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1 Usenix Conference, Boston, MA June, 2008 From Flapping Birds to Space Telescopes: The Modern Science of Origami Robert J. Lang Usenix Conference, Boston, MA June, 2008 Background Origami Traditional form Modern extension Most common version: One Sheet, No Cuts

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Page 1: From Flapping Birds to Space Telescopes - USENIX · 2019-02-25 · Modern Origami Reborn by Yoshizawa A. Yoshizawa, Origami Dokuhon I Usenix Conference, Boston, MA June, 2008 Origami

1

Usenix Conference, Boston, MAJune, 2008

From Flapping Birds to Space Telescopes:The Modern Science of Origami

Robert J. Lang

Usenix Conference, Boston, MAJune, 2008

Background• Origami• Traditional form

• Modern extension• Most common version: One Sheet, No Cuts

Page 2: From Flapping Birds to Space Telescopes - USENIX · 2019-02-25 · Modern Origami Reborn by Yoshizawa A. Yoshizawa, Origami Dokuhon I Usenix Conference, Boston, MA June, 2008 Origami

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Usenix Conference, Boston, MAJune, 2008

Evolution of origami• Right: origami

circa 1797.• The traditional“tsuru” (crane)

Usenix Conference, Boston, MAJune, 2008

Even earlier…• Japanese newspaper from 1734: Crane, boat, table, “yakko-san”

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Usenix Conference, Boston, MAJune, 2008

Modern OrigamiReborn by Yoshizawa

A. Yoshizawa, Origami Dokuhon I

Usenix Conference, Boston, MAJune, 2008

Origami Today

• “Black Forest Cuckoo Clock,”designed in 1987

• One sheet, no cuts• 216 steps

– not including repeats• Several hours to fold

Page 4: From Flapping Birds to Space Telescopes - USENIX · 2019-02-25 · Modern Origami Reborn by Yoshizawa A. Yoshizawa, Origami Dokuhon I Usenix Conference, Boston, MA June, 2008 Origami

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Usenix Conference, Boston, MAJune, 2008

Ibex

Usenix Conference, Boston, MAJune, 2008

Klein Bottle

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Usenix Conference, Boston, MAJune, 2008

What Changed?• Origami was discovered by mathematicians.• Or rather, mathematical principles• 1950-2000…

– From about 100…– …to over 36,000! (see http://www.origamidatabase.com).

Usenix Conference, Boston, MAJune, 2008

The Technical Revolution• The connection between art and science is made by

mathematics.

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Usenix Conference, Boston, MAJune, 2008

Origami Mathematics• The mathematics underlying origami addresses three areas:

– Existence (what is possible)– Complexity (how hard it is)– Algorithms (how do you accomplish something)

• The scope of origami math include:Plane GeometryTrigonometrySolid GeometryCalculus and Differential GeometryLinear AlgebraGraph TheoryGroup TheoryComplexity/ComputabilityComputational Geometry

Usenix Conference, Boston, MAJune, 2008

Geometric Constructions• What shapes and distances can be constructed entirely by

folding?• Analogous to “compass-and-straightedge,” but more general

Page 7: From Flapping Birds to Space Telescopes - USENIX · 2019-02-25 · Modern Origami Reborn by Yoshizawa A. Yoshizawa, Origami Dokuhon I Usenix Conference, Boston, MA June, 2008 Origami

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Usenix Conference, Boston, MAJune, 2008

The Delian Problems• Trisect an angle• Double the cube• Square the circle• All three are impossible with compass and unmarked

straightedge, but:

Usenix Conference, Boston, MAJune, 2008

Hisashi Abe’s Trisection

aa

θ

A

B

D

C

P

A

B

D

C

EF

G

H

P

A

B

D

C

E F

P

A

B

D

C

EF

G

H

P

A

B

D

C

E F

G H

P

A

B

D

C

E F

G H

JP

A

B

D

C

E F

G H

P

A

B

D

C

EK

G H

J

θ/3

θ/3

θ/3

P

1. Begin with thedesired angle (PBCin this example)marked within thecorner of a square.

2. Make a horizontalfold anywhere acrossthe square, definingline EF.

3. Fold line BC up toline EF and unfold,creating line GH.

4. Fold the bottomleft corner up so thatpoint E touches lineBP and the corner,point B, touches lineGH.

5. With the cornerstill up, fold alllayers through theexisting crease thathits the edge at pointG and unfold.

6. Unfold corner B. 7. Fold along thecrease that runs topoint J, extending itto point B. Fold thebottom edge BC upto line BJ and unfold.

8. The two creasesBJ and BK divide theoriginal angle PBCinto thirds.

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Usenix Conference, Boston, MAJune, 2008

Peter Messer’s Cube Doubling

aa

4. Fold corner C tolie on line AB whilepoint I lies on lineFG.

3. Fold the top edgedown along ahorizontal fold totouch the creaseintersection andunfold. Then fold thebottom edge up totouch this new creaseand unfold.

2. Make a creaseconnecting points Aand C and anotherconnecting B and E.Only make themsharp where theycross each other.

1. Make a small foldhalfway up the rightside of the paper.

A

B

D

C

E

A

B

D

C

E

A

B

D

C

E

A

B

D

C

EF G

H I

5. Point C divides edgeAB into two segmentswhose proportions are 1and the cube root of 2.

A

B

D

F

H

I

C

G

1

23

Usenix Conference, Boston, MAJune, 2008

Binary Approximation for Distance

• Any distance can be approximated to 1/N using log2N foldstaken from its binary expansion

• Example: 0.7813 ~ 25/32 = .110012

a

.11001

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Usenix Conference, Boston, MAJune, 2008

Generalize Constructions• The binary algorithm is a special answer to a general question:

• Starting with a blank square,• for a given point or line,• construct an folding sequence accurate to a specified error,• defining every fold in the sequence in terms of preexisting

points and lines.

Usenix Conference, Boston, MAJune, 2008

Building Blocks• Points and Lines (creases)

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Usenix Conference, Boston, MAJune, 2008

PointsA point (mark) can only be defined as the intersection of two lines.But a line (fold) can be made in many ways…

Usenix Conference, Boston, MAJune, 2008

Lines• For many years, it was thought that there were only six ways to

define a fold.• The six operations are called the Huzita “Axioms.”

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Usenix Conference, Boston, MAJune, 2008

Huzita Axioms 2

Usenix Conference, Boston, MAJune, 2008

Hatori’s Axiom• In 2002, Koshiro Hatori discovered a seventh “axiom.”

• In 2006, it was observed that Jacques Justin had identified all 7in 1989.

• It has since been proven that these seven are the only ways todefine a single fold.

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Usenix Conference, Boston, MAJune, 2008

Geometric Constructions• One-fold-at-a-time origami can solve exactly:

– All quadratic equations with rational coefficients– All cubic equations with rational coefficients– Angle trisection (Abe, Justin)– Doubling of the cube (Messer)– Regular polygons for N=2i3j{2k3l+1} if last term is prime (Alperin,

Geretschläger)• All regular N-gons up to N=20 except N=11

Usenix Conference, Boston, MAJune, 2008

Simultaneous Creases• If you allow forming

two creases at one time,higher-order equationsare possible.

• An angle quintisection!

• Quintisections areimpossible with onlyHuzita (one-fold-at-a-time) axioms.

• There are over 400 two-fold-at-a-time “axioms.”

a

D

A

E

G

C

FH I

J

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Usenix Conference, Boston, MAJune, 2008

More simultaneous• What about N-at-a-time folding?

Usenix Conference, Boston, MAJune, 2008

Crease Patterns• The design of an origami figure is encoded in the crease pattern• What constraints are there on such patterns?

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Usenix Conference, Boston, MAJune, 2008

Properties of Crease Patterns• 2-colorability• Every flat-foldable origami crease pattern can be colored so

that no 2 adjacent facets are the same color with only 2 colors.

Usenix Conference, Boston, MAJune, 2008

Mountain-Valley Counting• Maekawa Condition:

– At any interior vertex, M – V = ±2

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Usenix Conference, Boston, MAJune, 2008

Angles Around a Vertex• Kawasaki Condition:

– Alternate angles around a vertex sum to a straight line– Independently discovered by Kawasaki, Justin, and Huffman– Generalized to 3D by Hull & belcastro

a

12

3

4

56

1 23 4

5 6

Usenix Conference, Boston, MAJune, 2008

Layer Ordering• A complete description of a folded form includes the layer

ordering among overlapping facets (M-V is not enough!)• Four necessary conditions were enumerated by Jacques Justin• Pictorially, these are the “legal” layer orderings between layers,

folded creases, and unfolded (flat) creases

aa

CCFOFi,1Fi,2Fj

Ci

Fi,1 Fi ,2Fj,1 Fj ,2

Ci

Cj

CUUCO

Fi,1Fi,2Fj Cj

Ci

CFUCO

Fi,1Fi,2Fi,1Fi,2 Cj

Ci

CFFCOFi,1

Fi,2

Fi,1Fi,2

Cj

Ci

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Usenix Conference, Boston, MAJune, 2008

Complexity• Satisfying M-V=±2 is “easy”• Satisfying alternate angle sums is “easy”• Satisfying layer order (M-V assignment) is “hard”…

• How hard?

Usenix Conference, Boston, MAJune, 2008

Pleats as logical signals• Two parallel pleats must be opposite parity• For a specified direction, there are 2 allowed crease assignments

a

Valley on right = “true” Valley on left = “false”

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Usenix Conference, Boston, MAJune, 2008

Not-All-Equal• A particular crease pattern enforces the condition “Not-All-Equal” on

its incident pleats

• It is possible to create multiple such conditions, thereby encoding NAElogic problems as crease assignment problems

a

?

Valid

FalseTrue

True

Invalid

TrueTrue

True

Crease Pattern Crease PatternFolded Form

Usenix Conference, Boston, MAJune, 2008

Crease Assignment Complexity

• Marshall Bern and Barry Hayes showed in 1996 that any NAE-3-SAT problem can be encoded as a crease assignment problem

• NAE-3-SAT is NP-complete!

• Ergo, “Origami is hard!”•

But most problems of interest are polynomial (still hard, butsolvable)

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Usenix Conference, Boston, MAJune, 2008

P.S.– Even if you have the complete crease assignment, simply

determining a valid layer ordering is still NP-complete!

Usenix Conference, Boston, MAJune, 2008

Flat-Foldability• A crease pattern is “flat foldable” iff it satisfies:

– Maekawa Condition (M-V parity) at every interior vertex– Kawasaki Condition (Angles) at every interior vertex– Justin Conditions (Ordering) for all facets and creases

Within this description, there are many interesting and unsolvedproblems!

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Usenix Conference, Boston, MAJune, 2008

But is it useful, or just fun?• The mathematical progression:• Flat-foldability rules (math)…• lead to crease pattern matching rules (application)…• and thus, the generation of beauty (art)…• and even practical functional objects ($$$)!

Usenix Conference, Boston, MAJune, 2008

Textures• Patterns of intersecting pleats can be integrated with other

folds to create textures and visual interest

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Usenix Conference, Boston, MAJune, 2008

The recipient form

Usenix Conference, Boston, MAJune, 2008

(Scaled Koi)

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Usenix Conference, Boston, MAJune, 2008

Western Pond Turtle

Usenix Conference, Boston, MAJune, 2008

Flag

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Usenix Conference, Boston, MAJune, 2008

Rattlesnake

Usenix Conference, Boston, MAJune, 2008

Flap Generation• The most extensive and powerful origami tools deal with the

generation of flaps in a desired configuration.

• Why is this useful?

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Usenix Conference, Boston, MAJune, 2008

Origami design• The fundamental problem of origami design is: given a desired

subject, how do you fold a square to produce a representationof the subject?

Usenix Conference, Boston, MAJune, 2008

Stag Beetle

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Usenix Conference, Boston, MAJune, 2008

A four-step process• The fundamental concept of design is the base• The fundamental element of the base is the flap

– From a base, it is relatively straightforward to shape the flaps intothe appendages of the subject.

• The hard step is:– Given a tree (stick figure), how do you fold a Base with the same

number, length, and distribution of flaps as the stick figure?

a

Subject Tree ModelBase

easy Hard easy

Usenix Conference, Boston, MAJune, 2008

How to make a flap• To make a single flap, we pick a corner and make it narrower.• The boundary of the flap divides the crease pattern into:

– Inside the flap– Everything else

• “Everything else” is available to make other flaps

a

L L

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Usenix Conference, Boston, MAJune, 2008

Limiting process• What does the paper look like as we make a flap skinner and

skinnier?• A circle!

a

L L LL

Usenix Conference, Boston, MAJune, 2008

Other types of flap• Flaps can come from edges…• …and from the interior of the paper.

a

L L

a

L L

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Usenix Conference, Boston, MAJune, 2008

Circle Packing• In the early 1990s, several of us realized that we could design

origami bases by representing all of the flaps of the base bycircles overlaid on a square.

a

Subject HypotheticalBase

Circle Packing

Usenix Conference, Boston, MAJune, 2008

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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Usenix Conference, Boston, MAJune, 2008

Molecules• Crease patterns that collapse a polygon so that its edges form a

stick figure are called “bun-shi,” or molecules (Meguro)• Each polygon forms a piece of the overall stick figure (Divide

and conquer).• Different molecules are known from the origami literature.• Triangles have only one possible molecule.

aaa

A

CB

DD

D

a a

b

b c

c

E

A

C

BD

a

bc

E A

C

BD

a

bc

the “rabbit ear” molecule

Usenix Conference, Boston, MAJune, 2008

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

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Usenix Conference, Boston, MAJune, 2008

Circles and Rivers• Pack circles, which represent all the body parts.• Fill in with molecular crease patterns.• Fold!

Usenix Conference, Boston, MAJune, 2008

Circle-River Design• The combination of circle-river packing and molecules allows an

origami composer to construct bases of great complexity usingnothing more than a pencil and paper.

• But what if the composer had more…• Like a computer?

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Usenix Conference, Boston, MAJune, 2008

Formal Statement of the Solution

• The search for the largest possible base from a given squarebecomes a well-posed nonconvex nonlinear constrainedoptimization:– Linear objective function– Linear and quadratic constraints– Nonconvex feasible region

• Solving this system of tens to hundreds of equations gives thesame crease pattern as a circle-river packing:

aaa

optimize m subject to:

m lij − ui,x − uj ,x( )2+ ui, y − uj , y( )2[ ]

1/2

≤ 0 for all i, j

0 ≤ ui, x ≤ 1, 0 ≤ ui, y ≤1 for all i

Usenix Conference, Boston, MAJune, 2008

Computer-Aided Origami Design

• 16 circles (flaps)• 9 rivers of assorted lengths• 120 possible paths• 184 inequality constraints• Considerations of symmetry add

another 16 more equalities• 200 equations total!• Child’s play for computers.• I have written a computer

program, “TreeMaker,” whichperforms the optimization andconstruction.

a

body

tail

hind leg hind leg

foreleg foreleg

neckhead

ears

antlers (4 tines each side)

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Usenix Conference, Boston, MAJune, 2008

The crease pattern

Usenix Conference, Boston, MAJune, 2008

(The folded figure)

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Usenix Conference, Boston, MAJune, 2008

Roosevelt Elk

Usenix Conference, Boston, MAJune, 2008

Bull Moose

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Usenix Conference, Boston, MAJune, 2008

Tarantula

Usenix Conference, Boston, MAJune, 2008

Dragonfly

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Usenix Conference, Boston, MAJune, 2008

Praying Mantis

Usenix Conference, Boston, MAJune, 2008

Two PrayingMantises

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Usenix Conference, Boston, MAJune, 2008

Grizzly Bear

Usenix Conference, Boston, MAJune, 2008

Tree Frog

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Usenix Conference, Boston, MAJune, 2008

Murex

Usenix Conference, Boston, MAJune, 2008

Spindle Murex

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Usenix Conference, Boston, MAJune, 2008

12-Spined Shell

Usenix Conference, Boston, MAJune, 2008

Instrumentalists

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Usenix Conference, Boston, MAJune, 2008

Organist

Usenix Conference, Boston, MAJune, 2008

TreeMaker• Algorithms are described in

– R. J. Lang, “A Computational Algorithm for Origami Design,” 12thACM Symposium on Computational Geometry, 1996

– R. J. Lang, Origami Design Secrets (A K Peters, 2003)• Macintosh/Linux/Windows binaries and source available (free!)

from– http://www.langorigami.com/treemaker.htm

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Usenix Conference, Boston, MAJune, 2008

Origami on Demand• Tools for origami design allow one to create an origami version

of “almost anything”• Recent years have seen origami commissioned for graphics,

advertisements, commercials

Usenix Conference, Boston, MAJune, 2008

Mitsubishi Endeavor

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Usenix Conference, Boston, MAJune, 2008

Assembly

Usenix Conference, Boston, MAJune, 2008

Origami SoAware• TreeMaker (Lang) -- shapes with appendages• Origamizer (Tachi) -- arbitrary surfaces• ReferenceFinder (Lang) -- finds folding sequences• Tess (Bateman) -- constructs origami tessellations• Rigid Simulator (Tachi) -- flexible surface linkages• Oripa (Jun Mitani) -- crease pattern folder• …and more!

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Usenix Conference, Boston, MAJune, 2008

Tachi’s Teapot

The “Utah teapot” Computed crease pattern

Usenix Conference, Boston, MAJune, 2008

Geometric Origami• Mathematical descriptions have permitted the construction of

elaborate geometrical objects from single-sheet folding:– Flat Tessellations (Resch, Palmer, Bateman, Verrill)3-D faceted tessellations (Fujimoto, Huffman)Curved surfaces (Huffman, Mosely)…and more!

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Usenix Conference, Boston, MAJune, 2008

Spiral Tessellation

Usenix Conference, Boston, MAJune, 2008

Egg17 Tessellation

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Usenix Conference, Boston, MAJune, 2008

Ron Resch• Computer scientist and artist

Ron Resch designed (andpatented) 2- and 3-Dtessellations back in the 1960s

• See US Patent 3,407,588.

Usenix Conference, Boston, MAJune, 2008

Ron Resch

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Usenix Conference, Boston, MAJune, 2008

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.

• Application in technology: origami rules don’t matter• …but no-cut-folding can be driven by technological reasons!

Usenix Conference, Boston, MAJune, 2008

Muira-Ori, by Koryo Miura• First “origami in

space”• Solar array, flew

in 1995

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Usenix Conference, Boston, MAJune, 2008

James Webb Space Telescope• Multiply segmented mirror folds into thirds

Usenix Conference, Boston, MAJune, 2008

JWST Stowage

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The “Eyeglass” Telescope• Under development at

Lawrence LivermoreNational Laboratory

• 25,000 miles above theearth

• 100 meter diameter (afootball field)

• Look up: see planets arounddistant stars

• Look down…

Usenix Conference, Boston, MAJune, 2008

The lens and the problem• The 100-meter lens must

fold up to 3 meters(shuttle bay)

• Lens must be made fromultra-thin sheets of glasswith flexures alonghinges

• What pattern to use?

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Analysis• Analyzed several families of

collapsing structures,including “flashers” andumbrella-liked patterns

• Initial modeling inMathematica™ solving NLCOthat enforce isometrybetween folded andunfolded state, followed by3D modeling at LLNL

Usenix Conference, Boston, MAJune, 2008

Umbrella

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Usenix Conference, Boston, MAJune, 2008

Manufacturability• “Umbrella” was selected based on

manufacturability issues• Non-origami issues drive

applications of origami

Usenix Conference, Boston, MAJune, 2008

Foldable 3.7 meter Eyeglass

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Usenix Conference, Boston, MAJune, 2008

5-meter prototype• The 5-meter

prototype folds upto about 1.5 meterdiameter.

Usenix Conference, Boston, MAJune, 2008

Solar Sail• Japanese Aerospace Exploration Agency• Mission flown in August 2004• First deployment of a solar sail in space• Pleated when furled, expands into sail

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Solar Sail

http://www.isas.jaxa.jp/e/snews/2004/0809.shtml

Usenix Conference, Boston, MAJune, 2008

NASA Sail• NASA, too, is

developingunfolded andinflatable solarsails.

Video courtesy Dave Murphy, AEC-Able Engineering,developed under NASA contract NAS803043

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Usenix Conference, Boston, MAJune, 2008

Paper Airplanes• JAXA approved “paper

airplane” from space studies• Prototype has survived

Mach 7 and 446°Ftemperature!

• Tracking?

Usenix Conference, Boston, MAJune, 2008

Stents

www.tulane.edu/~sbc2003/pdfdocs/0257.PDF

• Origami Stent graA developed by Zhong You (OxfordUniversity) and Kaori Kuribayashi

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Optics• “Optigami” -- simulation of

optical systems using origamireverse folds

• --Jon Myer, Hughes ResearchLaboratories, Applied Optics,1969

Usenix Conference, Boston, MAJune, 2008

Lasers• “Folded Cavity Laser” produces higher brightness than

conventional broad-area semiconductor lasers

U.S. Patent 6,542,529 by Mats Hagberg and Robert J. Lang

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Airbags• A mathematical

algorithm developedfor origami designturned out to be theproper algorithm forsimulating the flat-folding of an airbag.

Animation courtesy EASi Enginering GmbH

Usenix Conference, Boston, MAJune, 2008

Airbag Algorithm• The airbag-flattening algorithm was derived directly from the

universal molecule algorithm used in insect design.• More complex airbag shapes (nonconvex) can be flattened

using derivatives of Erik Demaine’s fold-and-cut algorithm.• No one foresaw these technological applications.• (Not uncommon in mathematics!)

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Resources• Further information may be found at

http://www.langorigami.com, or email me [email protected]