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-d...
Modern Origami• Akira Yoshizawa (1911-  2005)• Inspired a worldwide  renaissance of origami                               ...
Origami Today• “Black Forest Cuckoo  Clock,” (1987)• One sheet, no cuts                             MOOC                  ...
Klein Bottle                   MOOC               December, 2012
What Changed?Math!Two forms:      “Origami Mathematics”        number fields        constructibility        origami in hig...
Basic Folds of OrigamiValley fold                      M u tain fo                                  on        ld          ...
Crease Patterns                 QuickTime™ and a       TIFF (Uncompressed) decompressor          are needed to see this pi...
Origami design• The fundamental equation:• given a desired subject, how do you fold a square to produce a  representation ...
Stag Beetle           MOOC       December, 2012
A four-step processScu tbj e         T           r           ee           B                        as                     ...
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                        ...
Other types of flap• Flaps can come from edges…• …and from the interior of the paper.                              MOOC   ...
Unify• They’re 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 p...
Divide and conquer•   The creases divide the square into distinct polygons that correspond to    pieces of the stick figur...
Molecules• Crease patterns that collapse a polygon so that its edges form a  stick figure are called “bun-shi,” or molecul...
Quadrilateral molecules•   There are two possible trees and several different molecules for a    quadrilateral.•   Beyond ...
Circles and Rivers• Pack circles, which represent all the body parts.• Fill in with molecular crease patterns.• Fold!     ...
MOOCDecember, 2012
Computer-Aided Origami Design•   16 circles (flaps)•   9 “rivers “ (connections)                                          ...
The crease pattern           MOOC       December, 2012
Whitetail Deer           MOOC       December, 2012
Mule DeerMule Deer                MOOC            December, 2012
Roosevelt Elk        MOOC    December, 2012
Bull Moose                 MOOC             December, 2012
Tarantula           MOOC       December, 2012
Dragonfly           MOOC       December, 2012
MOOCDecember, 2012
Kabuto Mushi “Samurai December, 2012          Helmet” Beetle              MOOC
Eupatorus gracilicornis           MOOC       December, 2012
Euthysanius BeetleRoosevelt Elk        MOOC    December, 2012
Praying Mantis           MOOC       December, 2012
Two PrayingMantises                  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-s...
Flanged sphere• Similar to  concept  demo’d by  Palmer in  2000  (inspiration for  this work)                         MOOC...
MOOCDecember, 2012
MOOCDecember, 2012
MOOCDecember, 2012
Mathematica Package          MOOC      December, 2012
Applications in the Real WorldMathematical origami has found many applications in solving real-  world technological probl...
Miura “map-fold”• A map of Venice  with one degree of  freedom                           MOOC                       Decemb...
Miura-Ori, by Koryo Miura• First “origami in  space”• Solar array, flew  in 1995                          MOOC            ...
Umbrella          MOOC      December, 2012
5-meter prototype• The 5-meter  prototype folds to  about 1.5 meter.                           MOOC                       ...
Stents• Origami Stent graft developed by Zhong You (Oxford  University) and Kaori Kuribayashi                             ...
Folding DNA    • Paul Rothemund at Caltech      developed techniques to fold DNA      into origami shapesPaul Rothemund, “...
Origami5• Based on the 5th  International Conference on  Origami in Science,  Mathematics, and  Education (Singapore,  201...
Pots   http://www.langorigami.com            MOOC        December, 2012
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  • This is the crease pattern I showed earlier.
  • 2012 mooc lecture

    1. 1. Mathematical methods in origami Robert J. Lang www.langorigami.com MOOC December, 2012
    2. 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. 3. Modern Origami• Akira Yoshizawa (1911- 2005)• Inspired a worldwide renaissance of origami MOOC December, 2012
    4. 4. Origami Today• “Black Forest Cuckoo Clock,” (1987)• One sheet, no cuts MOOC December, 2012
    5. 5. Klein Bottle MOOC December, 2012
    6. 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. 7. Basic Folds of OrigamiValley fold M u tain fo on ld MOOC December, 2012
    8. 8. Crease Patterns QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. MOOC December, 2012
    9. 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. 10. Stag Beetle MOOC December, 2012
    11. 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. 12. The hard step• How do you make a bunch of flaps? MOOC December, 2012
    13. 13. How to make a flap MOOC December, 2012
    14. 14. Limiting process• Skinnier flap leads to…• A (quarter) circle! MOOC December, 2012
    15. 15. Other types of flap• Flaps can come from edges…• …and from the interior of the paper. MOOC December, 2012
    16. 16. Unify• They’re all circles MOOC December, 2012
    17. 17. Circle Packing• Many flaps: use many circles. MOOC December, 2012
    18. 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. 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. 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. 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. 22. Circles and Rivers• Pack circles, which represent all the body parts.• Fill in with molecular crease patterns.• Fold! MOOC December, 2012
    23. 23. MOOCDecember, 2012
    24. 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. 25. The crease pattern MOOC December, 2012
    26. 26. Whitetail Deer MOOC December, 2012
    27. 27. Mule DeerMule Deer MOOC December, 2012
    28. 28. Roosevelt Elk MOOC December, 2012
    29. 29. Bull Moose MOOC December, 2012
    30. 30. Tarantula MOOC December, 2012
    31. 31. Dragonfly MOOC December, 2012
    32. 32. MOOCDecember, 2012
    33. 33. Kabuto Mushi “Samurai December, 2012 Helmet” Beetle MOOC
    34. 34. Eupatorus gracilicornis MOOC December, 2012
    35. 35. Euthysanius BeetleRoosevelt Elk MOOC December, 2012
    36. 36. Praying Mantis MOOC December, 2012
    37. 37. Two PrayingMantises MOOC December, 2012
    38. 38. Representational MOOC December, 2012
    39. 39. Dancing Crane Dancing Crane MOOC December, 2012
    40. 40. Barn Owl Barn Owl MOOC December, 2012
    41. 41. Grizzly Bear MOOC December, 2012
    42. 42. Tree Frog MOOC December, 2012
    43. 43. Instrumentalists MOOC December, 2012
    44. 44. Organist MOOC December, 2012
    45. 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. 46. Flanged sphere• Similar to concept demo’d by Palmer in 2000 (inspiration for this work) MOOC December, 2012
    47. 47. MOOCDecember, 2012
    48. 48. MOOCDecember, 2012
    49. 49. MOOCDecember, 2012
    50. 50. Mathematica Package MOOC December, 2012
    51. 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. 52. Miura “map-fold”• A map of Venice with one degree of freedom MOOC December, 2012
    53. 53. Miura-Ori, by Koryo Miura• First “origami in space”• Solar array, flew in 1995 MOOC December, 2012
    54. 54. Umbrella MOOC December, 2012
    55. 55. 5-meter prototype• The 5-meter prototype folds to about 1.5 meter. MOOC December, 2012
    56. 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. 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. 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. 59. Pots http://www.langorigami.com MOOC December, 2012

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