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Origami Design

Tree Theory for Uniaxial Bases

Robert J. Lang

robert@langorigami.com

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Context

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Background

Origami: Japanese paper-folding.

Traditional form: Decorative abstract shapes & childs craft

Modern extension: a form of sculpture in which the primary

means of creating the form consists of folding

Most common version: a figure folded from one sheet of paper,

usually a square, with no cuts.

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Traditional Origami

Japanese newspaper from 1734: Crane, boat, table,yakko-san

By 1734, it is already well-developed

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Modern Origami

The modern art form was reborn in the early 20th century

through the efforts of a Japanese artist, Akira Yoshizawa, who

created new figures of artistic beauty and developed a written

instructional language.

A. Yoshizawa, Origami Dokuhon I

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The Design Revolution

Creativeorigami caught on worldwide in the 1950s and 1960s.

Beginning in the 1970s, many geometric design techniques

were developed that enabled the creation of figures of

undreamed-of complexity.

The mathematical theory of origami was greatly expanded in

the 1990s, leading to computer-aided origami design.

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Origami

today Black Forest Cuckoo Clock,

designed in 1987

One sheet, no cuts

216 steps

not including repeats

Several hours to fold

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Ibex

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November,2004

Dragonfly

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November,2004

Scaled Koi

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November,2004

Western Pond Turtle

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November,2004

Rattlesnake

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November,2004

White-tailed Deer

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November,2004

Bull Moose

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November,2004

Bull Elephant

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November,2004

Hummingbird & Trumpet Vine

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(Hummingbird Close-up)

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November,2004

Grizzly Bear

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November,2004Roosevelt Elk

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November,2004

Tree Frog

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November,2004

Tarantula

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November,2004

Murex

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November,2004

Spindle Murex

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November,2004

12-Spined Shell

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November,2004

Banana Slug

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November,2004

Spiral Tessellation

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November,2004

Egg17 Tessellation

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November,2004

Molecular Tessellation

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November,2004

Chalk time

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A Tree & Active Polygons

1

2

34

56

7

8

910

11

12

13 14

15

16

17

181920

21

22

2324

25 26

(root)

9

15

16

9

15

14

5 8

11

15

16

17

20

21

23

25 26

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Subtrees

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Chalk time

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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.

A

CB

DD

D

a a

b

b c

c

E

A

C

BD

a

b

c

E A

C

B

D

a

b

c

the rabbit ear molecule

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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.

4-star sawhorse

Husimi/Kawasaki Maekawa Lang

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Four is enough

It is always possible to add flaps (circles) to a base so that the

only polygons are triangles and quadrilaterals, so these

molecules suffice.

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Universal molecule

An algorithm that produces the crease pattern to collapse an

arbitrary valid convex polygon into a base whose projection is a

specified tree.

0.449

0.354

0.194

0.325

0.393

0.372

0.627

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A pentagonal UM

T

A B

C

D

EF

P

pB

pC

pD

pE

pA

pF,2

pF,1

pF,4

pF,3

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Insetting

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Gusset formation

v2

v1

v3

v4

v5

M

v2

v1

v3

v4

v5

M

v1v2

v3

v4

v5

hmax

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Finished gussets

v2

v1

v3

v4

v5

M

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Creases & Folded Form

P

pB

pC

pA pF,1

pE

pF,3

pD

pF,4

pF,2

pF,5

pF,6

pBpC

pA

pEpD

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Chalk time

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Universal Molecule 1

1

2

3

4

5

6

7 8

1

1

1

3

5

5

7 85

5

3

3

6

1

42

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Universal Molecule 2

1

2

3 4 5

6 7

89

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

86

99

9

6

7

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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.

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More Resources

Origami Design Secrets, my new book teaching how to designorigami (and more), was published by A. K. Peters in October

2003.

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 atrobert@langorigami.com