1. Mathematical methods in origami Robert J. Lang www.langorigami.com MOOC December, 2012
2. Early (but not first)• Japanese newspaper from 1734: Crane, boat, table, “yakko- san”• By 1734, origami is already well-developed MOOC December, 2012
3. Modern Origami• Akira Yoshizawa (1911- 2005)• Inspired a worldwide renaissance of origami MOOC December, 2012
4. Origami Today• “Black Forest Cuckoo Clock,” (1987)• One sheet, no cuts MOOC December, 2012
5. Klein Bottle MOOC December, 2012
6. What Changed?Math!Two forms: “Origami Mathematics” number fields constructibility origami in higher dimensions, curved spaces QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. “Computational Origami” computability complexity algorithms for design and simulation MOOC December, 2012
7. Basic Folds of OrigamiValley fold M u tain fo on ld MOOC December, 2012
8. Crease Patterns QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. MOOC December, 2012
9. Origami design• The fundamental equation:• given a desired subject, how do you fold a square to produce a representation of the subject? MOOC December, 2012
10. Stag Beetle MOOC December, 2012
11. A four-step processScu tbj e T r ee B as e M o de l e as y H a rd e as y MOOC December, 2012
12. The hard step• How do you make a bunch of flaps? MOOC December, 2012
15. Other types of flap• Flaps can come from edges…• …and from the interior of the paper. MOOC December, 2012
16. Unify• They’re all circles MOOC December, 2012
17. Circle Packing• Many flaps: use many circles. MOOC December, 2012
18. Creases• The lines between the centers of touching circles are always creases.• But there needs to be more. Fill in the polygons, but how? MOOC December, 2012
19. Divide and conquer• The creases divide the square into distinct polygons that correspond to pieces of the stick figure. A E F B E F E F A A A B B A A E F 1 E F B B B B 1 1 C C C C 1 m.6 = 27 0 G H G H C C 1 1 G H D 1 G H A D D B C MOOC G H December, 2012 D
20. Molecules• Crease patterns that collapse a polygon so that its edges form a stick figure are called “bun-shi,” or molecules (Meguro)• Different molecules are known from the origami literature.• Triangles have only one possible molecule. A a a E A A D a a D E b B B c b D b D c c C CB C b D c te bem l h at a ou “ b r lc r i ” ee MOOC December, 2012
21. Quadrilateral molecules• There are two possible trees and several different molecules for a quadrilateral.• Beyond 4 sides, the possibilities grow rapidly. “-t r 4sa” “ a hr e s wos ” Hs i/ a a a i u imK ws k Me a a ak w Ln ag MOOC December, 2012
22. Circles and Rivers• Pack circles, which represent all the body parts.• Fill in with molecular crease patterns.• Fold! MOOC December, 2012
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24. Computer-Aided Origami Design• 16 circles (flaps)• 9 “rivers “ (connections) a tle (4 tin s e c sid ) n rs e ah e• 200 equations! e rs a ha ed nc ek bd oy tail fo le re g fo le re g h d le in g h d le in g MOOC December, 2012
45. Moving to 3D...• Mathematical descriptions have permitted the construction of elaborate geometrical objects from single-sheet folding: – Flat Tessellations (Fujimoto, Resch, Palmer, Bateman, Verrill) 3-D faceted tessellations (Fujimoto, Huffman) Curved surfaces (Huffman, Mosely) …and more! MOOC December, 2012
46. Flanged sphere• Similar to concept demo’d by Palmer in 2000 (inspiration for this work) MOOC December, 2012
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50. Mathematica Package MOOC December, 2012
51. Applications in the Real WorldMathematical origami has found many applications in solving real- world technological problems, in: – Space exploration (telescopes, solar arrays, deployable antennas) – Automotive (air bag design) – Medicine (sterile wrappings, implants) – Consumer electronics (fold-up devices) – …and more. MOOC December, 2012
52. Miura “map-fold”• A map of Venice with one degree of freedom MOOC December, 2012
53. Miura-Ori, by Koryo Miura• First “origami in space”• Solar array, flew in 1995 MOOC December, 2012
54. Umbrella MOOC December, 2012
55. 5-meter prototype• The 5-meter prototype folds to about 1.5 meter. MOOC December, 2012
56. Stents• Origami Stent graft developed by Zhong You (Oxford University) and Kaori Kuribayashi MOOC www.tulane.edu/~sbc2003/pdfdocs/0257.PDF December, 2012
57. Folding DNA • Paul Rothemund at Caltech developed techniques to fold DNA into origami shapesPaul Rothemund, “Folding DNA to createnanoscale shapes and patterns,” Nature, 2006 MOOC December, 2012
58. Origami5• Based on the 5th International Conference on Origami in Science, Mathematics, and Education (Singapore, 2010)• Next conference: Kobe, Japan, 2014 MOOC December, 2012
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