Putting thin sheets to work: selected topics in origami

metamaterialsZeb Rocklin

Georgia Institute of Technology

A brief history of origami

• Paper developed in first-third century CE

• Paper folding present in Japan by sixth century

• Took off slowly in Asia, Europe

• Religious, cultural, artistic significance, increasingly novelty, fun value for kids

• By 1970’s, 1980’s not just art, but science, math, engineering applications

Modern realization by Ben J. Wong of early Chinese paper folding

Miura ori

• Problem: solar panels for unmanned space mission need to fold up for flight, unfold by themselves in orbit

• Task: design rigid origami fold pattern that collapses down tightly and has only one unfolding mechanism

• 1970: astrophysicist Koryo Miura developed the “Miura ori”, parallelogram fold pattern

• Deployed on multiple missions

• Continues to inspire origami engineering

Origami metamaterials

• Lightweight and strong

• Collapsible and quickly deployable

• Large design space

• Operates at multiple scales (down to atomic thickness and 10’s of nm)

• Combinable with other technology (electronics, chemistry)

Origami Kevlar: collapsible, lightweight

Howell, BYU, 2017

Kirigamibuckling-induced stiffness, Rafsanjani, Bertoldi PRL(2017)

Origami metamaterials

• Lightweight and strong

• Collapsible and quickly deployable

• Large design space

• Operates at multiple scales (down to atomic thickness and 10’s of nm)

• Combinable with other technology (electronics, chemistry)

Mechanics of periodic, triangulated surfaces

• Consider a periodic mechanical structure…

• Consisting of rigid triangular faces free to bend about edges

• … but costing energy to stretch or shear faces

• How do we describe the mechanics of such a system?

Triangular faces via springs

• Describe state of triangle via positions of vertices

• Add “springs” along edges of triangles that cost energy to compress or extend

• Triangle (unlike square) is rigid, such that face needn’t distort so long as no springs stretch

• Isometries/zero modes of triangulated surface are those of ball and spring structure

Mechanical networks

• “Ball and spring”: minimal model of mechanical systems

Particles/sites can move

Stretching bonds/springs costs energy

But rotating bonds at “hinges” doesn’t

Other physical systems have same mathematical structure

The rigidity matrix and its friends

Bond extensions(tensions)

Site forces

Equilibrium matrix

Dynamical matrix

Site displace-mentsSite forces

Bond extensions

Site displace-ments

Rigidity matrix

Index Theorem

See Maxwell (1864), Calladine (1978), Kane Lubensky (2014)

Index theorem:

Zero modes

Self stresses

Site modes

Bonds

RigidFloppy but stressed

Zero mode: movement of particles without extension of bondsSelf stress: tension in bonds without force on particles

Index Theorem

See Maxwell (1864), Calladine (1978), Kane Lubensky (2014)

Index theorem:

Zero modes

Self stresses

Site modes

Bonds

RigidFloppy but stressed

Maxwell lattices (ν=0) : symmetry-generated zero modes (uniform translations) generate self stresses

Maxwell index for triangulated surface?

See Maxwell (1864), Calladine (1978), Kane Lubensky (2014)

Index theorem:

Zero modes

Self stresses

Site modes

Bonds

Is our triangulated surface over- or under-coordinated? How many constraints does it have, how many degrees of freedom?

Need a new (or very old) topological count

Euler formula for the polyhedron

• A polyhedral surface has some numbers F, E, V of faces, edges and vertices

• Quantity F-E+V is not arbitrary, but generic to all polyhedral of same Euler characteristic

• Another deep geometric meaning associated with Euler characteristic: Gauss Bonnet

𝐹 − 𝐸 + 𝑉 = 𝜒 = 2(1 − 𝑔)

Proof of Euler formula for disk: face elimination

• Consider arbitrary polyhedral disk

• Contract an n-gon to a single point, which doesn’t change F – E + V

• Eventually a single n-gon is left, with F – E + V = 1 – n +n =1

𝐹 → 𝐹 − 1

E → 𝐸 − 𝑛 − 1

V → 𝑉 − 𝑛

𝐹 − 𝐸 + 𝑉 → 𝐹 − 𝐸 + 𝑉

Higher-genus surfaces

• Gluing together two disks into a sphere eliminates n edges and n vertices, so F –E+V is unchanged and 𝜒𝑠𝑝ℎ𝑒𝑟𝑒 = 2 𝜒𝑑𝑖𝑠𝑘 = 2

• Torus can be split into four disk-shaped regions, adding four more vertices than edges, so 𝜒𝑡𝑜𝑟𝑢𝑠 = 4 𝜒𝑑𝑖𝑠𝑘 − 4 = 0

• Higher-genus surfaces handled similarly

𝐹 − 𝐸 + 𝑉 = 𝜒 = 2(1 − 𝑔)

Soccer balls: how many pentagons?

• Can relate numbers of faces, edges, vertices

• (3/2) edges per vertex (or (n/2) for an n-vertex)

• (n/2) edges per n-gon

• 𝐸 =6

2𝑁ℎ𝑒𝑥 +

5

2𝑁𝑝𝑒𝑛𝑡 = 3𝐹 −

1

2𝑁𝑝𝑒𝑛𝑡

• 𝐸 − 𝑉 = 𝐸 −2

3𝐸 =

1

3𝐸 = 𝐹 −

1

6𝑁𝑝𝑒𝑛𝑡

• 𝜒𝑠𝑝ℎ𝑒𝑟𝑒 = 2 = 𝐹 − 𝐸 − 𝑉 =1

6𝑁𝑝𝑒𝑛𝑡

• Every soccer ball has 12 pentagons (if it’s all pentagons, hexagons and three-vertices)

• Makes crystallization on a sphere tricky (Guerra et al., Nature 2018)

Triangulated surfaces: Euler count? Maxwell count?

• Periodic boundary conditions are like being on torus, 𝜒 = 𝜒𝑡𝑜𝑟𝑢𝑠 = 0

• Triangles means 𝐸 =3

2𝐹

• Triangulated periodic surface: 3𝑉 − 𝐸 = 0

• Degrees of freedom: 3V

• Constraints: E

• Triangulated surfaces are naturally isostatic/Maxwell!

Maxwell lattices and boundaries

• Consider a “periodic” Maxwell lattice with open boundaries formed by missing bonds

• Index theorem: zero modes created by removal of constraints on boundary

• Where are the zero modes? Expect them to be near boundaries where bonds are missing

• This is not always true!

Periodic systems and Bloch-type modes

• Bloch’s theorem: wavefunctions in periodic systems are composed of modes that vary only by phase between cells: 𝜓 𝑟 =𝑒𝑖𝑘⋅𝑟𝑢 𝑟 , 𝜓𝑗 𝑟 = 𝑒𝑖𝑘⋅𝑟𝑢𝑗 = 𝑧1

𝑛1𝑧2𝑛2𝑢𝑗

• Periodic boundary conditions require real wavevector

• Local constraints permit complex wavevector with modes growing or shrinking across system

Reciprocity and non-reciprocity

• Dynamical matrix is symmetric: 𝐷𝑖𝑗 = 𝜕𝑖𝜕𝑗𝐸 = 𝐷𝑗𝑖

• An external force 𝑓𝐴applied to mechanical structure results in displacements 𝑢 𝑓𝐴

• Reciprocity: external forces 𝑓𝐴, 𝑓𝐵result in displacements 𝑢𝐴, 𝑢𝐵satisfying 𝑓𝐴 ⋅ 𝑢𝐵 = 𝑓𝐵 ⋅ 𝑢𝐴

• Consequence: If force at site A leads to response at site B, force at site B leads to response at site A

Non-reciprocity: zero modes and self stresses

• Periodic structures: equilibrium matrix is transposed compatibility matrix with site indices reversed

• Zero mode on left edge ⇒self stress on right edge

• Shapes of zero modes, self stresses are generally unrelated

𝑸 𝑧 = 𝑹𝑇(1

𝑧)

Topological invariant

• Maxwell lattice: system has zero modes when det 𝑅 𝑧 = 0

• Fundamental theorem of algebra: each between cells generates a zero mode

• Intuition says cutting boundaries should result in balanced numbers of zero modes

• But this can’t be: some systems have one boundary mode but two boundaries

• Topological invariant determines placement of edge modes (Kane, Lubensky, Nat. Phys. (2013)) One missing bond per cell on both sides,

but both zero modes on left side!

Topological polarization: Bulk boundary correspondence• Zero modes on left edge: det 𝑅 𝑧 =0, 𝑧 < 1

• Complex analysis: we can count zeroes (and poles) in region by integrating over boundary:∫ 𝑑𝑧 arg(det R z =2𝜋𝑖(𝑁 − 𝑃)

• Boundary of region: 𝑧 = 𝑒𝑖𝑘 = 1:bulk modes

• Winding of bulk constraints determines number of boundary modes

Det = 0

Counting edge zero modes via bulk invariants

Consequences of topological polarization

• Differential edge stiffness: one edge has folding mode, one edge requires stretching to deform

• Directional bulk response: distorting lattice in interior produces deformation on only one side of lattice

Unpolarized:Polarized:

DZR, NJP(2017)

DZR, Zhou, Sun, Mao, Nat. Comm. (2016)

Prototype system: K’nex

Published in Nat. Comm. 2017

Triangulated origami doesn’t polarize!

• Numerical observation: every single zero mode at some “z” for triangulated origami comes with another at 1/z

• Topological polarization is impossible—why?

• What about surface structure modifies/restricts ball and spring systems?

• Does origami have a “hidden symmetry”?

• Only observed in triangulated origami ⇒need origami-specific description of zero modes

Isometry: distance-preserving transformation

• For vanishingly thin sheets, no cost to bending, but finite cost to stretching

• Low-energy modes of thin sheets don’t stretch them

• These modes preserve distance alongsheets, but not in embedding space

• Mathematically, they preserve the metric

• What sheet transformations areisometric? What transformations aren’t?

Graphic: Crane, “Discrete differential geometry”

A simple isometry: flat sheet to cylinder

• Consider a family of sheets rolled up into cylinders of various radii: 𝑓 𝑢, 𝑣 = (

1

𝑅sin

𝑢

𝑅,1

𝑅(1 − cos

𝑢

𝑅), 𝑣)

• 𝑔 =1 00 1

regardless of R!

• Same metric ⇒ same distance along sheet

• Same distances ⇒ same geodesics

• Gaussian curvature is how geodesics converge/diverge along surface ⇒Gaussian curvature preserved by isometries!

A simple isometry and curvature

• Second fundamental form describes curvature:

𝐼𝐼 =1/𝑅 00 0

• Principal curvatures: 0, 1/R

• Gaussian curvature: 0, regardless of curling radius

• All isometries preserve Gaussian curvature, but not all curvature-preserving transformations (e.g., stretching a flat sheet) are isometries

A simple non-isometry: the Mapmaker’s dilemma• Consider mapping a spherical surface to

a flat sheet

• Spheres have Gaussian curvature, so not an isometry

• What’s the best way to do the projection? What do we mean by best?

Graphics from Wikipedia’s “List of Map Projections”

Intrinsic vs. Extrinsic

• Our notion of a surface, 𝑓:𝑀 → 𝑅3, depends on position in embedding space: you give me coordinates (u,v) and I give you a point in 𝑅3

• But what if all you had was the metric 𝑔𝑖

𝑗?

• You’d know lengths, areas, angles, geodesics, but not positions in Euclidean space

• Such embedding-independent quantities are called I

Intrinsic vs. Extrinsic

• Could a bug trapped on the Earth’s surface infer the Earth’s curvature? What about a cylinder’s?

• Is curvature intrinsic or extrinsic? Does it change under isometries?

Gauss Bonnet is (almost) intrinsic

• Gauss Bonnet theorem relates Gaussian curvature, area and geodesic curvature of region

• As long as boundary is sharp corners and geodesics, curvature is all intrinsic

• Gaussian curvature of small patch is angle surplus (relative to flat triangle) divided by area

(Gaussian curvature)(Triangle area) + (sum of interior angles) = 2𝜋

Gauss’s remarkable theorem

• Modern language: Gaussian curvature is intrinsic– any isometric embedding of a surface preserves Gaussian curvature

• Gauss’s language (from Latin): If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.

• Why is this so remarkable? Surface was initially defined by its embedding, not clear that any thing is preserved by isometries (other than the metric itself)

Graphic modified from En. Britannica

Gauss’s theorem remarkably easy to extend to origami• Gaussian curvature is angle deficit at each

vertex: 2𝜋 − σ𝑗 𝜃𝑗

• Angles between edges on rigid faces are preserved under isometry

• Gauss-Bonnet can be checked by drawing small circular loops or straight-line paths along vertex and measuring geodesic curvature

• Loops can be deformed, combined as on smooth curves

Origami isometries

• How do we describe isometry of origami?

• Each face is rigid ⇒ it can rotate and displace, but not distort

• Each vertex has fixed curvature, determined by angle deficit (zero for initially flat sheet)

• Only change in curvature: dihedral angles of edges

• Origami isometries are a scalar theory living on edges

Collaborators

Bryan Chen,Penn

James McInerney,Georgia Tech

Louis Theran,St. Andrews

Chris Santangelo,U Mass Amherst

Constraints on origami isometries

• Does every set of dihedral angles correspond to an isometry?

• Relative orientation of two adjacent origami faces given by rotating one about shared axis by amount equal to dihedral angle, 𝑹(𝜌𝑖 , Ƹ𝑟𝑖)

• This element of rotation group SO(3) fully defines face for solid sheet

• Position also needs to be well-defined, but this is guaranteed if orientation is

Constraints on origami isometries

• Relative orientation of two sheets connected by edges i: ς𝑖𝑹(𝜌𝑖 , Ƹ𝑟𝑖)(careful: dihedral rotations modify edge orientations)

• Necessary condition for genuine isometry: set of rotations on closedpath must be the identity

• Closed paths decompose into loops around individual vertices

Linear isometries

• Suppose initial dihedral angles change by infinitesimal amounts 𝜙𝑖

• Constraints on changes in dihedral angles?

• Rotations of vectors come from skew-symmetric generators of SO(3): 𝑣 → 𝑣 +𝜙 Ƹ𝑟𝜇𝜎𝜇

Vertex condition for isometries:σ𝑖𝜙𝑖 Ƹ𝑟𝑖=0

Equivalence of kinematic, equilibrium vertex conditions

Vertex condition for isometries:σ𝑖𝜙𝑖 Ƹ𝑟𝑖=0

Vertex condition for self stresses:σ𝑖 𝑡𝑖 Ƹ𝑟𝑖=0Each linear

isometry can be mapped directly onto self stress! Both are scalar theories living on edges of known orientation. (Known to Tachi, others…)

Origami’s hidden symmetry

Mechanical criticality:Zero mode at some complex wavevector k, 𝑧 = 𝑒𝑖𝑘

Mechanical criticality:Self stress at some complex wavevector -k, 1/𝑧 = 𝑒−𝑖𝑘

Triangulated surface:Zero mode at some complex wavevector k, 𝑧 = 𝑒𝑖𝑘 via dihedral angles

Triangulated surface:Self stress at some complex wavevector k, 1/𝑧 = 𝑒𝑖𝑘 via dihedral angles → line tensions

Self stresses (and zero modes) come paired at z, 1/z and are balanced on edges ⇒ triangulated surfaces can’t polarize

Break the surface, break the symmetry

• Turn origami into kirigami: add holes

• Modify orientationalrequirements: isometries must preserve face orientation around every vertex and every hole

• New requirement: change in position along edges of hole induced by change in orientation must be zero

• No analog for states of self stress: now zero modes can’t be mapped to states of self stress, can they?

Surface with “blocks”

• To restore isostaticity, holes must come paired with “blocks” of faces with more than three sides

• Same isometry condition (orientation preserved around vertex), but what about condition for mechanical equilibrium?

• Forces on vertices and blocks must be zero, but also torques on blocks must be zero

A new symmetry: isometries to self stresses with holes to blocks• Surprisingly, position

closure around hole and force balance around block are still equivalent!

• Zero mode on kirigamistructure corresponds to self stress on block/hole swapped structure

• What does this mean for topological polarization?

Finbow, Ross, Whiteley, SIAM J Discrete Math 2012

Topologically polarized kirigami

• Topological polarization is now possible: kirigami structures can have topologically stiff edges where origami can’t

• Observed in Chen, Liu, Evans, Paulose, Cohen, Vitelli and Santangelo PRL (2016)

• New symmetry: polarization flips under block/hole swapping

Global modes of triangulated sheets

• Tachi: triangulated cylinders have two uniform deformation modes

• Can our hidden symmetries explain the origins of these modes?

• Global modes ⇒ same behavior in every cell (z=1)

• What are our zero modes?

Tachi, “Rigid folding of origami tessellations” (2015)

Global zero modes of triangulated origami

• Origami sheet exists in three dimensions

• Three translations

• Three rotations

• Rotations involve large displacements for large cells, don’t appear in linear theory

• What do our three translations tell us?

From translations to linear folds

• Mechanical criticality: three global translations ⇒ three global self stresses

• Triangulated surface vertex condition: three global self stresses ⇒ three linear folding modes

• So for triangulated sheets, rigid-body translations come paired with very non-rigid folding modes!

• Do our linear modes extend to nonlinear modes?

Linear vs. nonlinear modes

• If something violates a constraint to linear order, it violates the constraint

• But, if it doesn’t violate the constraint to linear order it could still violate it at higher orders

• Simple example: displacement of red particle attached to two rotors doesn’t stretch either to linear order, but nonlinearities prevent it

Folding modes, orientation and periodicity

• Consider relative orientation of a face and its analog in a different cell

• Our folding modes do nothing to ensure orientation is preserved; in general it won’t be

• Orientation is changing as we move across our sheet, is it still a crystal?

Same face, different orientations

Crystals and “screwy crystals”

Regular crystal: going from one cell to another by translating along global primitive vector

Screw theory: combine translations + rotations (or forces and torques, positions and orientations…) into single object. New objects have their own algebra suited to rigid-body dynamics

Screwy crystal: go from one cell to another by screw motion: translate in local basis then rotate basis

Generic periodic origami is not globally flat

Can we do any set of orientations? No, now we have a nonlinear orientation requirement: rotations in both lattice directions must be coaxial

Global modes of the screwy cylinder

• For a flat sheet, three translations

• Now, flat sheet is a pathological special case where cylinder axis has disappeared

• Two new global transformations: rotation about cylinder axis and translation along it

• Analysis carries over: two global rigid modes come paired with two linear folding modes

Global folding and unfolding

• Tachi’s 2D family of triangulated cylinder folding explained: comes from mechanical criticality and global modes

• Extension: mechanically critical kirigamicylinders should also have folding modes

• Such global modes can induce changes in topological polarization (DZR et al. 2017)

• How much control do we have over shape, response?

Triangular faces via springs via… magnets?

• Magnetic spins can have fixed length but variable orientation

• Resemble springs or origami edges

• Magnetic coupling that corresponds to mechanical structure?

Spin origami: frustrated kagomeantiferromagnets• Kagome triangle: three interacting spins

• Antiferromagnetism: spins want to anti-align

• Frustration: three spins can’t all anti-align

• Actual minimum-energy solution: sum of spin vectors is zero

• Vectors summing to zero along triangles is exactly what origami edges do.

• Zero-energy manifold of spin system is exactly nonlinear isometries of origami sheet!

Ritchey, Chandra, and Coleman PRB (1993)

Engineering magnets with Gaussian curvature

• Isotropic magnetic system corresponds to equilateral triangle faces

• What if we tweak couplings?

• Sometimes we get something that isn’t a triangulated surface (and can topologically polarize!)

• Sometimes we still get a flat surface

• Sometimes we get vertices with Gaussian curvature, not flat foldable!

• Interesting new topological properties

Roychowdhury, Rocklin, and Lawler arXiv:1705.00015

Summary

• Origami metamaterials have broad application, cool new properties

• Triangulated origami has rich mode structure from mechanical criticality

• Hidden symmetry between kinematics, statics permits topological polarization via kirigami

• Symmetry links rigid-body and folding modes into screwy cylindrical structures

• Triangulated surface can be constructed for magnetic (and other?) systems