geometry of flat origami triangulations · origami in nature and engineering saito et al, pnas 2017...

Geometry of Flat Origami Triangulations

Bryan Gin-ge Chen & Chris SantangeloUMass Amherst Physics

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Origami in nature and engineering

Andresen et al, PRE 2007Saito et al, PNAS 2017

Wood et al, Science 2015

J.-H. Na et al., Adv. Mat. 2015

Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/

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


But how much does a crease pattern really tell us?




What does it tell us near the flat state?


Triangulated Vb-gon : Vb boundary vertices

Rigid origami as bond-node structure

“bird base”



“bird base”Demaine et al, Graphs and Combinatorics, 2011




Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices




3Vint+Vb-3: # of folds





# of degrees of freedom?







N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)






N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)=Vb + 3

=6 + (Vb � 3)




rigid body motions




=6 + (Vb � 3)




generically Vb-3 dofrigid body motions




=6 + (Vb � 3)

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BGC and Santangelo, 2017

Configuration space near the flat state

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Configuration space near the flat state

4 branches

Flat is not generic!

Vint+1 linear motions!




generically 1dofvs ??




Toy example:

linear motion




Toy example:

linear motion


redundant constraints = ‘self stress’



Toy example:

linear motion

compression

tension


redundant constraints = ‘self stress’



Toy example:

Self stresses and second-order constraints

compression

tension

linear motion


compression

tension

linear motion

The second-order constraints are in 1 to 1 correspondence with self stresses!

Connelly and Whiteley, SIAM J Discrete Math 1996


compression

tension

linear motion



uT⌦u = 0 ⌦ symmetric “stress matrix”


compression

tension

linear motion



uT⌦u = 0 ⌦ symmetric “stress matrix”

u⌦

Self stresses in flat triangulations



“wheel stress”



“wheel stress”


u vertical displacements


“wheel stress”


uT⌦u = 0

⌦ symmetric stress matrix



“wheel stress”


uT⌦u = 0


↵1,2

↵2,3↵4,1

�1,2�2,3

�4,1

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Gaussian curvature vanishes at each vertex



“wheel stress”


uT⌦u = 0


↵1,2

↵2,3↵4,1

�1,2�2,3

�4,1

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Gaussian curvature vanishes at each vertex()


!!

origami n-vertex configuration space


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uT⌦u = 0

⌦(n+1)x(n+1) symmetric

stress matrix

u (n+1)-vector of vertical displacements

3-dim kernel from isometries


Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997


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uT⌦u = 0


stress matrix


Always exactly one negative eigenvalue!



Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997


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uT⌦u = 0


stress matrix


Always exactly one negative eigenvalue!


BGC, Theran and Nixon, 2017

What are the two nappes?

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Demaine et al, Proceedings of the IASS, 2016



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Negative eigenvectormaximizes Gaussian curvature



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The two nappes correspond to popped up and popped down configurations!


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Abel et al, JoCG, 2016; Streinu and Whiteley, 2005

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Multiple vertex configuration space

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Vint = 2 ⇒ 2 wheel stresses



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2 homogeneous quadratic equations in 3 unknowns


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2 homogeneous quadratic equations in 3 unknowns

Bézout’s theorem: at most 2^2 solutions


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Exactly 2^Vint solutions ???

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Exactly 2^Vint solutions ???2^Vint*

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Exactly 2^Vint solutions ???

vertex sign patterns seem to uniquely label

pairs of branches!

2^Vint*

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Yes, if the crease patternis constructed with Henneberg-Imoves from a pair of triangles!

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Demaine et al, Graphs and Combinatorics, 2011

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Demaine et al, Graphs and Combinatorics, 2011

How to show that all vertex sign patterns are realized twice?



# of branches ≤ 2^(Vint)




# of Mountain-Valley choices = 2#creases = 2^(3Vint+1)# of branches ≤ 2^(Vint)




# of Mountain-Valley choices = 2#creases = 2^(3Vint+1)Only a tiny fraction of MV’s can be realized!





# of Mountain-Valley choices = 2#creases = 2^(3Vint+1)

Maybe we’re in good shape…

Only a tiny fraction of MV’s can be realized!



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Hull and Tachi, J Mechanisms Robotics, 2017Abel et al, JoCG, 2016

Brunck et al, PRE, 2016

One crease pattern with fixed M-V labels : two branches!


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Same M-V labels, same vertex sign pattern : two branches!

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Thanks!

NSF PHY-1125915EFRI ODISSEI-1240441

Tom Hull, Louis Theran

2^Vint branches from popping vertices up / down?

The flat state is singular,but self stresses help us navigate…

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Summary:

Do these second-order motions generically extend

to continuous motions?

geometry of flat origami triangulations · origami in nature and engineering saito et al, pnas 2017...

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