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
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
Robert Lang Daniel Piker, after Ron Resch, Ben Parker and John Mckeevehttp://spacesymmetrystructure.wordpress.com/2009/03/24/origami-electromagnetism/
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?
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?
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
“bird base”
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary vertices
Rigid origami as bond-node structure
“bird base”Demaine et al, Graphs and Combinatorics, 2011
Triangulated Vb-gon : Vb boundary verticesVint : # of internal vertices
Rigid origami as bond-node structure
“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
“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
“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
“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)
“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)
“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)
“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)
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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
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BGC and Santangelo, 2017
Configuration space near the flat state
4 branches
BGC and Santangelo, 2017
BGC and Santangelo, 2017
Flat is not generic!
Vint+1 linear motions!
Flat is not generic!
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
Toy example:
linear motion
Vint+1 linear motions!
Flat is not generic!
generically 1dofvs ??
Toy example:
linear motion
Vint+1 linear motions!
redundant constraints = ‘self stress’
Flat is not generic!
generically 1dofvs ??
Toy example:
linear motion
compression
tension
Vint+1 linear motions!
redundant constraints = ‘self stress’
Flat is not generic!
generically 1dofvs ??
Toy example:
Self stresses and second-order constraints
compression
tension
linear motion
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
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”
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⌦
Self stresses in flat triangulations
BGC and Santangelo, 2017
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
u vertical displacements
Self stresses in flat triangulations
“wheel stress”
BGC and Santangelo, 2017
uT⌦u = 0
⌦ symmetric stress matrix
u vertical displacements
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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Gaussian curvature vanishes at each vertex
u vertical displacements
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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Gaussian curvature vanishes at each vertex()
u vertical displacements
!!
origami n-vertex configuration space
BGC and Santangelo, 2017
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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
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
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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
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
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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
origami n-vertex configuration space
Kapovich and Millson, Publ. RIMS Kyoto Univ, 1997
BGC and Santangelo, 2017
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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
What are the two nappes?
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What are the two nappes?
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BGC and Santangelo, 2017
What are the two nappes?
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BGC and Santangelo, 2017
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
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
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
What are the two nappes?
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Demaine et al, Proceedings of the IASS, 2016
BGC and Santangelo, 2017
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
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
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BGC and Santangelo, 2017
Multiple vertex configuration space
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BGC and Santangelo, 2017
Multiple vertex configuration space
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Vint = 2 ⇒ 2 wheel stresses
BGC and Santangelo, 2017
Multiple vertex configuration space
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Vint = 2 ⇒ 2 wheel stresses
BGC and Santangelo, 2017
2 homogeneous quadratic equations in 3 unknowns
Multiple vertex configuration space
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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
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Multiple vertex configuration space
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???2^Vint*
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Exactly 2^Vint solutions ???2^Vint*
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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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Exactly 2^Vint solutions ???2^Vint*
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???2^Vint*
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BGC and Santangelo, 2017
Exactly 2^Vint solutions ???2^Vint*
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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!
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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
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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?
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?
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)
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)
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)
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)
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(b)
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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!
(a)
(b)
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(c)
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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!
(a)
(b)
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(c)
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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!
(a)
(b)
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(c)
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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!
BGC and Santangelo, 2017
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(b)
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Same M-V labels, same vertex sign pattern : two branches!
BGC and Santangelo, 2017
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Same M-V labels, same vertex sign pattern : two branches!
BGC and Santangelo, 2017
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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?
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