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How To Fold It - The Math Of Linkages, Origami, Polyhedra
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Preface page ix PART I. Linkages 1 1 Robot Arms 3 1.1 Annulus 5 1.2 Reaching Angles 15 1.3 Above & Beyond 20 2 Straight-Line Linkages and the Pantograph 24 2.1 Straight-Line Linkages 24 2.2 Pantograph 28 2.3 Above & Beyond 36 3 Protein Folding and Pop-Up Cards 39 3.1 Fixed-Angle Chains 39 3.2 Protein Backbones 40 3.3 Maximum Span 42 3.4 Alignment 44 3.5 Piercing 46 3.6 Pop-Up Spinner 48 3.7 Above & Beyond 52 PART II. Origami 55 4 Flat Vertex Folds 57 4.1 Mountain and Valley Creases 57 4.2 Single-Vertex Flat Folds 58 4.3 The Maekawa-Justin Theorem 61 4.4 The Local Min Theorem 64 4.5 The Kawasaki-Justin Theorem 66 4.6 Above & Beyond 68 v vi Contents 5 Fold and One-Cut 72 5.1 Examples 72 5.2 Fold and One-Cut Theorem 78 5.3 Above & Beyond 81 6 The Shopping Bag Theorem 84 6.1 Two Rigid Origami Examples 85 6.2 Dihedral Angle Constraints 89 6.3 The Shopping Bag Theorem 93 6.4 Above & Beyond 96 PART III. Polyhedra 99 7 Dürer’s Problem: Edge Unfolding 101 7.1 Albrecht Dürer’s Nets 101 7.2 Convex Polyhedra 103 7.3 The Open Problem 106 7.4 Spanning Cut Tree 109 7.5 Some Polyhedra with Nets 112 7.6 Above & Beyond 115 8 Unfolding Orthogonal Polyhedra 119 8.1 Orthogonal Polyhedra 119 8.2 Orthogonal Terrains 120 8.3 Grid Unfoldings 125 8.4 Above & Beyond 126 9 Folding Polygons to Convex Polyhedra 130 9.1 Questions 132 9.2 Alexandrov’s Theorem 133 9.3 Folding Convex Polygons 135 9.4 The Foldings of the Latin Cross 138 9.5 Above & Beyond 140 10 Further Reading 142 Glossary 147 Answers to Exercises 151 Chapter 1 151 Chapter 2 155 Chapter 3 156 Chapter 4 158 Chapter 5 161 Chapter 6 162 Contents vii Chapter 7 165 Chapter 8 168 Chapter 9 170 Acknowledgments 173 Index 175 |
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2015-09-06 11:51:43, 16.66 M
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