Twists, Tilings, and Tessellations : Mathematical Methods for Geometric Origami book cover
1st Edition

Twists, Tilings, and Tessellations
Mathematical Methods for Geometric Origami

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ISBN 9781568812328
Published February 22, 2018 by A K Peters/CRC Press
776 Pages

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Book Description

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.

Table of Contents



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




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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.