An excursion through the art, math andapplications of origami

Francois Monard

SIMUW 2014University of Washington, July 16th 2014

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Outline

1 Origami: the art

2 Origami, math and engineering

3 Session I: Modules

4 Session II: Tesselations

Origami: the art

Historically. . .

Origami = Ori (folding) + kami (paper).

Started in 17th century Japan.

Origami butterflies were used in Shinto weddings to representthe bride and groom.

“Noshi” were offered to samurai warriors as good luck tokens.

Folding 1000 cranes: a symbolic ritual.

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Origami: the art

Representational origami: Akira Yoshizawa (1911 -2005, Jp)

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Origami: the art

Representational origami: Eric Joisel (1956-2010, Fr)

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Origami: the art

Representational origami: Satoshi Kamiya (1981- , Jp)

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Origami: the art

Representational Origami: Brian Chan

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Origami: the art

Representational Origami: Brian Chan

Crease pattern of Mens et Manus.

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Origami: the art

Modular origami: Tomoko Fuse (1951-, Jp)

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Origami: the art

Modular + street art: Mademoiselle Maurice (1984-,Fr)

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Origami: the art

Along the same lines

Butterfly decoration, MelinaHermsen.

Koi, Chang rok Yoo (Model:Sipho Mabona)

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Origami: the art

Tesselations - Melina Hermsen (1983-, De)

Flowers of May Variations Wheel of butterflies

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Origami: the art

Curved creases - Erik Demaine (1981-, US)

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Origami: the art

. . . and many more . . .

Chris Palmer, Paul Jackson, Thomas Hull, Jun Maekawa, MichaelLaFosse, Brian Chan, John Montroll, Sipho Mabona, Peter Engel,Eric Gjerde, Joel Cooper, Christine Edison, . . .

Watch ! Between the Folds, a film by Vanessa Gould.

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Outline

1 Origami: the art

2 Origami, math and engineering

3 Session I: Modules

4 Session II: Tesselations

Origami, math and engineering

Mathematical questions: Fold-and-cut problem

On your papers !

Cut out this rectangle using folds and a single straight cut.

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Origami, math and engineering

Mathematical questions: Fold-and-cut problem

Question: could one obtain a prescribed polygon out of many foldsand a single straight cut ?Answer: Yes ! It’s been proved in two ways.

By E. Demaine, M. Demaine and A. Lubiw using thestraight-skeleton method.

By M. Bern, E. Demaine, D. Eppstein and B. Hayes using thedisk-packing method.

Source: Erik Demaine’s website.

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Origami, math and engineering

Other mathematical questions: flat-foldability

Question: Can one fold a given crease pattern into a 2-dimensionalmodel ?Answer: It must obey four simple rules.

1 it must be two-colorable.2 Maekawa’s theorem: at any vertex the number of valley and

mountain folds always differ by two in either direction.3 Kawasaki’s theorem: at any vertex, the sum of all the odd

angles adds up to 180 degrees, as do the even.4 a sheet can never penetrate a fold.

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Origami, math and engineering

Improvements in transmission techniques

Direct use of crease patterns (instead of step-wise instructions) topass on models.

more challenging (yeah !),more direct.

Designs by Robert Lang.

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Origami, math and engineering

Computer assisted conception

Representational origami is usually a 2-step process:

1 Folding the base (“coarse” structure of the model),

2 Folding the finishes.

TreeMaker, a program developped by Robert Lang, allows to createthe crease pattern of a base given a simple “tree” of the model.

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Origami, math and engineering

Origami and Engineering 1/2

Eyeglass telescope Folding airbags optimally(using TreeMaker)

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Origami, math and engineering

Origami and Engineering 2/2

Blood stents.

Protein folding (Erik Demaine).

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Outline

1 Origami: the art

2 Origami, math and engineering

3 Session I: Modules

4 Session II: Tesselations

Session I: Modules

Main idea

Idea: Build a simple unit (or two or three. . . ) and combine manyinstances of it.

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Session I: Modules

Solids

Source: Ornamental Origami, Meenakshi Mukerji

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Session I: Modules

Solids

Source: Ornamental Origami, Meenakshi Mukerji23 / 35

Session I: Modules

Colorings

Source: Ornamental Origami, Meenakshi Mukerji

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Session I: Modules

Taking it further

Build a shape satisfying:

All edges have the same length,

All faces are triangles,

and complete it with the proposed unit.

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Outline

1 Origami: the art

2 Origami, math and engineering

3 Session I: Modules

4 Session II: Tesselations

Session II: Tesselations

Tesselations

Moorish designs (ex: the Alhambra, Granada, Spain)

Idea: use origami to replicate tilings of the plane.

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Session II: Tesselations

Tesselations - Eric Gjerde (1/2)

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Session II: Tesselations

Tesselations - Eric Gjerde (2/2)

42 1-meter long sheets of Dragon helix design.

More than one kilometer of creasing.

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Session II: Tesselations

Tesselations (+ sculpture) - Joel Cooper (1/2)

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Session II: Tesselations

Tesselations (+ sculpture) - Joel Cooper (2/2)

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Session II: Tesselations

Tesselations - Robert Lang vs. M.C. Escher

How about tiling the hyperbolic plane ?

Hyperbolic Limit, opus 600,Robert Lang.

Circle Limit, M.C. Escher.

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Session II: Tesselations

Tesselations - Square grid

(Diagrams: Eric Gjerde.)

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Session II: Tesselations

Tesselations - elementary twists

90 degree pleat intersection. Square Twist

(Diagrams: Eric Gjerde.)

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Session II: Tesselations

What to do with these two kinds of twist ?

Modern Blue, Christine Edison.

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