Introduction to the geometry and response of origami and

thin sheetsZeb Rocklin

Georgia Institute of Technology

Brandeis IGERT Summer Institute 2018

Outline

• Part 1: Mechanical response and uses of thin sheets

• Part 2: Quantitative description of thin sheets

• Part 3: Deformations of thin sheets

Mechanical response

• How do systems respond to external mechanical forces and constraints?

• Elastic (linear and nonlinear). Also, viscoelastic, fluid, plastic/damage

2 cm

Elastic response: material dependence

• Solid materials hold their shape until subjected to forces of some threshold

• Elastic moduli of solids can vary by orders of magnitude

• Composition isn’t the only way to control mechanical response

Chart: Granta Design Ltd. (2018)

Mechanical response via structure

• Same material can have completely different responses based on structure

• Thinness key to interesting folding mechanisms

Solid objects and elastic response

• Solid object: retains its shape

• Resists deformations, even ones that preserve density: has finite shear modulus

• Result of applying small forces to solid? Small deformations

Solid objects and deformation

• Point r maps to r’ = r + u(r)

• Displacement field includes rigid-body rotations, translations

• Strain field 𝑢𝑖𝑗(𝑟) describes how deformation in one direction varies

• Response to external force and confinement is global strain

See, e.g., Landau and Lifshitz Theory of elasticity

Solid objects and bending

• Thick objects compress under confinement

• Thin objects can also bend

• Bending energy cost ~(thickness)3

• Compression cost: (thickness)1

• System’s tendency to bend rather than stretch can be achieved with thinness

• Small strains but large deformations: can be quite elastic

Three-dimensional structure via thin sheets

• Origami crease pattern fixes curvature along thin lines

• 2D faces rotate rigidly

• Flat sheet plus fold pattern can approximate arbitrary curves

• Can you see what the sheet folds into?

Tachi’s folding pattern

Folding for deployment

• Many devices fold flat or collapse for shipping or storage

• Mechanism extends area/volume for usage

• General problem: assuming different configurations for multifunctional materials

Extreme deployment: solar panels for space missions• Storage volume needs to be small for

launch

• Deployed area needs to be large for collection

• Panel faces must be rigid

• Mis-folding can ruin mission

Contreras, Trease and Sherwood (2013)

• Origami to the rescue: rigid faces hold panels, hinges permit only one unfolding motion

Origami and the human heart

• But origami can also touch the human heart like nothing else…

Origami and the human heart

• But origami can also touch the human heart like nothing else …literally

• Origami heart stent collapses for insertion

• Expands to hold passageway open in heart

• Made of stainless steel

Zhong You and Kaori Kuribayashi-Shigetomi

Folding in nature has more curves

• Leaves are biological solar cells, folding for deployment

• Nature incorporates more curves than technology does

• Biology’s ultimate functional folders: proteins

Video: Neil Bromhall, Youtube

DNA origami

• Technological folding molecules: controlling binding of DNA molecules

• Can do 2D, 3D, rigid beams, flexible hinges

• Functionality: drug delivery, frames for plasmonics

Hinge mechanism

Marras, Zhou, Su, Castro PNAS (2014)

Triggered release

Douglas, Bachelet Church Science (2012)

Castro et al. Nat. Methods (2011)

Human folds at the micro-scale: graphene kirigami

Blees et al., Nature (2015), McEuen group

• Graphene ultimate thin sheet: one atom thick

• Kirigami: cutting pattern for spring behavior

• Nonlinear stretching motion• Electronic components and

variable force meters• How do we describe states of

thin sheets?

The mathematical technology of thin mechanics• How do we describe a thin

sheet?

• What structure does it have?

• How can we deform thin sheets?

• What properties of sheets are intrinsic?

Graphic: Crane

Differential geometry of curved surfaces

• A surface is a 2D structure embedded in 3D (R3)

• It could have a boundary (disk) or no boundary (sphere)

• It could have holes in it (torus) or not (sphere)

• It could be smooth (sphere) or not (tetrahedron, origami)

• Let’s start with a smooth surface without a boundary

Surface as map

• Consider 𝑓:𝑀 → 𝑅3 as a map from a 2D space to 3D Euclidean space

• We need two coordinates (u,v) to describe a point p on our surface, and f tells us where this point lies in the embedding space

Graphic: Crane

Surface as manifold

• Manifold: a space that locally looks Euclidean

• Consider a point p on surface manifold M: the partial derivatives w.r.t. the two manifold coordinates, 𝜕𝑢𝑓, 𝜕𝑣𝑓, are two tangent vectors

• Tangent vectors are not generally orthogonal (or even normalized) but can be made linearly independent

• The space spanned by those two vectors is a 2D slice of 3D space, which is equivalent to R2

• Curved surfaces have different tangent planes at different points

Surface normal vectors

• The orthogonal complement of the tangent space is the normal vector. Unnormalized, it is 𝑛 = 𝜕𝑢𝑓 × 𝜕𝑣𝑓

• After normalization, the normal vector has an overall sign ambiguity, 𝑛 ↔ −𝑛, which can be chosen for orientablesurfaces

• For non-orientable surfaces, closed paths can flip normal vector

• Normal vector maps from manifold to surface of sphere, called Gauss map

Mobius strip art: Todd Davidson

Building up curvature

• Normal vector is local orientation of surface

• How fast normal vector changes is curvature 𝜅 (signed scalar).

• Curvature is inverse radius of “osculating” circle

• Locally, height above tangent line is ℎ 𝑥 =

𝜅

2𝑥2 + 𝑂(𝑥3)

Graphic: Crane

One-dimensional curves

• Consider a curve from a one-dimensional manifold to R3

• We have tangent line 𝑡 = 𝜕𝑢𝑓, and choose units so that this has unit magnitude

• “Principal normal” vector n is direction of change of tangent vector

• Their cross product gives a third vector to fill out the basis

t

n

Graphic modified from Crane

Curvature of 2D surfaces

• Any point p on 2D surface has curvature associated with each direction in its 2D tangent plane

• Height function above the plane takes the coordinate-dependent form:

ℎ 𝑥, 𝑦 =𝑎

2𝑥2 + 𝑏𝑥𝑦 +

𝑐

2𝑦2

• We can always rotate coordinates x,y to eliminate “b” and express in terms of principal curvatures:

• Is curvature a function of embedding or is it “intrinsic” to the surface?

ℎ 𝑥′, 𝑦′ =𝜅12𝑥′2 +

𝜅22𝑦′2

Graphic: Crane

Surface metric: how to determine the distance

• We want to generalize the notion of distance to curved spaces

• In flat space, the squared distance along a vector x is 𝑥2 = 𝑥𝑖𝑥

𝑖 = 𝑥𝑖𝑥𝑖, (repeated indices imply summation)

• For curves, the tangent vectors aren’t orthonormal: what then are the distances?

Graphic modified from Crane

Surface metric: determining the distance

• Consider two points very close to one another on a curved surface

• For a smooth surface, they approach one another’s tangent planes, so the separation is a linear combination of tangent vectors 𝑥1𝜕𝑢𝑓 + 𝑥2𝜕𝑣𝑓

• The squared distance is then 𝑥𝑖𝑔𝑖

𝑗𝑥𝑗, with 𝑔𝑖

𝑗the metric

tensor

• The metric tensor is also called the first fundamental form

Squared distance:

𝑥12 𝜕𝑢𝑓

2 + 2 𝑥1𝑥2𝜕𝑢𝑓𝜕𝑣𝑓+ 𝑥2

2 𝜕𝑣𝑓2 ⇒

𝑔 =(𝜕𝑢𝑓)2 𝜕𝑢𝑓𝜕𝑣𝑓𝜕𝑢𝑓𝜕𝑣𝑓 (𝜕𝑣𝑓)2

Geodesics

• Straight lines have perhaps two defining features: constant direction and minimum length, which coincide in flat (Euclidean) space

• In curved space, our “straight” lines have the second virtue, and are called geodesics

• Think of putting a loose string on a surface between two points and gradually pulling it tight

• Geodesics exist and are locally unique

Graphic modified from Crane

Finding geodesics: flat space

• Map: f(u,v) = (u,v,0)

• Metric: 𝑔𝑖𝑗= 𝛿𝑖

𝑗

• Length of path: L ~𝑡1

𝑡2 𝑑𝑡(𝑥′ 𝑡 2 + 𝑦′ 𝑡 2)

• We need functional calculus: if x was a variable we’d extremize the length by requiring 𝜕𝐿/𝜕𝑥 = 0. If it was several variables xi, we’d set each derivative equal to zero.

• Instead, it’s a function with a continuous “index” t, so we need to say that altering the position x (or y) at any time doesn’t change the length (to lowest order)

• We write that: 𝛿𝐿

𝛿𝑥 𝑡= 0 for all t.

Can this be a geodesic?

Functional calculus: quick and dirty introduction/review• Conceptually, exactly like multivariable calculus: x(t) is a variable with

a continuous index

• First-order conditions: our functional (function of function) is extremized when varying x(t) at any t doesn’t change it (to lowest order)

• How do we change x(t) by a tiny bit at a particular time t’? 𝑥 𝑡 →𝑥 𝑡 + 𝜖𝛿 𝑡 − 𝑡′

• Derivative of functional is the 𝑂(𝜖) change in the functional that results

Some useful functional derivatives

•𝛿

𝛿𝑥 𝑡′𝑥 𝑡 ?

•𝛿

𝛿𝑥 𝑡′ 𝑑𝑡 𝑓(𝑡)𝑥 𝑡 ?

•𝛿

𝛿𝑥 𝑡′ 𝑑𝑡 𝑓(𝑥 𝑡 )?

•𝛿

𝛿𝑥 𝑡′𝑥′ 𝑡 ?

•𝛿

𝛿𝑥 𝑡′ 𝑑𝑡 𝑓(𝑡)𝑥′ 𝑡 ?

𝛿(𝑡 − 𝑡′)

𝑓(𝑡′)

𝑓′(𝑥(𝑡′))

𝛿′(𝑡 − 𝑡′)

−𝑓′(𝑡′)

Finding straight lines in flatland

• Length of path: L

𝑡0=𝑡𝑓 𝑑𝑡(𝑥′ 𝑡 2 + 𝑦′ 𝑡 2)

• Functional derivatives say x’’(t) = y’’(t) = 0

• 𝑥 𝑡 = 𝑥0 +𝑡−𝑡0

𝑡𝑓−𝑡0(𝑥𝑓 − 𝑥0),

y 𝑡 = 𝑦0 +𝑡−𝑡0

𝑡𝑓−𝑡0(𝑦𝑓 − 𝑦0)

Surface of constant curvature: shortest distance on the sphere• Map to surface of sphere: f(u,v) =

(sin(u)cos(v),sin(u)sin(v),cos(u))

• Curved metric: 𝑔𝑢𝑢 = 1, 𝑔𝑢𝑣 = 0, 𝑔𝑣𝑣 = sin2(𝑢)

• Length: L 𝑡1~

𝑡2 𝑑𝑡(𝑢′ 𝑡 2 + sin2(𝑢)𝑣′ 𝑡 2)

• First-order conditions in v(t) tell us v’’(t) = 0

• First-order conditions in u:

𝑢′′ 𝑡 = sin 𝑢 cos 𝑢 𝑣′ 𝑡2

Is this a geodesic?

Surface of constant curvature: shortest distance on the sphere• First-order conditions in u:

𝑢′′ 𝑡 = sin 𝑢 cos 𝑢 𝑣′ 𝑡2

• Choose coordinates so that both points lie in the equator, so 𝑢 0 =𝑢 1 =

𝜋

2⇒ 𝑢 𝑡 =

𝜋

2

• Shortest path should lie on the great circle, but which way?

• Answer lies with higher-order terms, stability analysis

More general geodesics

• We can always use our metric tensor to write down a path length:

𝐿~ 𝑑𝑡 𝑢′𝑖𝑔𝑖𝑗𝑢𝑗′

• Functional differentiation gives Euler-Lagrange type equations for the path in curved space (not analytically solvable in general)

• General relativity: geodesics in spacetime are wordlines of particles

• Geodesic concept also applies to non-geometric systems (e.g., graphs)

Graphic modified from Crane

Geodesics and curvature

• On flat space, parallel geodesics maintain their distance

• On a sphere, they eventually join up

• Geodesics seem to measure curvature, but first fundamental form can’t tell us curvature

• We need a form that tells us change in geodesics

Second fundamental form and curvature

• Earlier, we had a “height” function depending on coordinates in transverse plane

• In terms of coordinates u, v, can define a “two-form” (II)

• To make this height in transverse plane, need second derivatives 𝜕𝑢𝑢𝑓, etc.

• Also need to dot them with normal vector n

• 𝐿 = 𝑛 ⋅ 𝜕𝑢𝑢𝑓, etc.

ℎ 𝑥, 𝑦 =𝑎

2𝑥2 + 𝑏𝑥𝑦 +

𝑐

2𝑦2

𝐼𝐼 = 𝐿 𝑑𝑢2 + 2𝑀 𝑑𝑢 𝑑𝑣 + 𝑁 𝑑𝑣2

Properties of second fundamental form

• For a unit vector w along surface, matrix multiplication 𝑤𝑇 𝐼𝐼 𝑤 gives normal curvaturealong vector

• Eigenvalues of form are principle curvatures

• Trace gives (twice) mean curvature

• Determinant gives Gaussian curvature

as form:

𝐼𝐼 = 𝐿 𝑑𝑢2 + 2𝑀 𝑑𝑢 𝑑𝑣 + 𝑁 𝑑𝑣2

As matrix:

𝐼𝐼 =𝑛 ⋅ 𝜕𝑢𝑢𝑓 𝑛 ⋅ 𝜕𝑢𝑣𝑓𝑛 ⋅ 𝜕𝑢𝑣𝑓 𝑛 ⋅ 𝜕𝑣𝑣𝑓

Normal vs. geodesic curvature

• Consider a non-geodesic path on a curved surface

• Curvature is inverse radius of osculating circle

• Normal curvature gives curvature of surface

• Geodesic curvature gives curvature of path relative to surface (how far it is from being a geodesic)

• Because of orthogonality, this leads to total curvature 𝜅2 = 𝜅𝑔

2 + 𝜅𝑛2

In flat space, geodesic curvature is just curvature

Smooth, local Gauss-Bonnet theorem

• For a smooth, disk-like region on a curved surface, the integral of the Gaussian curvature is 2𝜋 minus the integral of the geodesic curvature over the boundary

Graphic: Nguyen-Schäfer, Schmidt, “Elementary Differential Geometry”

Incorporating kinks in boundary

• Suppose our path turns at a sharp angle

• Integrals require smoothness, so is the sharp path a well-defined limit of a smooth one?

• Replace sharp turn of angle 𝜃 with circular path radius R

• Geodesic curvature is 1/R and continues over length R 𝜃, so good limit as 𝑅 → 0.

• Normal curvature irrelevant because surface itself is smooth even when path isn’t (by assumption)

• Hence, integrals of geodesic curvature over kinked paths pick up sum of all turn angles in path, σ𝑗 𝜃𝑗

Gauss-Bonnet and high-genus surfaces

• Consider a torus. We can divide the surface into four quadrants, each topologically a disk

• Quadrants have 2𝜋 of discrete rotations, so once the geodesic curvatures cancel out total Gaussian curvature is zero

• Every topologically toroidal surface has regions of positive and negative curvature

Gauss-Bonnet and even higher-genus surfaces • What about double- or triple-tori?

• We can make these by gluing together multiple tori after dividing them into quadrants

• Each time we do, a quadrant with four 𝜋/2 rotations turns into one with eight, changing the integrated curvature by 2𝜋

• This gives us the full Gauss-Bonnet theorem

Gauss-Bonnet theorem

• The integrated Gaussian curvature of a region plus geodesic curvature of its boundary (including discrete curvature) is 2𝜋times the region’s Euler characteristic

• “Euler characteristic”: one for disks, two minus two times the genus for surfaces of well-defined genus

• Deep relationship between two types of curvature (geometric, continuous) and genus (topological, discrete).

From continuum surfaces to discrete

• Origami and other real-world systems are faceted

• We no longer have well-defined surface normal, finite curvatures

• We have moved from differential geometry to discrete differential geometry

• Is the discrete surface a well-defined limit of the continuous one?

Graphic: Crane

Origami face: all the area, none of the curvature• Origami is made up of two-

dimensional faces

• Faces contact one another at one-dimensional edges (creases)

• Edges contact one another at zero-dimensional vertices

• Faces possess all the surface’s area, but in rigid origami they have no curvature

The origami edge

• Origami edge has a dihedral angle, angle between the two planes defined by the adjoining faces

• In origami, dihedrals are dynamic, can be changed

• Dihedrals are angles between surface normals or between tangent vectors parallel to edge

Origami edge as cylindrical section

• Consider replacing origami edge with dihedral angle 𝜃 with cylindrical section of radius r

• Principal curvatures across and along edge: 1/r, 0. Gaussian curvature 0

• Integrated mean curvature

r𝜃

𝑟= 𝜃, well-defined as 𝑟 → 0+

The origami vertex

• Last chance to see Gaussian curvature

• Smooth version of vertex? Spherical cap of radius r lying between cylindrical edges

• Area of the cap?

Area of spherical cap

• Boundary of spherical cap travels in straight lines along n edges: cap is spherical n-gon

• Area of spherical triangle: sum of interior angles minus 𝜋

• Area of spherical n-gon: sum of interior angles minus 𝑛 − 2 𝜋

• Area of cap of radius R ~ R2, Gaussian curvature ~R-2.

• Gaussian curvature of vertex cap is angle deficit

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

• What if there was another way?

• What could you work out about the Earth’s curvature from measuring distances (and angles) along the Earth?

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

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

Summary

• Thinness of real materials permit folding, bending

• Geometric design leads to (multi-)functional materials

• Differential geometry describes intrinsic, extrinsic properties of thin sheets