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Mathematical methods in origami Robert J. Lang www.langorigami.com MOOC December, 2012
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Early (but not first)• Japanese newspaper from 1734: Crane, boat, table, “yakko- san”• By 1734, origami is already well-developed MOOC December, 2012
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Modern Origami• Akira Yoshizawa (1911- 2005)• Inspired a worldwide renaissance of origami MOOC December, 2012
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Origami Today• “Black Forest Cuckoo Clock,” (1987)• One sheet, no cuts MOOC December, 2012
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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
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Basic Folds of OrigamiValley fold M u tain fo on ld MOOC December, 2012
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Crease Patterns QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. MOOC December, 2012
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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
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Other types of flap• Flaps can come from edges…• …and from the interior of the paper. MOOC December, 2012
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Unify• They’re all circles MOOC December, 2012
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Circle Packing• Many flaps: use many circles. MOOC December, 2012
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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
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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
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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
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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
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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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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
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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
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Miura “map-fold”• A map of Venice with one degree of freedom MOOC December, 2012
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Miura-Ori, by Koryo Miura• First “origami in space”• Solar array, flew in 1995 MOOC December, 2012
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5-meter prototype• The 5-meter prototype folds to about 1.5 meter. MOOC December, 2012
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Stents• Origami Stent graft developed by Zhong You (Oxford University) and Kaori Kuribayashi MOOC www.tulane.edu/~sbc2003/pdfdocs/0257.PDF December, 2012
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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
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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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