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
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
