MITNovember, 2004

Origami DesignTree Theory for Uniaxial Bases

Robert J. Langrobert@langorigami.com

MITNovember, 2004

Context

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.

MITNovember, 2004

Traditional Origami• Japanese newspaper from 1734: Crane, boat, table, “yakko-san”• By 1734, it is already well-developed

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

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.

MITNovember, 2004

Origamitoday

• “Black Forest Cuckoo Clock,”designed in 1987

• One sheet, no cuts• 216 steps

– not including repeats• Several hours to fold

MITNovember, 2004

Ibex

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Dragonfly

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Scaled Koi

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Western Pond Turtle

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Rattlesnake

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White-tailed Deer

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Bull Moose

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Bull Elephant

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Hummingbird & Trumpet Vine

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(Hummingbird Close-up)

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Grizzly Bear

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Roosevelt Elk

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Tree Frog

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Tarantula

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Murex

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Spindle Murex

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12-Spined Shell

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Banana Slug

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Spiral Tessellation

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Egg17 Tessellation

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Molecular Tessellation

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Chalk time…

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

MITNovember, 2004

Subtrees

MITNovember, 2004

Chalk time…

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

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

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.

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

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

MITNovember, 2004

Insetting

MITNovember, 2004

Gusset formation

aa

v2v 1

v3

v4

v5

Mv2v 1

v3

v4

v5

M

′v1 ′v2

′v3′v4

′v5hmax

MITNovember, 2004

Finished gussets

aa

v2v1

v3

v4

v5

M

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

MITNovember, 2004

Chalk time…

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

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

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.

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 atrobert@langorigami.com