2. Early (but not first)
• Japanese newspaper from 1734: Crane, boat, table, “yakko-
san”
• By 1734, origami is already well-developed
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December, 2012
3. Modern Origami
• Akira Yoshizawa (1911-
2005)
• Inspired a worldwide
renaissance of origami
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December, 2012
4. Origami Today
• “Black Forest Cuckoo
Clock,” (1987)
• One sheet, no cuts
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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
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December, 2012
7. Basic Folds of Origami
Valley fold M u tain fo
on ld
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December, 2012
8. Crease Patterns
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
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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?
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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?
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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
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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
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December, 2012
22. Circles and Rivers
• Pack circles, which represent all the body parts.
• Fill in with molecular crease patterns.
• Fold!
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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
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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.
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December, 2012
52. Miura “map-fold”
• A map of Venice
with one degree of
freedom
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December, 2012
53. Miura-Ori, by Koryo Miura
• First “origami in
space”
• Solar array, flew
in 1995
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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
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December, 2012