Mathematical Origami Geo Hagopian College of the Desert [email protected] October 25, 2016 Fall 2016 : Math & Science Lecture Series The College of the Desert School of Math and Science Presents: The Mathematics and Algorithms of Origami On Tuesday, October 25th in MSTC 250, College of the Desert Math and Com- puter Science Professor Geo Hagopian will give an introduction to the exciting new science of computational origami. Figure 1: Robert Lang's Fiddler Crab. From simple folds on 1-dimensional pa- per to squash folds in hyperspace, learn 13 how new computer algorithms are de- 12 signed to create crease patterns to fold 11 lizards and lions; elk, shrimp, praying 10 mantises and the dollar velociraptor. 9 8 7 6 5 4 3 2 1 0 This is a potent new area of research and 012345678910111213 development with applications ranging Figure 2: The crease pattern of the ddler crab. from biology (folding proteins) to astro- physics (deploying NASA's gliders in the Martian atmosphere!) I Maekawa's Theorem I Kawasaki's Theorem 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. I Maekawa's Theorem I Kawasaki's Theorem 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. I Maekawa's Theorem I Kawasaki's Theorem 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget I Maekawa's Theorem I Kawasaki's Theorem 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) I Maekawa's Theorem I Kawasaki's Theorem 6. Two Dimensional Flat Folding 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. I Maekawa's Theorem I Kawasaki's Theorem 7. Single vertex Folding. 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 11. Origamizer. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 12. Hyperbolic Paraboloid Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. Overview 1. Mathematical denitions for origami. 2. Universality. 3. Turn Gadget 4. Example (dodecahedron) 5. One Dimensional Flat Folding. 6. Two Dimensional Flat Folding 7. Single vertex Folding. I Maekawa's Theorem I Kawasaki's Theorem 8. Tesselation Folds 9. Straight Cuts 10. Treemaker. 11. Origamizer. 12. Hyperbolic Paraboloid • Crease: line segment or curve on the paper. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Folded state: nished origamiunfolding! crease pattern • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Valley crease: top sides touch • Mountain crease: bottom sides touch Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Folded state: nished origamiunfolding! crease pattern • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Valley crease: top sides touch • Mountain crease: bottom sides touch Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease: line segment or curve on the paper. • Folded state: nished origamiunfolding! crease pattern • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Valley crease: top sides touch • Mountain crease: bottom sides touch Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease: line segment or curve on the paper. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Valley crease: top sides touch • Mountain crease: bottom sides touch Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease: line segment or curve on the paper. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Folded state: nished origamiunfolding! crease pattern Valley crease: top sides touch • Mountain crease: bottom sides touch Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease: line segment or curve on the paper. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Folded state: nished origamiunfolding! crease pattern • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Origami Denitions • Piece of paper: A 2-D polygon with distinguished top and bottom faces. • Crease: line segment or curve on the paper. • Crease pattern: a set of creases; that is, a planar graph drawn on the paper. • Folded state: nished origamiunfolding! crease pattern • Flat folding: folded state lying in a planeif all creases are folded, the crease pattern is called at foldable Valley crease: top sides touch • Mountain crease: bottom sides touch • Mountain-Valley pattern: crease pattern + mountain-valley assignment. • Simple fold: fold along a single mountain/valley line by ±180◦ This presents the choice of how many layers to fold. • One-layer simple fold: just top or bottom layer All-layers simple fold: fold all the layers • Strip: a long, narrow rectangle • Non-simple strip folding: a strip knot More Origami Terminology • Mountain-Valley assignment: specify which creases are mountains and which are valleys. • Simple fold: fold along a single mountain/valley line by ±180◦ This presents the choice of how many layers to fold. • One-layer simple fold: just top or bottom layer All-layers simple fold: fold all the layers • Strip: a long, narrow rectangle • Non-simple strip folding: a strip knot More Origami Terminology • Mountain-Valley assignment: specify which creases are mountains and which are valleys. • Mountain-Valley pattern: crease pattern + mountain-valley assignment. • One-layer simple fold: just top or bottom layer All-layers simple fold: fold all the layers • Strip: a long, narrow rectangle • Non-simple strip folding: a strip knot More Origami Terminology • Mountain-Valley assignment: specify which creases are mountains and