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2012 mooc lecture

Betty Hurley
Betty Hurley

The document discusses the history and modern applications of mathematical methods in origami. It describes how origami evolved from a traditional Japanese art form to a field that uses mathematical principles and algorithms. Modern origami artists now design complex 3D shapes and engineers use origami techniques to develop advanced technologies in fields like aerospace, biomedicine, and consumer electronics.

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Mathematical methods in origami Robert J. Lang www.langorigami.com MOOC December, 2012
Early (but not first) Japanese newspaper from 1734: Crane, boat, table, yakko- san By 1734, origami is already well-developed MOOC December, 2012
Modern Origami Akira Yoshizawa (1911- 2005) Inspired a worldwide renaissance of origami MOOC December, 2012
Origami Today Black Forest Cuckoo Clock, (1987) One sheet, no cuts MOOC December, 2012
Klein Bottle MOOC December, 2012
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
Basic Folds of Origami Valley fold M u tain fo on ld MOOC December, 2012
Crease Patterns QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. MOOC December, 2012
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
Stag Beetle MOOC December, 2012
A four-step process Sc u t bj e T r ee B as e M o de l e as y H a rd e as y MOOC December, 2012
The hard step How do you make a bunch of flaps? MOOC December, 2012
How to make a flap MOOC December, 2012
Limiting process Skinnier flap leads to A (quarter) circle! MOOC December, 2012
Other types of flap Flaps can come from edges and from the interior of the paper. MOOC December, 2012
Unify Theyre all circles MOOC December, 2012
Circle Packing Many flaps: use many circles. MOOC December, 2012
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
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
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 C B C b D c te bem l h at a ou b r lc r i ee MOOC December, 2012
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
Circles and Rivers Pack circles, which represent all the body parts. Fill in with molecular crease patterns. Fold! MOOC December, 2012
MOOC December, 2012
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
The crease pattern MOOC December, 2012
Whitetail Deer MOOC December, 2012
Mule Deer Mule Deer MOOC December, 2012
Roosevelt Elk MOOC December, 2012
Bull Moose MOOC December, 2012
Tarantula MOOC December, 2012
Dragonfly MOOC December, 2012
MOOC December, 2012
Kabuto Mushi Samurai December, 2012 Helmet Beetle MOOC
Eupatorus gracilicornis MOOC December, 2012
Euthysanius Beetle Roosevelt Elk MOOC December, 2012
Praying Mantis MOOC December, 2012
Two Praying Mantises MOOC December, 2012
Representational MOOC December, 2012
Dancing Crane Dancing Crane MOOC December, 2012
Barn Owl Barn Owl MOOC December, 2012
Grizzly Bear MOOC December, 2012
Tree Frog MOOC December, 2012
Instrumentalists MOOC December, 2012
Organist MOOC December, 2012
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
Flanged sphere Similar to concept demod by Palmer in 2000 (inspiration for this work) MOOC December, 2012
MOOC December, 2012
MOOC December, 2012
MOOC December, 2012
Mathematica Package MOOC December, 2012
Applications in the Real World Mathematical 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
Miura map-fold A map of Venice with one degree of freedom MOOC December, 2012
Miura-Ori, by Koryo Miura First origami in space Solar array, flew in 1995 MOOC December, 2012
Umbrella MOOC December, 2012
5-meter prototype The 5-meter prototype folds to about 1.5 meter. MOOC December, 2012
Stents Origami Stent graft developed by Zhong You (Oxford University) and Kaori Kuribayashi MOOC www.tulane.edu/~sbc2003/pdfdocs/0257.PDF December, 2012
Folding DNA Paul Rothemund at Caltech developed techniques to fold DNA into origami shapes Paul Rothemund, Folding DNA to create nanoscale shapes and patterns, Nature, 2006 MOOC December, 2012
Origami5 Based on the 5th International Conference on Origami in Science, Mathematics, and Education (Singapore, 2010) Next conference: Kobe, Japan, 2014 MOOC December, 2012
Pots http://www.langorigami.com MOOC December, 2012

More Related Content

2012 mooc lecture

  • 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 Origami Valley 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 process Sc u t bj 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
  • 13. How to make a flap MOOC December, 2012
  • 14. Limiting process Skinnier flap leads to A (quarter) circle! 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 Theyre 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 C B 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
  • 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
  • 25. The crease pattern MOOC December, 2012
  • 26. Whitetail Deer MOOC December, 2012
  • 27. Mule Deer Mule Deer MOOC December, 2012
  • 28. Roosevelt Elk MOOC December, 2012
  • 29. Bull Moose MOOC December, 2012
  • 30. Tarantula MOOC December, 2012
  • 31. Dragonfly MOOC December, 2012
  • 33. Kabuto Mushi Samurai December, 2012 Helmet Beetle MOOC
  • 34. Eupatorus gracilicornis MOOC December, 2012
  • 35. Euthysanius Beetle Roosevelt Elk MOOC December, 2012
  • 36. Praying Mantis MOOC December, 2012
  • 37. Two Praying Mantises MOOC December, 2012
  • 38. Representational MOOC December, 2012
  • 39. Dancing Crane Dancing Crane MOOC December, 2012
  • 40. Barn Owl Barn Owl MOOC December, 2012
  • 41. Grizzly Bear MOOC December, 2012
  • 42. Tree Frog MOOC December, 2012
  • 43. Instrumentalists MOOC December, 2012
  • 44. Organist 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 demod by Palmer in 2000 (inspiration for this work) MOOC December, 2012
  • 50. Mathematica Package MOOC December, 2012
  • 51. Applications in the Real World Mathematical 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 shapes Paul Rothemund, Folding DNA to create nanoscale 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
  • 59. Pots http://www.langorigami.com MOOC December, 2012

Editor's Notes

  1. This is the crease pattern I showed earlier.