building with butterflies: folding concentric pleated polygons

Building with Butterflies: Folding Concentric Pleated Polygons

Bryan Gin-ge ChenLeiden University

Thursday, January 10, 13

Origami is an artform aimed at creating 3D objects from 2D paper:

Tomohiro Tachis Origamizer, 2011

Satoshi Kamiya

rigid origami


Daniel Piker, after Ron Resch, Ben Parker and John Mckeeve

http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

Origami as metamaterial


http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/http://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

Origami corrugations on flickr.comThursday, January 10, 13

The study of origami as a method of making new materials is still in its infancy.

Origami artists have been mostly alchemists; we should attempt to develop the corresponding chemistry.

Robert Langs Treemaker / Universal Molecule algorithm

Bowers, Streinu, Proc. ADG 2012


The butterflies of my first slide turn out to be a useful family of atoms for thinking about rigid origami.

I will discuss mostly the geometry of the substance / molecule known as the hypar.


erikdemaine.org/hypar


erikdemaine.org/hypar

Christmas tree star


Demaine, Demaine, Lubiw 1999

Polyhedral sculptures


Fernando Sierra, 2009

http://www.flickr.com/photos/elelvis/3179640715/in/photostream

p- and d- surfaces


http://www.flickr.com/photos/elelvis/3179640715/in/photostreamhttp://www.flickr.com/photos/elelvis/3179640715/in/photostream

????

...


Demaine et al, 2011

Key ideas:Interior polygons must remain planar.

A vertex with all mountain or all valley folds coming out cannot be folded with planar faces.

However, any folded internal ring would induce such a vertex by extending the paper.


What actually happens?


http://www.grasshopper3d.com/forum/topics/folding-paper

Mrten Nettelbladt, 2010


http://www.grasshopper3d.com/forum/topics/folding-paperhttp://www.grasshopper3d.com/forum/topics/folding-paper

Demaine et al, 2011

????

Non-alternating Alternating


These two do exist for all angles and all layers! (But they might self-intersect)

BGC 2011Thursday, January 10, 13

BGC 2011

First, these crease patterns can be decomposed into butterflies:


We build the hypar from the innermost layer outwards.

The first stage is just a square folded by some angle along its diagonal.

Can we glue the first butterfly?


Each butterfly has a single degree of freedom which changes the distance between the wingtips.

This distance is maximized (or minimized) when the butterfly is planar.

If the distance between the tips of the target is in between these values, we can always glue the butterfly either above or below the plane through ABC.

Can we glue the first butterfly?


The distance shrinks!

|~a+~b| < |~a|+ |~b|


Distances always shrink!

Therefore, these exist (as possibly self-intersecting immersions) for all angles and all numbers of layers!


x

a

2

y

b

2= 2z

How close are they to the real Hyperbolic Paraboloid?


x

a

2

y

b

2= 2z

x

a

yb

=2z

u

x

a

+y

b

= u

x

a

+y

b

=2z

v

x

a

yb

= v

The real Hyperbolic Paraboloid

Doubly ruled:


x

a

2

y

b

2= 2z

x(u, v) = a

u+ v

2

y(u, v) = b

u v2

z(u, v) =uv

2


Parametric form:


x

a

2

y

b

2= 2z



x

a

2

y

b

2= 2z


(0, bk,k2/2)(ak, 0, k2/2)

(ak, 0, k2/2)

(0,bk,k2/2)

4-bar linkage


x

a

2

y

b

2= 2z


(0, bk,k2/2)(ak, 0, k2/2)

(ak, 0, k2/2)

(0,bk,k2/2)

4-bar linkage

BGC 2011

Fit the creases!


x

a

2

y

b

2= 2z


(0, bk,k2/2)(ak, 0, k2/2)

(ak, 0, k2/2)

(0,bk,k2/2)

4-bar linkage

+(0, 0,)

BGC 2011

Fit the creases!


I computed the change in a, b, from layer to layer as an asymptotic series in 1/k...

(0,bnkn,k2n/2 +n)(ankn, 0, k

2n/2 +n)

(0,bn+1kn+1,k2n+1/2 +n+1)(an+1kn+1, 0, k

2n+1/2 +n+1)

d(Vi,j , Vi,j1) = pi,j1

d(Vi,j , Vi,j) = pi,j

d(Vi,j , Vi,j+1) = pi,j1


nonalternating:

alternating:n+2 n

=a2 + b2p

2k2 a+ 2b

2k3+O(k4)

kn+2 k

=

p2

k

k3+O(k4)

an+2 a

=

k3+

a(b2 a2)p2k4

+O(k5)

bn+2 b

=b(a2 b2)p

2k4+

(2a2 + ab+ 2b2)

2k5+O(k6)

n+2 n

=p2(a2 b2) +

2k2 3(a+ b)

4k3+O(k4)

kn+2 k

=

p2

k

k3+O(k4)

an+2 a

=

2k3+

p2a(b2 a2) + (b 2a)

2k4+O(k5)

bn+2 b

= 2k3

+

p2b(b2 a2) + (a 2b)

2k4+O(k5)


(I dropped most subscripts when they are n) BGC 2011Thursday, January 10, 13

kn+2 k

p2

kkn

q2p2n+ C

Nonalternating:

n+2 n

p2(a2 b2) +

2k2

=O(1/n)

an+2 a

=O(n3/2)

bn+2 b

=O(n3/2)



kn+2 k

p2

kkn

q2p2n+ C

Nonalternating:

n+2 n

p2(a2 b2) +

2k2

=O(1/n)

an+2 a

=O(n3/2)

bn+2 b

=O(n3/2)


n = O(log n)


kn+2 k

p2

kkn

q2p2n+ C

Nonalternating:

n+2 n

p2(a2 b2) +

2k2

=O(1/n)

an+2 a

=O(n3/2)

bn+2 b

=O(n3/2)


n = O(log n)

Folding the nonalternating pattern doesnt approximate a hyperbolic paraboloid! BGC 2011



kn+2 k

p2

kkn

q2p2n+ C

Alternating:We can show that an bn = O(1/k)

n+2 n

=a2 + b2p

2k2 a+ 2b

2k3+O(k4)

=O(k3)

=O(n3/2)

an+2 a

=O(k3) = O(n3/2)

bn+2 b

=b(a2 b2)p

2k4+O(k5)

=O(k5) = O(n5/2)BGC 2011



kn+2 k

p2

kkn

q2p2n+ C

Alternating:We can show that an bn = O(1/k)

n+2 n

=a2 + b2p

2k2 a+ 2b

2k3+O(k4)

=O(k3)

=O(n3/2)

an+2 a

=O(k3) = O(n3/2)

bn+2 b

=b(a2 b2)p

2k4+O(k5)

=O(k5) = O(n5/2)

Everything converges! Whew!



5000 10000 15000 20000 25000layer

0.5

1.0

1.5

2.0

2.5

3.0a,b

alternating

5000 10000 15000 20000 25000layer

0.3

0.4

0.5

0.6

Dalternating Heven>oddL

5000 10000 15000 20000 25000layer

0.5

1.0

1.5

2.0

2.5

3.0a,b

nonalternating Ha>bL

5000 10000 15000 20000 25000layer

-0.4

-0.2

0.2

0.4

0.6

Dnonalternating Heven>oddL

Non-alternating

Measurements:


0.5 1.0 1.5 2.0 2.5 3.0q

-4

-2

2

4

6

a,bAspect Ratio for nonalternating pattern

=0.1

=/2-.01BGC 2011


The argument for existence was really general; rigid foldings exist for these patterns too!


Concentric hexagons:


Concentric octagons:


Concentric decagons:


z = Im(x+ iy)

nn1

Rob Kusner suggested

However, at long distances, z doesnt seem to scale the right way...?

n d

2 2

3 ~1.33

4 ~1.17

5 ~1.11

z rd

This works when 2n=4, and the branch structure seems to work for all pleated 2n-gons!

nonalternating BGC 2011Thursday, January 10, 13

Other initial conditions....

There are limiting polygon shapes:


Constant ratio offsets move us in shape space nicer:


Vertex trails of the normalized polygons

BGC 2011

A cycle!


Two quick applications


Miura-ori

Wei et al, arXiv:1211.6396Thursday, January 10, 13

Extra creases in Miura-ori: How many degrees of freedom?

Wei et al, arXiv:1211.6396Thursday, January 10, 13

Wu, You, Proc. R. Soc A 2011

Balkcom et al, 2009

Rigidly folding a shopping bag?


Wu, You, Proc. R. Soc A 2011


Ive presented butterflies as a useful tool for understanding the possible rigid motions of folding patterns:

They should also be useful as atoms for understanding bending energies as well!

The Future:


Thanks!

2011 IMA workshop on strain induced shape formation

Acknowledgements:

NSF DMR05-47230

Chris Santangelo, Marcelo DiasRob Kusner


building with butterflies: folding concentric pleated polygons

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