What to Expect, What You Need


    Modeling Origami

    Crease Patterns

    Creases and Folds


    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




    Line Intersection

    2D Developability

    2D Flat-Foldability

    Analytic vs Numerical



    Repeating Vertices

    1D Periodicity

    Periodicity and Symmetry


    Linear Chains

    2D Periodicity

    Hu_man Grid

    Yoshimura Pattern


    Miura-ori Variations

    Barreto's Mars

    Generalized Mars

    Partial Periodicity

    Yoshimura-Miura Hybrids

    Semigeneralized Miura-ori


    Tachi-Miura Mechanisms

    Triangulated Cylinders

    Triangulated Cylinder Geometry

    Waterbomb Tessellation

    Troublewit and Pleats

    Corrugations and More


    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


    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



    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


    Primal-Dual Tessellations


    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


    Woven Tessellations

    Woven Concepts

    Simple Woven Patterns

    Woven Algorithm

    Flat Unfoldability

    Woven Algorithm, Continued

    Woven Examples


    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


    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


    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


    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


    Artists of Revolution

    Variations on the Theme

    Twist Lateral Shifts

    Triangulated Gores





    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.