Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017...
Transcript of Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017...
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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
![Page 3: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/3.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
![Page 4: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/4.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
![Page 5: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/5.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
![Page 6: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/6.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
![Page 7: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/7.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://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?
![Page 8: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/8.jpg)
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
![Page 9: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/9.jpg)
“bird base”
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
![Page 10: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/10.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
![Page 11: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/11.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
Rigid origami as bond-node structure
![Page 12: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/12.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
3Vint+Vb-3: # of folds
Rigid origami as bond-node structure
![Page 13: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/13.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
3Vint+Vb-3: # of folds
Rigid origami as bond-node structure
![Page 14: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/14.jpg)
“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?
Rigid origami as bond-node structure
![Page 15: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/15.jpg)
“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?
Rigid origami as bond-node structure
N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)
![Page 16: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/16.jpg)
“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?
Rigid origami as bond-node structure
N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)=Vb + 3
=6 + (Vb � 3)
![Page 17: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/17.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
3Vint+Vb-3: # of folds
rigid body motions
# of degrees of freedom?
Rigid origami as bond-node structure
N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)=Vb + 3
=6 + (Vb � 3)
![Page 18: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/18.jpg)
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
3Vint+Vb-3: # of folds
generically Vb-3 dofrigid body motions
# of degrees of freedom?
Rigid origami as bond-node structure
N0 = 3(Vint + Vb)� (3Vint + Vb � 3 + Vb)=Vb + 3
=6 + (Vb � 3)
![Page 19: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/19.jpg)
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BGC and Santangelo, 2017
Configuration space near the flat state
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BGC and Santangelo, 2017
Configuration space near the flat state
![Page 21: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/21.jpg)
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Configuration space near the flat state
4 branches
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BGC and Santangelo, 2017
![Page 23: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/23.jpg)
BGC and Santangelo, 2017
![Page 24: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/24.jpg)
Flat is not generic!
![Page 25: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/25.jpg)
Vint+1 linear motions!
Flat is not generic!
![Page 26: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/26.jpg)
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
![Page 27: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/27.jpg)
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
Toy example:
![Page 28: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/28.jpg)
linear motion
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
Toy example:
![Page 29: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/29.jpg)
linear motion
Vint+1 linear motions!
redundant constraints = ‘self stress’
Flat is not generic!
generically 1dofvs ??
Toy example:
![Page 30: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/30.jpg)
linear motion
compression
tension
Vint+1 linear motions!
redundant constraints = ‘self stress’
Flat is not generic!
generically 1dofvs ??
Toy example:
![Page 31: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/31.jpg)
Self stresses and second-order constraints
compression
tension
linear motion
![Page 32: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/32.jpg)
Self stresses and second-order constraints
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
![Page 33: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/33.jpg)
Self stresses and second-order constraints
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
uT⌦u = 0 ⌦ symmetric “stress matrix”
![Page 34: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/34.jpg)
Self stresses and second-order constraints
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
uT⌦u = 0 ⌦ symmetric “stress matrix”
u⌦
![Page 35: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/35.jpg)
Self stresses in flat triangulations
BGC and Santangelo, 2017
![Page 36: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/36.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
![Page 37: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/37.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
![Page 38: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/38.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
u vertical displacements
![Page 39: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/39.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
uT⌦u = 0
⌦ symmetric stress matrix
u vertical displacements
![Page 40: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/40.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
uT⌦u = 0
⌦ symmetric stress matrix
↵1,2
↵2,3↵4,1
�1,2�2,3
�4,1
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3
4
Gaussian curvature vanishes at each vertex
u vertical displacements
![Page 41: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/41.jpg)
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
uT⌦u = 0
⌦ symmetric stress matrix
↵1,2
↵2,3↵4,1
�1,2�2,3
�4,1
1
2
3
4
Gaussian curvature vanishes at each vertex()
u vertical displacements
!!
![Page 42: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/42.jpg)
origami n-vertex configuration space
BGC and Santangelo, 2017
1 2
3
4
5
6
uT⌦u = 0
⌦(n+1)x(n+1) symmetric
stress matrix
u (n+1)-vector of vertical displacements
3-dim kernel from isometries
![Page 43: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/43.jpg)
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
1 2
3
4
5
6
uT⌦u = 0
⌦(n+1)x(n+1) symmetric
stress matrix
u (n+1)-vector of vertical displacements
Always exactly one negative eigenvalue!
3-dim kernel from isometries
![Page 44: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/44.jpg)
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
1 2
3
4
5
6
uT⌦u = 0
⌦(n+1)x(n+1) symmetric
stress matrix
u (n+1)-vector of vertical displacements
Always exactly one negative eigenvalue!
3-dim kernel from isometries
BGC, Theran and Nixon, 2017
![Page 45: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/45.jpg)
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
1 2
3
4
5
6
uT⌦u = 0
⌦(n+1)x(n+1) symmetric
stress matrix
u (n+1)-vector of vertical displacements
Always exactly one negative eigenvalue!
3-dim kernel from isometries
BGC, Theran and Nixon, 2017
![Page 46: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/46.jpg)
What are the two nappes?
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3
4
5
6
BGC and Santangelo, 2017
![Page 47: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/47.jpg)
What are the two nappes?
1 2
3
4
5
6
BGC and Santangelo, 2017
![Page 48: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/48.jpg)
What are the two nappes?
1 2
3
4
5
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BGC and Santangelo, 2017
![Page 49: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/49.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
![Page 50: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/50.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
Negative eigenvectormaximizes Gaussian curvature
BGC and Santangelo, 2017
![Page 51: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/51.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
![Page 52: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/52.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
![Page 53: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/53.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
![Page 54: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/54.jpg)
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
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BGC and Santangelo, 2017
![Page 55: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/55.jpg)
What are the two nappes?
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The two nappes correspond to popped up and popped down configurations!
Demaine et al, Proceedings of the IASS, 2016
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Abel et al, JoCG, 2016; Streinu and Whiteley, 2005
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BGC and Santangelo, 2017
![Page 56: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/56.jpg)
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BGC and Santangelo, 2017
Multiple vertex configuration space
![Page 57: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/57.jpg)
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BGC and Santangelo, 2017
Multiple vertex configuration space
![Page 58: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/58.jpg)
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Vint = 2 ⇒ 2 wheel stresses
BGC and Santangelo, 2017
Multiple vertex configuration space
![Page 59: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/59.jpg)
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Vint = 2 ⇒ 2 wheel stresses
BGC and Santangelo, 2017
2 homogeneous quadratic equations in 3 unknowns
Multiple vertex configuration space
![Page 60: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/60.jpg)
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Vint = 2 ⇒ 2 wheel stresses
BGC and Santangelo, 2017
2 homogeneous quadratic equations in 3 unknowns
Bézout’s theorem: at most 2^2 solutions
Multiple vertex configuration space
![Page 61: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/61.jpg)
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BGC and Santangelo, 2017
Multiple vertex configuration space
![Page 62: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/62.jpg)
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
![Page 63: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/63.jpg)
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
![Page 64: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/64.jpg)
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
![Page 65: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/65.jpg)
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Exactly 2^Vint solutions ???2^Vint*
![Page 66: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/66.jpg)
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Exactly 2^Vint solutions ???2^Vint*
![Page 67: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/67.jpg)
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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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![Page 69: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/69.jpg)
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Exactly 2^Vint solutions ???2^Vint*
![Page 70: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/70.jpg)
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Exactly 2^Vint solutions ???2^Vint*
![Page 71: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/71.jpg)
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Exactly 2^Vint solutions ???2^Vint*
![Page 72: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/72.jpg)
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Exactly 2^Vint solutions ???2^Vint*
Yes, if the crease patternis constructed with Henneberg-Imoves from a pair of triangles!
![Page 73: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/73.jpg)
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???2^Vint*
Yes, if the crease patternis constructed with Henneberg-Imoves from a pair of triangles!
Demaine et al, Graphs and Combinatorics, 2011
![Page 74: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/74.jpg)
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???2^Vint*
Yes, if the crease patternis constructed with Henneberg-Imoves from a pair of triangles!
Demaine et al, Graphs and Combinatorics, 2011
How to show that all vertex sign patterns are realized twice?
![Page 75: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/75.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
![Page 76: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/76.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
# of branches ≤ 2^(Vint)
![Page 77: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/77.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
# of Mountain-Valley choices = 2#creases = 2^(3Vint+1)# of branches ≤ 2^(Vint)
![Page 78: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/78.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
# of Mountain-Valley choices = 2#creases = 2^(3Vint+1)Only a tiny fraction of MV’s can be realized!
# of branches ≤ 2^(Vint)
![Page 79: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/79.jpg)
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
But how much does a crease pattern really tell us?
# 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!
# of branches ≤ 2^(Vint)
![Page 80: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/80.jpg)
(a)
(b)
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(c)
3
21
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678
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BGC and Santangelo, 2017
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!
![Page 81: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/81.jpg)
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(b)
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(c)
3
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678
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BGC and Santangelo, 2017
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!
![Page 82: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/82.jpg)
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678
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BGC and Santangelo, 2017
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!
![Page 83: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/83.jpg)
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(b)
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(c)
3
21
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678
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BGC and Santangelo, 2017
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!
![Page 84: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/84.jpg)
BGC and Santangelo, 2017
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Same M-V labels, same vertex sign pattern : two branches!
![Page 85: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/85.jpg)
BGC and Santangelo, 2017
(a)
(b)
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Same M-V labels, same vertex sign pattern : two branches!
![Page 86: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/86.jpg)
BGC and Santangelo, 2017
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Same M-V labels, same vertex sign pattern : two branches!
![Page 87: Geometry of Flat Origami Triangulations · Origami in nature and engineering Saito et al, PNAS 2017 Andresen et al, PRE 2007 Wood et al, Science 2015 J.-H. Na et al., Adv. Mat. 2015](https://reader036.fdocuments.net/reader036/viewer/2022070706/5e9dcc5791ed977beb2b1ff1/html5/thumbnails/87.jpg)
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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?