Origami design, the theory for uniaxial bases
-
Upload
geralt-rivia -
Category
Documents
-
view
74 -
download
7
description
Transcript of Origami design, the theory for uniaxial bases
![Page 2: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/2.jpg)
MITNovember, 2004
Context
![Page 3: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/3.jpg)
MITNovember, 2004
Background• Origami: Japanese paper-folding.
• Traditional form: Decorative abstract shapes & child’s craft
• Modern extension: a form of sculpture in which the primarymeans of creating the form consists of folding
• Most common version: a figure folded from one sheet of paper,usually a square, with no cuts.
![Page 4: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/4.jpg)
MITNovember, 2004
Traditional Origami• Japanese newspaper from 1734: Crane, boat, table, “yakko-san”• By 1734, it is already well-developed
![Page 5: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/5.jpg)
MITNovember, 2004
Modern Origami• The modern art form was reborn in the early 20th century
through the efforts of a Japanese artist, Akira Yoshizawa, whocreated new figures of artistic beauty and developed a writteninstructional language.
A. Yoshizawa, Origami Dokuhon I
![Page 6: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/6.jpg)
MITNovember, 2004
The Design Revolution• “Creative” origami caught on worldwide in the 1950s and 1960s.
• Beginning in the 1970s, many geometric design techniqueswere developed that enabled the creation of figures ofundreamed-of complexity.
• The mathematical theory of origami was greatly expanded inthe 1990s, leading to computer-aided origami design.
![Page 7: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/7.jpg)
MITNovember, 2004
Origamitoday
• “Black Forest Cuckoo Clock,”designed in 1987
• One sheet, no cuts• 216 steps
– not including repeats• Several hours to fold
![Page 8: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/8.jpg)
MITNovember, 2004
Ibex
![Page 9: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/9.jpg)
MITNovember, 2004
Dragonfly
![Page 10: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/10.jpg)
MITNovember, 2004
Scaled Koi
![Page 11: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/11.jpg)
MITNovember, 2004
Western Pond Turtle
![Page 12: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/12.jpg)
MITNovember, 2004
Rattlesnake
![Page 13: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/13.jpg)
MITNovember, 2004
White-tailed Deer
![Page 14: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/14.jpg)
MITNovember, 2004
Bull Moose
![Page 15: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/15.jpg)
MITNovember, 2004
Bull Elephant
![Page 16: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/16.jpg)
MITNovember, 2004
Hummingbird & Trumpet Vine
![Page 17: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/17.jpg)
MITNovember, 2004
(Hummingbird Close-up)
![Page 18: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/18.jpg)
MITNovember, 2004
Grizzly Bear
![Page 19: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/19.jpg)
MITNovember, 2004
Roosevelt Elk
![Page 20: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/20.jpg)
MITNovember, 2004
Tree Frog
![Page 21: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/21.jpg)
MITNovember, 2004
Tarantula
![Page 22: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/22.jpg)
MITNovember, 2004
Murex
![Page 23: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/23.jpg)
MITNovember, 2004
Spindle Murex
![Page 24: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/24.jpg)
MITNovember, 2004
12-Spined Shell
![Page 25: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/25.jpg)
MITNovember, 2004
Banana Slug
![Page 26: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/26.jpg)
MITNovember, 2004
Spiral Tessellation
![Page 27: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/27.jpg)
MITNovember, 2004
Egg17 Tessellation
![Page 28: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/28.jpg)
MITNovember, 2004
Molecular Tessellation
![Page 29: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/29.jpg)
MITNovember, 2004
Chalk time…
![Page 30: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/30.jpg)
MITNovember, 2004
A Tree & Active Polygons
aaa
1
234
56 7 8
91011
1213 14
1516
17
181920
21
222324
25 26
(root)
9
15
16
9
15
14
5 8
11
15
1617
20
21
23
25 26
![Page 31: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/31.jpg)
MITNovember, 2004
Subtrees
![Page 32: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/32.jpg)
MITNovember, 2004
Chalk time…
![Page 33: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/33.jpg)
MITNovember, 2004
Molecules• Crease patterns that collapse a polygon so that all edges lie on a
single line are called “bun-shi,” or molecules (Meguro)• Different bun-shi are known from the origami literature.• Triangles have only one possible molecule.
aaa
A
CB
DD
D
a a
b
b c
cE
A
C
BD
a
bc
E A
C
BD
a
bc
the “rabbit ear” molecule
![Page 34: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/34.jpg)
MITNovember, 2004
Quadrilateral molecules• There are two possible trees and several different molecules for a
quadrilateral.• Beyond 4 sides, the possibilities grow rapidly.
a
“4-star” “sawhorse”
Husimi/Kawasaki Maekawa Lang
![Page 35: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/35.jpg)
MITNovember, 2004
Four is enough• It is always possible to add flaps (circles) to a base so that the
only polygons are triangles and quadrilaterals, so thesemolecules suffice.
![Page 36: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/36.jpg)
MITNovember, 2004
Universal molecule• An algorithm that produces the crease pattern to collapse an
arbitrary valid convex polygon into a base whose projection is aspecified tree.
aa
0.449
0.354
0.194
0.325
0.393
0.372
0.627
![Page 37: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/37.jpg)
MITNovember, 2004
A pentagonal UM
aa
TA B
C
D
E F
P
pB
pC
pD
pE
pA
pF,2
pF,1
pF,4
pF,3
![Page 38: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/38.jpg)
MITNovember, 2004
Insetting
![Page 39: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/39.jpg)
MITNovember, 2004
Gusset formation
aa
v2v 1
v3
v4
v5
Mv2v 1
v3
v4
v5
M
′v1 ′v2
′v3′v4
′v5hmax
![Page 40: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/40.jpg)
MITNovember, 2004
Finished gussets
aa
v2v1
v3
v4
v5
M
![Page 41: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/41.jpg)
MITNovember, 2004
Creases & Folded Form
aa
P
pB
pC
pA pF,1
pE
pF,3
pD
pF,4
pF,2
pF,5
pF,6
pBpC
pA
pEpD
![Page 42: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/42.jpg)
MITNovember, 2004
Chalk time…
![Page 43: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/43.jpg)
MITNovember, 2004
Universal Molecule 1
a
1
2
3
45
6
7 8
1
1
1
3
5
5
7 85
5
3
3
6
1
42
![Page 44: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/44.jpg)
MITNovember, 2004
Universal Molecule 2
a
1
23 4 5
6 7
8 9 10 11
12 13 14 15
1
1 11
2 3
4
4
4
5
7
7
11
10
1514
13
12
6
8
8
8 6 9 9
9
6
7
![Page 45: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/45.jpg)
MITNovember, 2004
Resources• Origami design software TreeMaker (with 170 pp manual) can
be downloaded from:– http://origami.kvi.nl/programs/treemaker
• …or Google-search for “TreeMaker”• Version 5.0 (Mac/Linux/Windows) is under construction.• Other origami-related software, including ReferenceFinder, is
at the same site• Theory described in 12 ACM SCG paper, “An Algorithm for
Origami Design” (1996) by Robert J. Lang.
![Page 46: Origami design, the theory for uniaxial bases](https://reader033.fdocuments.us/reader033/viewer/2022050721/55cf91a1550346f57b8f1697/html5/thumbnails/46.jpg)
MITNovember, 2004
More Resources• Origami Design Secrets, my new book teaching how to design
origami (and more), was published by A. K. Peters in October2003.
• Origami Insects II, my latest, contains a collection of fairlychallenging insect designs
• Both (and other books) available from the OrigamiUSA Source(www.origami-usa.org).
• Further information may be found athttp://www.langorigami.com, or email me [email protected]