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Introduction 1 Genesis * 2 What to Expect and What You Need * I Vertices 1.1 Modeling Origami * 1.1.1 Crease Patterns * 1.1.2 Creases and Folds * 1.2 Vertices * 1.2.1 Kawasaki-Justin Theorem * 1.2.2 Justin Ordering Conditions * 1.2.3 Three Facet Theorem * 1.2.4 Big-Little-Big Angle Theorem * 1.2.5 Maekawa-Justin Theorem * 1.2.6 Vertex Type * 1.2.7 Vertex Validity * 1.3 Degree-2 Vertices * 1.4 Degree-4 Vertices * 1.4.1 Unique Smallest Sector * 1.4.2 Two Consecutive Smallest Sectors * 1.4.3 Four Equal Sectors * 1.4.4 Constructing Degree-4 Vertices * 1.4.5 Half-Plane Properties * 1.5 Multivertex Flat-Foldability ** 1.5.1 Isometry Conditions and Semifoldability * * 1.5.2 Injectivity Conditions and Non-Twist Relation ** 1.5.3 Local Flat-Foldability Graph ** 1.6 Vector Formulations of Vertices * * * 1.6.1 Vector Notation: Points * * * 1.6.2 Vector Notation: Lines *** 1.6.3 Translation *** 1.6.4 Rotation * * * 1.6.5 Reflection * ** 1.6.6 Line Intersection ** * 1.6.7 2D Developability * * * 1.6.8 2D Flat-Foldability *** 1.6.9 Analytic versus Numerical * * * 1.7 Terms * 2 Periodicity 2.1 Repeating Vertices * 2.2 1D Periodicity * 2.2.1 Periodicity and Symmetry * 2.2.2 Tiles * 2.2.3 Linear Chains * 2.3 2D Periodicity * 2.3.1 Huffman Grid * 2.3.2 Yoshimura Pattern * 2.3.3 Miura-ori * 2.3.4 Miura-ori Variations * 2.3.5 Barreto's Mars * 2.3.6 Generalized Mars * 2.4 Partial Periodicity *, **, * * * 2.4.1 Yoshimura-Miura Hybrids * 2.4.2 Semigeneralized Miura-ori * 2.4.3 Predistortion ** 2.4.4 Tachi-Miura Mechanisms * 2.4.5 Triangulated Cylinders * 2.4.6 Triangulated Cylinder Geometry * * * 2.4.7 Waterbomb Tessellation * 2.4.8 Troublewit and Pleats * 2.4.9 Corrugations and More * 2.5 Terms * 3 Simple Twists 3.1 Twist-Based Tessellations * 3.2 Folding a Twist * 3.2.1 Diagrams versus Crease Patterns * 3.2.2 A Square Twist Tessellation * ¡­¡­ 4 Twist Tiles 5 Tilings 6 Primal-Dual Tessellations 7 Rigid Foldability 8 Spherical Vertices 9 3D Analysis 10 Rotational Solids