From flat sheets to curved geometries: Origami and kirigami approaches

Abstract

Transforming flat sheets into three-dimensional structures has emerged as an exciting manufacturing
paradigm on a broad range of length scales. Among other advantages, this technique permits the use of
functionality-inducing planar processes on flat starting materials, which after shape-shifting, result in
a unique combination of macro-scale geometry and surface topography. Fabricating arbitrarily
complex three-dimensional geometries requires the ability to change the intrinsic curvature of initially
flat structures, while simultaneously limiting material distortion to not disturb the surface features.
The centuries-old art forms of origami and kirigami could offer elegant solutions, involving only
folding and cutting to transform flat papers into complex geometries. Although such techniques are
limited by an inherent developability constraint, the rational design of the crease and cut patterns
enables the shape-shifting of (nearly) inextensible sheets into geometries with apparent intrinsic
curvature. Here, we review recent origami and kirigami techniques that can be used for this purpose,
discuss their underlying mechanisms, and create physical models to demonstrate and compare their
feasibility. Moreover, we highlight practical aspects that are relevant in the development of advanced
materials with these techniques. Finally, we provide an outlook on future applications that could
benefit from origami and kirigami to create intrinsically curved surfaces.

Full text

Sebastien J.P. Callens
⇑
, Amir A. Zadpoor
Department of Biomechanical Engineering, Delft University of Technology (TU Delft), Mekelweg 2, Delft 2628CD, The Netherlands
Transforming flat sheets into three-dimensional structures has emerged as an exciting manufacturing
paradigm on a broad range of length scales. Among other advantages, this technique permits the use of
functionality-inducing planar processes on flat starting materials, which after shape-shifting, result in
a unique combination of macro-scale geometry and surface topography. Fabricating arbitrarily
complex three-dimensional geometries requires the ability to change the intrinsic curvature of initially
flat structures, while simultaneously limiting material distortion to not disturb the surface features.
The centuries-old art forms of origami and kirigami could offer elegant solutions, involving only
folding and cutting to transform flat papers into complex geometries. Although such techniques are
limited by an inherent developability constraint, the rational design of the crease and cut patterns
enables the shape-shifting of (nearly) inextensible sheets into geometries with apparent intrinsic
curvature. Here, we review recent origami and kirigami techniques that can be used for this purpose,
discuss their underlying mechanisms, and create physical models to demonstrate and compare their
feasibility. Moreover, we highlight practical aspects that are relevant in the development of advanced
materials with these techniques. Finally, we provide an outlook on future applications that could
benefit from origami and kirigami to create intrinsically curved surfaces.
Introduction
The many developments in additive manufacturing (AM) over
the last decades have significantly increased the attractiveness
of this manufacturing technique to fabricate arbitrarily complex
three-dimensional (3D) geometries at the nano-, micro-, and
macro-scales. Examples include bone-substituting biomaterials
[1], penta-mode mechanical metamaterials [2,3], triply periodic
minimal surfaces [4], and energy-absorbing cellular architectures
[5]. Despite many advantages of AM, one major limitation is the
incompatibility with planar surface patterning and imprinting
processes, which are crucial for imbuing surfaces with specific
functionalities such as hydro- or oleophobicity [6], integration
of electronic circuits [7], or control over cell interaction in the
case of biomaterials [8]. The ability to combine arbitrarily com-
plex surface features with arbitrarily complex geometries could
enable development of advanced materials with an unprece-
dented set of functionalities.
A potential solution to this deadlock is provided by another
manufacturing paradigm, which has been of growing interest
to the scientific community during recent years: the shape-
shifting of thin, planar sheets (which we consider 2D) into 3D
structures [9–12]. The planar sheets could first be decorated using
planar patterning processes, after which they are transformed
into complex 3D geometries. Additional advantages are the fast
and inexpensive production methods of the 2D sheets and their
efficient packing for storage and transportation [11,13]. More-
over, the 2D-to-3D paradigm is particularly interesting for the
development of micro- or nanoscale 3D constructs (e.g. micro-
and nanoelectromechanical devices [14]) since conventional
macroscale techniques cannot easily be scaled down to allow fab-
rication of such small structures [15,16]. As such, the out-of-
plane transformation of 2D sheets opens up new opportunities
From flat sheets to curved geometries:
Origami and kirigami approaches
⇑
Corresponding author.
E-mail address: S.J.P. Callens (s.j.p.callens@tudelft.nl).
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for the development of complex 3D structures with functional-
ized surfaces, especially at small length scales.
An important parameter governing the complexity of 3D
structures is the surface curvature and the variation thereof
throughout the structure. In order to create arbitrarily complex
3D structures from 2D sheets, the curvature of the initially flat
sheets should be altered in a controllable manner. The simplest
curved shapes could be obtained through bending or rolling of
flat sheets. However, more complex target shapes are character-
ized by “double curvature”and exhibit spherical (dome-
shaped) or hyperbolic (saddle-shaped) geometries, which cannot
be realized with inextensional deformations of a flat sheet (this is
readily understood when attempting to wrap a sphere or saddle
with paper). Instead, the flat sheet would need to be subjected
to in-plane distortions in order to achieve double-curved parts.
At the macro-scale, for example, flat sheet-metal is plastically
stretched to create double-curved shells (e.g. using (multi-
point) stretch-forming [17]), and fiber-reinforced composite lam-
inates are subjected to in-plane shearing deformations [18].At
smaller scales, researchers have recently used stimulus-
responsive materials that exhibit in-plane distortions in the form
of differential shrinkage [13,19] or swelling [20,21] to achieve
complex curved shapes from initially flat sheets, which is closely
related to non-uniform growth processes in initially planar
shapes observed in nature, resulting in wavy patterns at the edges
of plant leaves [22,23] and enabling the blooming of the lily
flower [24].
Subjecting 2D sheets to in-plane distortions is, therefore, a
feasible strategy to achieve complex curvature in 3D. Significant
downsides are that the strategy is primarily applicable to soft
elastic materials (such as gel sheets) and requires complicated
programming of the shape-shifting or complex external stimuli
to achieve the target shapes. Moreover, the in-plane distortions
are likely to disturb any of the surface features that were
imprinted on the 2D sheets, hence partially eliminating one of
the major advantages that the 2D-to-3D shape-shifting offers.
Fortunately, an alternative strategy that is more compatible with
rigid materials and delicate surface features exists at the intersec-
tion of art and science: the use of origami (traditional Japanese
paper folding) and kirigami (extended version of origami, also
allowing cuts) to create, or at least approximate, complex curved
shapes. Simply by imposing specific fold patterns, extended with
cuts in the case of kirigami, initially flat sheets could be trans-
formed into 2D or 3D geometries. Owing to their predictability,
controllability, and scalability, origami and kirigami techniques
have gained traction among scientists and engineers to develop
deployable structures [25–27], reconfigurable metamaterials
[28–32], self-folding robots [9,33], biomedical devices [34–37],
and stretchable electronics [38–40].Fig. 1 presents some exam-
ples of the potential applications of origami and kirigami across
a range of length scales. By folding or cutting along the right pat-
terns, origami and kirigami could transform planar sheets to
approximate complex curved geometries, without the need for
in-plane distortions.
In this review, the different origami and kirigami approaches
to approximate surfaces with “double”(or “intrinsic”) curvature
are discussed. We begin by providing a closer look at differential
geometry and its links to origami, providing a more formal defi-
nition of the concepts “surface curvature”and “flat sheets”. The
several origami techniques proposed to approximate curved sur-
faces are reviewed in the following section, followed by a section
on recent advances in kirigami. We conclude by comparing the
different techniques in terms of their suitability to approximate
curved surfaces and discussing the practical aspects as well as pro-
viding an outlook on future directions and applications.
Geometry of surfaces and origami
The notion of curvature could be somewhat ambiguous and may
be applied to a range of geometrical objects, such as curves, sur-
faces, or higher dimensional manifolds (i.e. higher dimension
generalizations of surfaces). Here, we are interested in the curva-
ture of surfaces, meaning how much the surface deviates from its
tangent plane at a certain point. We briefly introduce the con-
cepts that are crucial for understanding surface curvature in gen-
eral and its relation to origami/kirigami.
Defining surface curvature
It is useful to start the discussion of surface curvature by intro-
ducing the principal curvatures,j
1
and j
2
, at a given point on
the surface. The principal curvatures are the maximum and min-
imum values of all the normal curvatures at that point (Fig. 2a).
The directions corresponding to these principal curvatures are
the principal directions [41]. The principal curvatures and direc-
tions are a convenient way of indicating how the surface curves
in the vicinity of a point on the surface. It is important to note
that the principal curvatures cannot be uniquely determined in
points where the normal curvatures are all equal. Such a point
is called an umbilical point. The plane and the sphere are the only
two surfaces that are entirely composed of umbilical points
[41,42]. The principal curvatures can be combined to obtain
two well-known measures of the curvature at a given point on
the surface. The first measure is the mean curvature H, which is
simply the mean of both principal curvatures:
H¼1
2ðj
1
þj
2
Þ
Aflat plane has H= 0 since all normal curvatures are zero.
However, when the plane is bent into a wavy shape (see
Fig. 2b), the mean curvature becomes non-zero at certain loca-
tions since one of the principal curvatures becomes non-zero.
The second useful measure of curvature is the Gaussian curva-
ture K,defined as the product of the principal curvatures:
K¼j
1
j
2
This concept was introduced in Gauss’landmark paper on the
Theorema Egregium (“remarkable theorem”), considered by some
to be the most important theorem within differential geometry.
It is clear from this definition that the Gaussian curvature at a
certain point vanishes as soon as one of the principal curvatures
is zero, in which case the point is called a “parabolic point”.
When the principal curvatures are both non-zero and positive,
the Gaussian curvature is positive and the point is termed “ellip-
tic”. Finally, when the principal curvatures are non-zero and of
opposite signs (i.e. the surface curves upwards in one direction
and downwards in the other), the Gaussian curvature is negative,
which corresponds to a “hyperbolic”point [41,42].
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While both the mean and Gaussian curvatures could be
defined in terms of the two principal curvatures, they represent
a fundamentally different perspective on surface curvature. The
mean curvature is an extrinsic measure of the surface curvature.
This means that it depends on the way the surface is embedded
in the surrounding three-dimensional space (which is Euclidean
3-space within the context of this paper). On the other hand, the
Gaussian curvature is an intrinsic measure of the surface curva-
ture, meaning that it is independent of the surrounding space
and can be determined solely by measuring distances and angles
within the surface itself [42–44]. In other words, the mean
(extrinsic) curvature of the surface could only be determined
by an observer outside of the surface that has knowledge of its
surroundings, while the Gaussian (intrinsic) curvature of the sur-
face could be also determined by a 2D resident living on the sur-
face that has no perception of the surrounding 3D space.
The distinction between these two types of curvature is
important, as some surfaces might be extrinsically curved, yet
remain intrinsically flat. For example, it was already mentioned
that bending of a flat plane into a wavy shape gives the surface
a non-zero mean curvature. However, the Gaussian curvature
of the surface is still zero since one of the principal curvatures
is zero (Fig. 2b). Therefore, while the extrinsic curvature of the
flat plane could be changed by bending it, its intrinsic curvature
remains zero everywhere. Such a surface, having zero Gaussian
curvature everywhere, is called a developable surface. In addition
to the plane, three fundamental types of developable surfaces
exist in 3D: the generalized cone, the generalized cylinder, and
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FIGURE 1
Various examples of scientific and engineering applications of origami and kirigami. (a) An artist’s impression of an origami-based deployable solar array for
space satellites (adapted with permission from ASME from Ref. [27]). (b) A cardboard prototype of a reconfigurable origami-based metamaterial (top), which
has two degrees of freedom (middle and bottom). Adapted by permission from Macmillan Publishers Ltd: Nature [31], copyright 2017. (c) A deployable paper
structure based on origami zipper tubes. Reproduced from Ref. [29]. (d) Paper versions of cellular metamaterials combining aspects from origami and
kirigami. Reproduced from Fig. 3G and Fig. 3H of Ref. [32]. (e) A stretchable electrode based on a fractal kirigami cut-pattern, capable of wrapping around a
spherical object while lighting an LED. Reproduced from Ref. [40]. (f) A centimeter-scale, crawling robot that is self-folded from shape-memory composites.
Reproduced with permission from AAAS from Ref. [33]. (g) A biomedical application of origami: a self-folding microscale container that could be used for
controlled drug delivery. Reproduced from Ref. [35] with permission from Elsevier. (h) Another biomedical application: a self-deployable origami stent graft
based on the waterbomb pattern, developed by K. Kuribayashi et al. [34] (Reproduced with permission from Ref. [37]).
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FIGURE 2
Measures of surface curvature. (a) The principal curvatures are calculated from the intersections between the normal planes at a point and the surface. The
intersections form curved lines in the surface, with a certain “normal curvature”. The maximum and minimum values of all possible normal curvatures are the
principal curvatures, which are of opposite sign in this case given the fact that the surface curves “upward”in one direction and “downward”in the other. The
color bar indicates the Gaussian curvature. (b) The mean and Gaussian curvatures could be calculated from the principal curvatures. Bending a planar surface
could change the mean, or extrinsic, curvature (left figure, color bar indicates mean curvature), but not the Gaussian, or intrinsic, curvature (right figure, color
bar indicates Gaussian curvature). (c) Transforming a planar surface (color bar indicates the Gaussian curvature). Top row: three types of developable surfaces,
which could be flattened onto the plane through bending. From left to right: a cylindrical surface, a conical surface, and the tangent developable surface to a
space curve (a helix in this case). Bottom row: three types of intrinsically curved surfaces. From left to right: a sphere with K>0, a saddle with K<0, and a
vase surface with varying K. The sum of the internal angles of a triangle drawn on an intrinsically curved surface does not equal . (d) Interpreting the
relationship between the metric and the Gaussian curvature (color bar indicates Gaussian curvature). Creating the bell-shaped surface from an initially flat
plane requires distortion of the grid on the plane.
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the tangent developable to a space curve (Fig. 2c) [42,45,46]. The
key feature of developable surfaces is that they could be con-
structed by bending a planar surface, without requiring exten-
sional deformations. The observation that the Gaussian
curvature of a flat plane does not change when bending the
plane also holds more generally and forms the essence of Gauss’
remarkable theorem: the Gaussian curvature is bending-
invariant [42,47,48]. Arbitrary bending of any surface could
therefore change its extrinsic (mean) curvature, but it cannot
change the intrinsic (Gaussian) curvature of the surface. Conse-
quently, a flat plane cannot be transformed into a spherical or
saddle-shaped surface by bending deformations alone since these
surfaces have non-zero intrinsic curvature (Fig. 2c).
We have stated that the Gaussian curvature is an intrinsic
property of the surface, yet the classical definition that we have
given above relies on extrinsic concepts, namely the principal
curvatures. This is in fact the “remarkable”aspect in the Theo-
rema Egregium of Gauss [42,47]. However, Gauss showed that
the Gaussian curvature could also be defined on the basis of
angle and distance measurements within the surface itself (i.e.
intrinsically). A first understanding of this intrinsic description
is obtained when considering a triangle drawn on the various
surfaces, as shown in Fig. 2c. A triangle on the surface of the
plane or any other developable surface (K= 0) will always have
the sum of its internal angles, a
i
, equal to p. However, on the
surface of a sphere (K>0), the sum of the angles is larger than
p, while on the surface of a saddle (K<0), the sum of the
angles is smaller than p[43,47]. This angle measurement clearly
relates to the intrinsic geometry of the surfaces, as a 2D resi-
dent of the surface that has no knowledge of the space in
which the surface is situated could determine whether the sur-
face has positive, negative, or zero Gaussian curvature, simply
by measuring the angles of a triangle [43]. However, the resi-
dent would not be able to distinguish, for example, a flat plane
from a cylinder surface since they both have the same (zero)
intrinsic curvature.
A more formal “intrinsic”description of the Gaussian curva-
ture requires the introduction of another important concept
within differential geometry: the metric tensor, or simply metric.
The metric of a surface describes the distances between the
neighboring points on a surface which could be given as follows
(in Einstein summation convention) [49]:
ds
2
¼g
ij
dx
i
dx
j
where ds represents the distance between points and g
ij
represents the
metric components. In the case of a flat plane, the metric tensor
(which is called a “Euclidean metric”in this case) is simply repre-
sented in Cartesian coordinates as:
g¼10
01

In which case ds
2
reduces to the standard expression:
ds
2
¼dx
2
þdy
2
Physically, the metric could be interpreted as a grid on the sur-
face [50,51].Onaflat plane, this would be a regular grid consist-
ing of equally spaced, perpendicular lines. When the flat plane is
subjected to pure bending (see Fig. 2c), the grid is not distorted
and all distances and angles are preserved. For this reason, bend-
ing is called an isometric deformation, i.e. it leaves the metric
unaffected. However, if the plane is deformed into, for example,
a bell-shaped surface with regions of positive and negative Gaus-
sian curvatures (Fig. 2d), the grid becomes distorted, i.e. the met-
ric changes and becomes “non-Euclidean”. This simple
interpretation of the metric as a grid on the surface, though
not mathematically rigorous, does provide the important insight
that we are aiming for: changing the Gaussian curvature requires
a change in the surface metric (this leads to the inherent chal-
lenge that map-makers face: any map of the Earth will show
some level of distortion [52]). Moreover, this change in metric
(and thus in Gaussian curvature) cannot be achieved through
bending alone but requires stretching or shrinking of the surface.
Gauss showed that the Gaussian curvature could be defined
entirely in terms of the components of the metric tensor and
its derivatives, thereby proving the intrinsic character of this cur-
vature measure [47]. As a simple example, consider a non-
Euclidean metric defined in Cartesian coordinates of the form:
g¼10
0cðxÞ

The function cðxÞdescribes the distances between points in y-
direction as a function of x-position (ds
2
¼dx
2
þcðxÞdy
2
). In this
case, the Gaussian curvature is indeed defined entirely in terms
of the metric as [47,49,50]:
K¼ 1
ffiffiffi
c
p@
2
ffiffiffi
c
p
@x
2
Following the above definition, the Euclidean, “flat”metric
introduced earlier would indeed result in K¼0, or zero intrin-
sic curvature. The general definition of the Gaussian curvature
in terms of the metric components and their derivatives will
not be stated here, as this requires more advanced concepts
from differential geometry which are outside the scope of this
review. The reader interested in more detailed mathematical
accounts of Gauss’results is referred to several excellent sources
[47,48,52].
The direct relation between the surface metric and Gaussian
curvature has been harnessed by several researchers to control-
lably transform flat sheets into intrinsically curved geometries
[19–21,53,54]. As explained by Klein et al., this could be achieved
by prescribing a non-Euclidean “target metric”in the flat sheets,
which essentially means that a non-uniform expansion or con-
traction distribution is “programmed”into the sheets [19]. Upon
activation by an external stimulus, differential swelling/shrink-
ing occurs, which is accommodated by deforming into a curved,
3D geometry in accordance with the newly imposed metric.
Hence, these “metric-driven”[20] approaches represent a suc-
cessful application of Gauss’results to the shape-shifting of
advanced materials. However, an important remark is that these
approaches deal with real sheet materials of a small but finite
thickness, while our discussion thus far has only considered
mathematical surfaces of zero thickness. The presence of this
thickness forces researchers to consider the elastic energy of the
curved sheets, consisting of a stretching component (E
s
) and a
bending component (E
b
), both of which depend on the sheet
thickness [19,53,55]:
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E¼E
s
þE
b
When a non-Euclidean target metric is prescribed in the flat
sheet with finite thickness, the sheet will adopt a shape that min-
imizes its elastic energy E. This leads to a competition between
both components of the energy: the bending energy E
b
is zero
when the sheet remains flat, while the stretching energy E
s
is
zero when the sheet achieves the curved, 3D geometry with
the prescribed target metric [19,49,53]. The final shape corre-
sponds to a balance between both contributions, which is deter-
mined by the sheet thickness t. Since the stretching energy scales
with tand the bending energy scales with t
3
, there will be a cer-
tain thickness that marks a transition between bending energy
domination and stretching energy domination [49,56]. Conse-
quently, the thinner a sheet becomes, the more energetically
favorable it becomes to bend than to stretch [49,51,53]. In other
words, the bending energy decreases more rapidly with decreas-
ing thickness than the stretching energy does, meaning that
when given the choice between bending or stretching to accom-
modate local shrinking/swelling, it will “cost”much more energy
for the thin sheets to stretch than to bend (which is why very
thin sheets are often considered inextensible membranes [55]).
The sheets will thus bend in 3D to adopt the target metric (if a
suitable embedding of the target metric exists), although the
exact target metric will not be achieved for finite thickness since
there will always be some energetic cost to bending a sheet
[19,53].
In summary, the concept of surface curvature could be dis-
cussed from an extrinsic and an intrinsic perspective, using the
mean and Gaussian curvature respectively. Some surfaces might
be curved from an extrinsic view, yet intrinsically remain flat (a
developable surface). When the aim is to achieve extrinsic curva-
ture from a flat surface, this could be easily achieved by an inex-
tensional bending (isometric) deformation of the surface, which
leaves the Gaussian curvature unaffected. However, achieving
intrinsic (Gaussian) curvature from a flat surface is more compli-
cated, as it requires the distances between points on the surface
to change (i.e. the metric should change). This cannot be
achieved through bending alone, but requires in-plane stretch-
ing or shrinking of the surface. The geometrical aspects of ori-
gami introduced in the following section are better understood
within the context of the ideas presented here.
Geometrical aspects of origami
Origami has inspired artists for hundreds of years to transform
ordinary sheets of paper into intricate yet beautiful 2D or 3D
geometries. Recently, engineers and scientists have also become
attracted to origami and have studied the paper-folding art from
a more mathematical perspective, giving rise to the field of com-
putational origami [57]. Origami offers many interesting mathe-
matical challenges, such as the folding of an arbitrary
polyhedron from a flat piece of paper [58] or the question of flat
foldability, i.e. whether a crease pattern results in a folded state
having all points lying in a plane [59]. Another aspect that has
received broad attention and that is of greater relevance to the
folding of 3D engineering structures is the question of rigid-
foldability. An origami design is rigid-foldable if the transition
from the flat to the folded state occurs smoothly through bend-
ing at the creases only, thus, without bending or stretching of
the faces in between the creases. In other words, a rigid origami
design could be folded from rigid panels connected with hinges,
which is desirable for deployable origami structures made from
rigid materials, such as solar panels, medical stents, or robots
[60,61].
Classical origami starts with a flat sheet of paper, which could
be considered a developable surface and by definition has zero
Gaussian curvature. Folding this flat sheet along predefined
crease lines essentially means bending the sheet at a very high
radius of curvature. Since bending does not change the metric
of the sheet, the Gaussian curvature will remain zero at (nearly)
all points on the folded sheet. In other words, no matter how a
sheet is folded, it remains intrinsically flat. It must, however,
be noted that some degree of stretching is involved in the folding
of paper. More specifically, Witten [55] has explained that sharp
folds must involve some stretching, as they would otherwise
result in an infinitely high bending energy. Nevertheless, this
stretching is only confined to the small fold lines and it is typi-
cally neglected, i.e. a non-stretchable sheet with idealized sharp
folds is assumed [62].
When discussing origami and polyhedral surfaces, it is useful
to introduce yet another definition of the Gaussian curvature,
known as Gauss’spherical representation. Gauss introduced this
description in his original paper on the Theorema Egregium. This
concept has been used, for example, by Miura [63] and Huffman
[64] in the analysis of origami. Gauss’spherical representation
could be obtained by first considering a closed, oriented contour
Caround a point Pon an arbitrary surface (Fig. 3a). Let us collect
the unit vectors on Cthat are normal to the surface and translate
them to the center of a unit sphere (the Gauss sphere), effectively
tracing out a new oriented contour C
0
on the surface of the
sphere and obtaining the “Gauss map”of the original contour.
Both contours enclose a certain area on their respective surfaces:
say Cencloses Fand C
0
encloses G. The Gaussian curvature Kis
then defined as the ratio of Gto F, in the limit that Capproaches
P[42,65]:
K¼lim
C!P
G
F
While calculating the Gaussian curvature on an arbitrary sur-
face might not be trivial using the above definition, Gauss’spher-
ical representation does provide an additional interpretation of
the intrinsic curvature. For example, the Gauss map of a closed
contour on a developable surface encloses zero area on the Gauss
sphere (G¼0), indeed corresponding to zero Gaussian curvature
following the above definition (Fig. 3). On the other hand, the
closed contours on spherical or saddle surfaces map into closed
contours with non-zero enclosed areas on the Gauss sphere, indi-
cating the non-zero Gaussian curvature of these surfaces. Note
that G<0 when the orientation of C
0
is opposite to that of C,
resulting in a negative Gaussian curvature (Fig. 3a) [65].
Applying Gauss’spherical representation to the simplest type
of origami, a single straight crease crossing a flat sheet of paper,
once again proves that folding has no intrinsic effect on the sur-
face. The normals on each face map into a single point on Gauss’
sphere, while the normals on the crease between the faces are not
uniquely defined and map into an arc connecting both points,
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resulting in G¼0 and, thus, no Gaussian curvature (Fig. 3b) [44].
Miura [63] used Gauss’spherical representation to analyze differ-
ent configurations of fold lines joining at a common vertex and
showed that some combinations cannot be folded rigidly [63].
For example, a vertex of valency three (three creases joining at
the vertex) is never rigid-foldable: the three faces surrounding
the vertex have normals in different directions, tracing out a
spherical triangle on the Gauss sphere with non-zero area
(Fig. 3d). This would imply that K–0, which is not possible when
rigidly folding a flat surface. Similarly, Miura showed that a four-
valent vertex with all mountain (“upwards”) or valley (“down-
wards”) folds cannot be folded rigidly, while a four-valent vertex
with three mountain folds and one valley fold (and vice versa)
could be rigidly folded [63,65]. It must be, however, emphasized
that a three-valent vertex or a four-valent vertex with all moun-
tain folds could be folded when the rigid folding requirement is
relaxed, i.e. when the faces are allowed to bend.
Based on the above insights, it might be argued that origami
is not a suitable approach to create intrinsic curvature from flat
sheets as origami deals with isometric deformations. However,
applying the right fold and cut patterns could alter the “glo-
bal”or “apparent”Gaussian curvature, without the need for
in-plane stretching or shrinking of the flat sheet. In essence,
origami and kirigami techniques allow to approximate intrinsi-
cally curved surfaces through developable deformations of
many small faces connected through fold lines. The specific
origami and kirigami techniques that have been used by other
researchers for this purpose are described in the next sections
of this review. Note that we will restrict to the traditional form
of origami in which flat (Euclidean) sheets are folded. It is,
however, also possible to apply origami to non-Euclidean
paper, as shown by Alperin et al. [66] who folded an origami
crane from hyperbolic paper, i.e. paper with constant negative
curvature.
RESEARCH: Review
FIGURE 3
Gauss’spherical representation of the Gaussian curvature. (a) Definition of the Gauss map using a closed, oriented contour on a point on an intrinsically
curved surface. The unit normals on are translated to the center of a unit sphere and trace out a new contour . (b) Folding along a simple straight crease
does not change the Gaussian curvature (note the zero enclosed area on the unit sphere). In the flat state (left pane), all the normals point in the same
direction, resulting in no enclosed area on the unit sphere. In the folded configuration, both are connected on the unit sphere through a zero-area arc. (c) The
Gauss map applied to a unit cell of the rigid-foldable Miura-ori pattern. In the partially folded configurations (middle and right pane), the normals trace out a
bowtie contour on the Gauss sphere, with one-half of the bowtie classifying as “positive”area (clockwise tracing) and the other half as “negative”area
(counter-clockwise trace), resulting in zero net area. (d) The Gauss map applied to the three-valent vertex of a tetrahedron. The transformation of the flat state
(left pane) to the folded state (middle and right pane) induces a change in the Gaussian curvature (non-zero area on the unit sphere), showing that a three-
valent vertex cannot be achieved in rigid origami.
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Origami approaches
In this section, we will review different origami approaches that
have been used to approximate intrinsically curved surfaces. The
four different approaches discussed here, namely origami tessel-
lations, tucking molecules, curved-crease origami, and concen-
tric pleating, all start from a flat, uncut sheet that is folded
along predefined creases.
Origami tessellations
Origami tessellations, characterized by a periodic crease pattern
or “tiling”of a flat sheet, have inspired artists for many decades
[67] but have also found their way into scientific and engineering
applications such as compliant shell mechanisms [68] and
mechanical metamaterials [30,69]. Moreover, the rational design
of the tessellation pattern allows for changing the apparent cur-
vature of flat sheets without requiring local stretching of the
faces.
Miura-ori
The most widely studied origami tessellation is the herringbone
pattern known as Miura-ori, originally introduced as an efficient
packing of solar sails [25] but also observed in spontaneous wrin-
kling of thin, stiff films on thick, soft substrates subjected to biax-
ial compression [70]. A Miura-ori unit cell consists of a four-
valent vertex connecting four parallelograms using three moun-
tain folds and one valley fold (Fig. 4). An important property of
this origami design is that it is rigid-foldable, as indicated by
Gauss’spherical representation [63,65] (Fig. 3c). Schenk and
Guest studied the geometry and kinematics of Miura-ori and
showed that purely rigid Miura-ori has only a single degree of
freedom, i.e. in-plane folding and unfolding [28]. Based on the
analysis of a single unit cell, they concluded that a Miura-ori
sheet is an auxetic material, characterized by a negative in-
plane Poisson’s ratio. In a later study, however, Lv et al. showed
that some specific configurations could exhibit a positive in-
plane Poisson’s ratio as well [71]. While rigid-foldability of
Miura-ori allows only in-plane deformations, experiments with
simple paper models reveal that the folded sheets could also
deform out-of-plane (Fig. 4d-e). Indeed, Schenk and Guest iden-
tified saddle and twist deformation modes and showed that these
are only possible in non-rigid Miura-ori, where the individual
faces are allowed to bend [28]. This leads to an interesting prop-
FIGURE 4
Miura-ori tessellation. (a) A crease pattern of a Miura-ori tessellation. Solid lines are mountain folds and dashed lines are valley folds. (b) Two different unit
cells for a Miura-ori tessellation, consisting of four parallelograms connected through three mountain folds and one valley fold. (c) A paper model of the
Miura-ori tessellation in a partially folded state (left) and collapsed state (right), showing the flat-foldability of this origami pattern (Scale bar is 2 cm). (d)
Saddle-shaped and (e) twisted out-of-plane deformations of the (non-rigid) Miura-ori sheet.
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erty of Miura-ori (and origami tessellations in general): through
developable deformations at unit cell level, the global Gaussian
curvature of the sheet could be changed, making this origami tes-
sellation an interesting candidate for compliant shell mecha-
nisms [68,72].
Several researchers have sought for generalizations or varia-
tions of the Miura-ori pattern that approximate a curved surface
when folded, without requiring out-of-plane deformations.
Tachi investigated quadrilateral mesh origami consisting of
quadrilateral faces joined at four-valent vertices and established
rules for rigid-foldability. Starting from a regular Miura-ori pat-
tern, he explored variations that could fit a freeform surface
(e.g. a dome-shape), while remaining rigidly foldable [73]. Gattas
et al. parameterized the Miura-ori to be able to systematically
compare different pattern variations. They investigated five
rigidly foldable “first-level derivatives”obtained by changing a
single characteristic such as the crease orientation [74]. Depend-
ing on the derivative, geometries with an overall single or double
curvature could be achieved, although the latter seemed to be
limited to a non-developable crease pattern [74]. Sareh and Guest
considered Miura-ori as one of the seventeen plane crystallo-
graphic or “wallpaper”groups (a pmg group) and provided a
framework to obtain flat foldable symmetric generalizations of
the Miura-ori, some of which could result in globally curved
geometries when folded [75]. More recently, Wang et al. pro-
posed a design method to obtain Miura-ori generalizations that
approximate cylindrical surfaces upon folding [76]. While their
method takes into account rigid folding and the thickness of
the faces, which is useful for practical applications, it is restricted
to cylindrical geometries and hence single curvature [76]. Per-
haps the most complete and successful approach to approximate
arbitrarily curved surfaces with Miura-ori generalizations is the
one recently proposed by Dudte et al. (Fig. 5)[77]. Dudte et al.
employed constrained optimization algorithms to solve the
inverse problem of fitting an intrinsically curved surface with a
generalized Miura-ori tessellation. They showed that the surfaces
of generalized cylinders could be approximated using flat fold-
able and rigid-foldable tessellations, which could not be guaran-
teed for intrinsically curved surfaces. In the latter case, snapping
transitions were required during the action of folding or unfold-
ing, although the final configuration was strain-free [77]. The
researchers also showed that a higher number of unit cells results
in a more accurate fitting of the surface but at the cost of a higher
folding effort [77].
Other periodic tessellations
In addition to Miura-ori, several other tessellations are well
known among origami artists and scientists. Examples are the
tessellations obtained when tiling the plane with a six- or
eight-crease waterbomb base, the former of which has been used
to create an origami stent [34,78]. Other famous origami tessella-
tions were developed by Ron Resch [79,80] and have inspired sci-
entists to create origami-based mechanical metamaterials [71] or
freeform surface approximations [81]. Origami tessellations that
are rigid-foldable are of particular interest for engineering appli-
cations. Evans et al. used the method of fold angle multipliers
to analyze the existing flat foldable tessellations and identify
those that were rigid-foldable [61]. Furthermore, the researchers
presented rigid-foldable origami “gadgets”, local modifications
to a crease pattern, to develop new rigidly foldable tessellations
FIGURE 5
Generalized Miura-ori tessellations fitting curved surfaces. The top row depicts simulations, while the bottom row shows physical models. Adapted by
permission from Macmillan Publishers Ltd: Nature Materials [77] copyright 2016.
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[61]. Tachi studied the rigid foldability of “triangulated”origami
tessellations, in which the quadrangular faces are divided into tri-
angles (essentially capturing the bending of faces of non-
triangulated origami) [82]. Through numerical simulations (by
means of a truss model), it was observed that most periodic trian-
gulated origami tessellations exhibit two (rigid) degrees of free-
dom, a folding/unfolding motion and a twisting motion, with
Miura-ori being an exception as it only shows the folding/
unfolding motion [82].
Applying a periodic tessellation such as Resch’s triangular pat-
tern to a flat sheet essentially means texturing the sheet with
small-scale structures that give rise to unusual properties on a
global scale [72]. Due to the strong interaction between the local
kinematics and global shape, these tessellated sheets have earned
the name “meta-surfaces”, in analogy with 3D metamaterials
[68,72]. An interesting property of the textured sheets is that
folds may partially open or close locally, effectively simulating
local stretching or shrinking. As a consequence, the sheets could
undergo large deformations and change their global Gaussian
curvature, without stretching of the actual sheet material
[68,72] (Fig. 6). However, the opportunities to approximate
curved surfaces using this approach are limited despite the ease
with which small paper models could be manipulated. That is
because approximating saddle shapes might involve some facet
and crease bending that make it difficult, if not impossible, to
achieve anticlastic geometries (negative Gaussian curvature)
through rigid folding [81]. Moreover, for large tessellated sheets
(with many unit cells), even synclastic geometries (positive Gaus-
sian curvature) might not be rigid-foldable. Tachi showed that
smooth rigid folding of (triangulated) periodic tessellations in
dome shapes is obstructed once the tessellated sheet becomes
too large, making only cylindrical surfaces feasible [82]. A similar
conclusion was reached by Nassar et al. [83]. Indeed, the tessella-
tion shown in Fig. 6c naturally adopts a cylindrical shape in the
partially folded configuration.
Although several standard origami tessellated sheets could
conform to curved surfaces, the achievable geometries are lim-
ited. In order to obtain more complex 3D shapes, Tachi devel-
oped the Freeform Origami method to generalize existing
tessellations, essentially building further on his work on quadri-
FIGURE 6
Origami tessellations. (a and b) Triangular Ron Resch and square waterbomb tessellations, respectively. Top: crease pattern (solid lines are mountain folds,
dashed lines are valley folds), middle: partially folded state, bottom: fully folded state (Scale bar is 2 cm). (c) Natural resting state of the partially folded square
waterbomb tessellation (for a large enough sheet), adopting a cylindrical shape (Scale bar is 2 cm). (d) Various configurations with global intrinsic curvature of
the same square waterbomb tessellated sheet, obtained through locally opening and closing the unit cells.
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lateral mesh origami [84]. He provided mathematical descrip-
tions of the conditions that apply to the traditional tessellations
such as developability and flat-foldability, and numerically calcu-
lated perturbations of these tessellations while preserving those
conditions. The algorithm was implemented in a software pack-
age that allows the user to actively disturb an existing folded ori-
gami tessellation and observe the corresponding changes to the
crease pattern in real-time [84]. However, the software is not cap-
able of solving the inverse problem of finding the crease pattern
that belongs to a given 3D surface. Other researchers have also
used mathematical methods to calculate new tessellations that
could fold into 3D geometries. Zhou et al. proposed the “vertex
method”to inversely calculate a developable crease pattern
based on the Cartesian coordinates of a given 3D geometry
[85]. While their method is versatile enough to develop the
crease pattern for a structure that fits between two (single-)
curved surfaces, a major limitation is that it is not applicable to
intrinsically curved surfaces [85]. More recently, Song et al. built
further on this work and developed a mathematical framework to
create trapezoidal crease patterns that rigidly fold into axisym-
metric double-curved geometries [86]. More specifically, their
method calculates the crease pattern that fits both an inner
and outer target surface with the same symmetry axis. While
intrinsic curvatures could in this way be achieved, the proposed
method is limited to very specific ring-like geometries, possessing
rotational symmetry and having relatively small curvatures [86].
Tucking molecules
Other approaches to approximate intrinsically curved surfaces
could be obtained from the field of computational origami design,
in which one searches for the crease pattern that belongs to a
given shape, typically a 3D polyhedron. The first well-known
computational tool facilitating origami design was proposed by
Lang and is based on tree-like representations of the desired
shapes (“stick-figures”)[87]. However, the method is restricted
to calculating origami bases that need to be shaped afterward
into the desired geometry. To enable the construction of crease
patterns for arbitrary 3D polyhedrons, Tachi developed his
well-known “origamizing”approach based on tucking molecules
[59,88]. The starting point is a polyhedral representation of an
arbitrary surface, which is made topologically equivalent to a
disk (such that it is not a closed polyhedron but has a cut that
allows it to open). The basic idea of the approach is to map all
the surface polygons of the polyhedron onto a plane and to fill
the gaps in between with tucking molecules, which are flat fold-
able segments, creating a 2D crease pattern in doing so [88]. The
resulting crease pattern is not considered an origami tessellation
in the context of this review as it is highly non-periodic and com-
prises polygons of different shapes and sizes. The tucking mole-
cules connect adjacent surface polygons and are tucked away
behind the visible surface upon folding (Fig. 7a). Tachi defined
edge-tucking and vertex-tucking molecules, respectively bringing
edges or vertices together in the folded configuration. In order to
fit the desired 3D shape with the surface polygons, crimp folds
are also employed to locally adjust the tucking angle [88]. The
entire procedure was implemented in a software package for
which the input is a polygon mesh and the output is a 2D crease
pattern, allowing the creation of complex origami structures that
were never folded before, such as the origami Stanford bunny
(Fig. 7c) [59]. Tachi attributed the increased practicality of this
approach compared to earlier origami design methods to three
reasons: multi-layer folds rarely occur, the crimp folds offer struc-
tural stiffness by keeping vertices closed, and the method has a
relatively high efficiency, defined by the ratio of polyhedral sur-
face area to required paper area [59]. Despite its versatility and
generality, the origamizing method has the drawback that some
3D polyhedrons cannot be mapped into a 2D pattern, a problem
that was recently addressed by Demaine and Tachi [89], or that
the proposed pattern is inefficient. Moreover, the flat folding
requirement of the tucking molecules significantly reduces the
applicability of this method to the folding of stiff, thicker mate-
rials [81]; and the presence of crimp folds obstructs smooth fold-
ing [59], making this method intractable for industrial
applications.
Tachi also proposed a more practically applicable method to
approximate curved surfaces, combining aspects from his earlier
work on freeform origami [84] and the origamizing approach
[88]. The basic idea is that generalizations of Resch’s tessellations
are calculated that fit a given polyhedral surface [81]. Contrary to
the origamizing approach, the surface polygons do not have to
be mapped onto the plane; but a first approximation of the tes-
sellation is directly obtained from the 3D structure, after which
it is numerically optimized to become developable and to avoid
collisions of faces. As such, the implementation of this method
is related to the earlier work on Freeform Origami [84], but it is still
considered in this section since (simple) tucking molecules are
used. Tachi defined a “star tuck”and variations thereof as build-
ing blocks to tessellate the given 3D surface [81]. The fundamen-
tal difference between conventional tucks and the star tucks is
that the latter could also exist in a semi-folded state and do not
have to be folded flat (Fig. 7b). The surface polygons could be
arranged to locally fit the desired shape through partial opening
of the star tucks, while this required crimp folds in the origamiz-
ing approach. The algorithm was implemented in a software
package, allowing users to interactively design the generalized
Resch tessellations corresponding to a given surface (Fig. 7d).
However, for highly complex surfaces, which would require sig-
nificant stretching and shrinking to become developable, crease
patterns cannot always be generated. Moreover, smooth rigid
folding of the tessellations is not guaranteed for all cases.
Nonetheless, the proposed method has significant potential for
the practical folding of advanced materials into intrinsically
curved surfaces.
Curved-crease origami
When discussing curvature and origami, it is natural to also con-
sider curved-crease origami. While this variant of traditional ori-
gami has interested artists for several decades, the mathematics
of curved-crease folding is underexplored and practical applica-
tions of origami have been primarily limited to straight creases
[90]. Straight-crease and curved-crease origami is fundamentally
different since both faces adjacent to a curved crease always have
to bend in order to accommodate the folding motion [91,92].In
other words, curved-crease origami is never rigid-foldable. As a
consequence, curved-crease origami cannot be reduced to a mat-
ter of tracking vertex coordinates as in the case of rigid origami,
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and the bending stiffness of the folded sheet becomes an impor-
tant parameter [90]. Moreover, the bending induced in the faces
necessitates simultaneous folding along multiple creases, compli-
cating automated folding processes [93].
One of the first and most influential analyses of curved creases
was performed by Huffman, using Gauss’spherical representa-
tion to examine the local folding behavior [64]. Indeed, the
Gauss map of a closed contour crossing a straight-crease maps
into a zero-area arc while the map of a contour encompassing a
curved crease has non-zero area due to the facet bending, which
is indicative of its non-rigid-foldability. The geometry of curved-
crease folding has been further explored by Duncan and Duncan
[91] and Fuchs and Tabachnikov [94]. They presented several
theorems relating the properties of the curved crease to those
of the adjacent faces, which are outside the scope of this review.
The most important take-away is that folding along a curved
crease satisfies the developability of the sheet, meaning that a
curved-folded origami consists of developable patches of either
a generalized cylinder, a generalized cone, or a tangent devel-
opable to a space-curve [91,95,96]. Hence, folding along curved
creases cannot alter the intrinsic curvature of the sheet, as no
in-plane distortion of the faces occurs. However, curved-crease
origami does provide means to alter the global intrinsic curvature
of flat sheets, in a manner similar to straight-crease folding. More
specifically, we identify two approaches to approximate non-zero
Gaussian curvature: the use of curved-crease couplets and folding
along concentric curved creases. In the current sub-section, only
the first approach is discussed, as the latter fits within the broader
concept of concentric pleating that is treated in the next sub-
section.
Curved-crease couplets, a term introduced by Leong [97], are
pairs of curved and straight creases that have been employed
by origami artists to create 3D origami with apparent positive
and negative intrinsic curvatures. Typically, axisymmetric struc-
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FIGURE 7
Origami involving tucking molecules. (a) The definition of a tucking molecule used in the “Origamizer”approach. (I) A section of a polyhedral surface, (II) the
flattened polyhedral surface, showing the surface polygons connected through edge- and vertex-tucking molecules. (III) The folded configuration, showing
the excess material being tucked away behind the outer surface. Reproduced with permission from IEEE from Ref. [59]. (b) Comparison between tucking
molecules in the “Origamizer”approach and the generalized Ron Resch tessellation approach. Reproduced with permission from ASME from Ref. [81]. (c)
Origami Stanford bunny, folded from a single-sheet crease pattern created using the “Origamizer”software. Reproduced from Fig. 1 from Ref. [89]. (d)
Comparison between the surface approximations of the “Origamizer”approach (top) and the generalized Ron Resch pattern approach (bottom) to an
intrinsically curved bell-shaped surface. Note the partially opened tucking molecules in the latter approach. Figures and crease patterns were obtained using
“Origamizer”[59] and “Freeform Origami”[73] (red lines are mountain folds, blue lines are valley folds).
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tures are created from relatively simple crease patterns (Fig. 8). A
design method and software tool to generate crease patterns
based on “rotational sweep”was proposed by Mitani [98] and a
very similar tool was created by Lang [99]. The basic idea is that
aflat sheet is “wrapped”around the desired cylindrical or conical
geometry and that the excess material is folded into flaps, sec-
tions of material that are only connected at one edge [100],
on the outside surface. This is different from Tachi’s origamiz-
ing approach [59] in which excess material is tucked away
inside the geometry, resulting in more complicated crease pat-
terns [98].Aflap in Mitani’s method consists of a kind of
curved-crease couplet, containing a straight mountain crease
and a piecewise linear valley crease, approximating a curved
line. The latter crease represents half of the vertical cross sec-
tion of the desired shape and, when revolved around the verti-
cal axis, traces out this shape [97,98]. To create the crease
pattern, the curved-crease couplets are simply repeated Ntimes
and arranged on a rectangular sheet (for cylindrical geometries)
or on an N-gon (for conical geometries), with higher values of
Nresulting in higher rotational symmetry. Using this method,
double-curved shapes could be approximated, as shown in
Fig. 8. More recently, Mitani also proposed a variant of this
method in which the flaps are replaced by “triangular prism
protrusions”[101]. Again, the excess material that results when
wrapping the desired geometry is placed on the outside surface,
this time in a slightly different manner involving four creases
instead of two.
Although recent efforts have been made to understand
curved-crease origami from a more mathematical perspective
[102], it remains a field that is primarily reserved for artists. As
such, only limited work has been done that explores the capabil-
ities of curved-crease folding to approximate intrinsically curved
surfaces. Nonetheless, it is expected that the use of curved creases
has a significant potential in practical origami not only to
FIGURE 8
Curved-crease origami. (a) Simple examples of curved-crease origami (Scale bar is 2 cm). (b) An origami sphere (positive Gaussian curvature) and an origami
hyperboloid (negative Gaussian curvature) created with the method described by Mitani [98] (Scale bar is 2 cm). (c and d) Top views of the origami
hyperboloid and sphere (left panes) and the associated crease patterns (right panes, solid lines are mountain folds and dashed lines are valley folds). (e)
Comparison between the standard origami sphere obtained with Mitani’s method [98] and the “smooth”variant (Scale bar is 2 cm). (f) Closer view of a
wrinkle in the smooth origami sphere, indicative of the frustration between curved and straight creases (left) and the associated crease pattern without
horizontal creases (right) (Scale bar is 1 cm).
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achieve complex geometries but also for kinetic architectures [96]
and shape-programmable structures [103].
Concentric pleating
As a final category of folding strategies to approximate intrinsi-
cally curved surfaces, we consider origami based on concentric
pleating, i.e. alternately folding concentric shapes into moun-
tains and valleys. Geometries with apparent negative Gaussian
curvature spontaneously result after folding the remarkably sim-
ple crease patterns (Fig. 9). The original crease patterns for this
origami consisted of equally spaced concentric squares or circles,
although variants with ellipses or parabolas have also been
folded [90]. One might classify the concentric pleating as a type
of origami tessellation, however, we consider it separately due to
its remarkable properties.
The classical model with the concentric squares is called the
pleated hyperbolic paraboloid or simply hypar, after the nega-
tively curved surface it seems to approximate. As explained by
Demaine et al., the 3D shape naturally arises due to the paper’s
physics that balances the tendency of the uncreased paper to
remain flat and that of the creased paper to remain folded
[104]. Seffen explained the geometry by considering the pleating
as a “corrugation strain”toward the center without causing an
axial contraction of the hinge lines, thereby forcing the model
to deform out-of-plane [68]. Thus, the pleating introduces a dis-
tortion of the flat sheet that is relieved by settling on an energy-
minimizing 3D configuration. This principle of anisotropic strain
(shrinking) has been recently used by van Manen et al. to pro-
gram the transformation of flat shape memory polymer sheets
into an approximation of a hypar, using thermal activation
[105].
Although paper models of the pleated hypar are ubiquitous
and well-known among origamists, mathematicians have ques-
tioned whether the standard crease pattern of concentric squares
could actually result in the pleated hypar. More specifically: does
aproper folding (folding angles between 0 and p) along exactly
these creases exist? Demaine et al. proved the surprising fact that
it does not, hence the folding along the standard crease pattern
cannot result in the hypar without some stretching or additional
creasing of the paper [104,106]. The problem lies in the twisting
of the interior trapezoidal faces of the hypar. While a standard
piece of paper could be effortlessly twisted and curled in space,
Demaine et al. proved that this twisting should not occur for
the interior faces of the hypar. Using aspects of differential geom-
etry, such as the properties of torsal ruled surfaces, the research-
ers proved two theorems: straight creases must remain straight
after folding and a section of the paper bounded by straight
creases must remain planar and cannot bend or twist [104].
FIGURE 9
Concentric pleating origami. (a) Origami hyperbolic paraboloid (“hypar”), obtained by pleating concentric squares. Top left: top view, middle left: side view,
bottom left: Closer view of the twisted faces in the standard hypar model, right: side view showing the global saddle-shaped geometry of the hypar (Scale bar
is 2 cm). (b) Circular variant of the hypar, obtained by pleating concentric circles (with center hole). Left: side view of an annulus with three creases (Scale bar
is 2 cm), top right: top view of an annulus with three creases, bottom right: side view of an annulus with eight creases (Scale bar is 1 cm). (c) Different crease
patterns. Left: standard crease pattern for the square hypar, middle: triangulated crease pattern for the square hypar, white lines represent the additional
creases that enable proper folding of the hypar (adapted from Ref. [104]), right: crease pattern for the circular hypar (solid lines are mountain folds, dashed
lines are valley folds).
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Demaine et al. conjecture that the actual folding of the hypar
from the standard crease pattern is enabled through additional
creases in the paper, potentially many small ones. Alternatively,
some stretching at the material level might occur [106]. In any
case, it is clear that the folding of the standard crease pattern into
the pleated hypar is highly non-rigid, as could also be intuitively
understood when drawing a closed curve in one of the square
rings: the curve crosses four folds with the same mountain or val-
ley assignment, which cannot fold rigidly according to Gauss’
spherical representation [65].
The theorems of Demaine et al. [104] have implications for
straight crease origami that exhibits non-rigid behavior, such as
the Miura-ori discussed above. It was explained that a (partially)
folded Miura-ori could deform out-of-plane through facet bend-
ing, which should not be possible according to the above theo-
rems. However, facet bending may be enabled by an additional
spontaneous crease in the quadrangular facets, making the facet
piecewise planar and satisfying the theorems in doing so. This
“triangulation”of the facets has been employed by researchers
to capture facet bending in mathematical origami models
[69,72,77,107,108]. Moreover, Demaine et al. [104] proved that
a triangulation of the standard hypar crease pattern also renders
the pattern rigid-foldable, making it possible to create hypars
from more rigid materials such as sheet metal [104]. The insights
into the hypar crease pattern are also relevant for the curved-
crease couplets that were introduced in the previous section, con-
sisting of alternating curved and straight creases. The curved
creases require the adjacent faces to bend, while the straight
creases do not allow bending. Furthermore, the straight creases
do not remain straight in the folded 3D model. Following the
theorems of Demaine et al. [104], it thus seems that these
curved-crease couplets cannot be folded. This problem is allevi-
ated in Mitani’s method [98] by including horizontal creases
and representing the curved crease as a piecewise linear crease
(Fig. 8c and d), thereby ensuring that the model remains piece-
wise planar. It must also be noted that the theorems of Demaine
et al. [104] are applicable to interior faces (not at the boundary)
and fold angles between 0 and p, while the curved-crease cou-
plets end at the paper boundaries and the straight creases seem
to be folded at an angle of p. Indeed, Mitani states that the hor-
izontal creases can be omitted when the straight creases are
folded close to p[98], resulting in a smooth folded geometry
(Fig. 8e). However, closer inspection of such models still reveals
the occurrence of small, spontaneous kinks or wrinkles, indica-
tive of the frustration between the straight and curved crease
(Fig. 8f).
In addition to the pleated hypar, another classical model is
obtained by pleating concentric circles with a hole in the middle
(Fig. 9b). Similar to the hypar, this pleated annulus deforms into
a saddle, with the degree of curvature depending on the fold
angles. Mouthuy et al. attributed this specific deformation to
the “overcurvature”of the ring, which is a measure of how much
the curvature of the ring exceeds that of a circle with the same
circumference [109]. Indeed, the pleating causes the curvature
of the concentric creases to increase while their length is pre-
served, resulting in overcurvature. While Demaine et al. proved
that the standard hypar crease pattern cannot be folded without
additional creases or stretching, it remains unknown whether
this is also the case for the pleated annulus [104]. However, the
researchers conjecture that the annulus could be folded from
exactly the given crease pattern and that additional creases nor
stretching are required. Dias et al. investigated the mechanics
of the simplest type of pleated annulus: a paper strip with a sin-
gle circular crease [110]. The researchers provided analytical
expressions for the elastic energy of the annulus, to which both
the faces and the crease contribute. While the incompatibility
between the pleating and the resistance to in-plane stretching
forces the model to buckle out-of-plane, the actual shape it settles
on is determined by the minimization of this elastic energy
[110]. Later, Dias and Santangelo extended the work to a pleated
annulus with several concentric circles and investigated poten-
tial singularities that might arise when attempting to fold the
model from the given crease pattern [111]. The researchers did
not prove that the crease pattern is exactly foldable, but their
results indicated that singularities do not occur for a sufficiently
narrow crease spacing, supporting the conjecture of Demaine
et al. [104,111].
Concentric pleating is a captivating type of origami, as very
simple crease patterns result in geometries with apparent intrin-
sic curvature. Although the mechanisms of this technique are
not yet fully understood, particularly for curved creases, concen-
tric pleating could offer an interesting pathway to achieve intrin-
sic curvature. Especially when extreme values of overcurvature
are induced or when different types of hypars or annuli are com-
bined, complex geometries may arise, examples of which are the
“hyparhedra”proposed by Demaine et al. [112].
Kirigami approaches
Kirigami is an art form that is closely related to origami but also
involves cutting the paper at precise locations. Kirigami has not
been explored to the same extent as origami but has recently
gained traction among scientists as a promising paradigm toward
stretchable electronics [38,39,113,114] or advanced honeycomb
structures [115–117]. In this section, we review two distinct kiri-
gami approaches that could be employed to approximate intrin-
sically curved surfaces, namely lattice kirigami and kirigami-
engineered elasticity.
Lattice kirigami
Lattice kirigami is a relatively novel and promising cutting and
folding technique that was introduced by Castle et al. [62]. Its
essence lies in removing some areas from the sheet through cut-
ting, after which the resulting gaps are closed through folding
along prescribed creases. Lattice kirigami has its roots in crystal-
lography, particularly in the defects arising in crystal lattices.
Understanding the essentials of this technique therefore requires
some terms and concepts from crystallography.
The starting point is the honeycomb lattice, represented as a
2D tessellation of regular hexagons. An infinite flat plane (zero
Gaussian curvature) could be tiled using only hexagons, but this
is not possible for intrinsically curved surfaces such as spheres or
saddles [118]. For example, a soccer ball cannot be tiled with
hexagons entirely but requires twelve pentagons to conform to
the spherical shape. The insertion of a pentagon or a heptagon
within a tiling of hexagons is known as a lattice disclination,
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which is a type of topological defect that disrupts the orienta-
tional order of the lattice [118,119]. These local lattice distortions
cause the surfaces to deform out-of-plane to relieve in-plane
strains, reminiscent of the metric-driven principles discussed
before. The disclinations themselves form concentrated sources
of Gaussian curvature: pentagons result in positive Gaussian cur-
vature and heptagons in negative Gaussian curvature [118].
Fig. 10 illustrates the effect of disclinations in a hexagonal weave
when a single hexagon is replaced by a pentagon or a heptagon,
a technique which has long been employed by basket weavers to
create complex shapes [120]. Another type of topological defect
is a dislocation, which disturbs the translational symmetry of
the lattice and is formed by a dipole of disclinations (with oppo-
site topological charge) [118,119]. While disclinations and dislo-
cations are considered defects in a topological sense, they are
often necessary distortions of the crystal lattice in natural pro-
cesses. For example, Sadoc et al. showed that these defects are
crucial elements in phyllotaxis, the efficient packing algorithm
that nature uses in self-organizing growth processes, such as
the spiral distribution of florets in flowers [119,121]. The work
of Sadoc et al. [119] was in fact a direct inspiration for Castle
et al. to develop lattice kirigami [62].
The basic idea behind lattice kirigami is to strategically
remove areas from a honeycomb lattice, paste the newly formed
edges together and fold along prescribed creases to create discli-
nation dipoles, resulting in a stepped 3D surface with local con-
centrations of Gaussian curvature. Fig. 11a provides a simple
example showing two disclination dipoles at the ends of the
cut [62]. Inspection of a single disclination dipole reveals that
the cutting and pasting transforms one hexagon into a pentagon
(by removing a wedge of p=3) and combines two partial hexa-
gons into a heptagon (i.e. adding a wedge of p=3), see also
Fig. 11c. By systematically exploring cutting and pasting on the
honeycomb and its dual lattice, Castle et al. established the basic
rules for lattice kirigami that satisfy a no-stretching condition
and preserve edge lengths on the lattices [62]. The researchers
identified the basic units of lattice kirigami: i.e. a 5–7 disclination
pair and a 2–4 disclination pair, with the values indicating the
coordination number of the vertices (Fig. 11c and d). The gaps
that are left after cutting are closed through “climb”or “glide”
moves, or a combination of both, in order to result in a 3D
stepped surface (Fig. 11a–d). Additionally, the researchers inves-
tigated the “sixon”(Fig. 11e), which is obtained by removing
an entire hexagon from the honeycomb and closing the gap
using appropriate mountain and valley folds in the adjacent
hexagons. Through their basic rules, Castle et al. [62] constructed
the foundations for an elegant and new approach toward
stepped approximations of freeform surfaces.
Sussman et al. built upon these foundations and demon-
strated that lattice kirigami is well-suited to obtain stepped
FIGURE 10
Lattice disclinations in a hexagonal lattice. (a) Inserting a single pentagon in a hexagonal weave induces positive Gaussian curvature (Scale bars are 2cm). (b)
A hexagonal weave without lattice disclinations remains flat. (c) Inserting a single heptagon in the hexagonal weave induces negative Gaussian curvature (the
physical models photographed were made based on the work presented in Ref. [120]).
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approximations of arbitrarily curved surfaces, using a relatively
simple inverse design algorithm [122]. Key to their approach is
that the kirigami motifs presented by Castle et al. [62] could be
folded into several configurations by “popping”the plateaus
up or down (Fig. 11e). By connecting many of these motifs
together in admissible configurations and properly assigning
the plateau heights, complex stepped structures could be
obtained. Sussman et al. first considered the use of standard 5–
7 climb pairs, but this approach has the important limitation
that every target structure requires a new fold and cut pattern
[122]. In order to obtain a truly pluripotent kirigami pattern that
could fit several target shapes, the researchers used sixon motifs.
These sixons could be conveniently arranged on a triangular lat-
tice with the centers of the excised hexagons on the lattice points
(Fig. 11f). The hexagonal gaps are then closed by folding the
remaining hexagons in either one of their allowed configurations
(i.e. popping the plateaus upward or downward). As a conse-
quence, a myriad of stepped surfaces could be achieved from this
basic kirigami tessellations, simply by varying the mountain/val-
ley assignment of the fold lines while making sure that adjacent
plateaus only differ by one sidewall height (i.e. one step at a
time), as shown in Fig. 11f for a saddle-like geometry [122].
The mountain/valley assignment for every fold line could be
easily determined from a “height map”of the target surface. Suss-
man et al. showed that surfaces with arbitrary curvature could be
approximated, provided that the gradient with which the surface
rises or falls is not too steep (depending on the ratio of plateau
width to plateau height). Their results showed that lattice kiri-
gami constitutes a very versatile approach to approximate intrin-
sically curved surfaces, and has great potential for self-folding
due to its simplicity as compared to conventional origami tech-
niques [122].
The most recent progress into lattice kirigami has been made
by Castle et al. [123] who generalized their earlier work by relax-
ing some of the initially imposed rules and restrictions. Natural
generalizations included the removal of larger wedges from the
sheet and cutting and folding along different angles than in
the original method. Additionally, the researchers demonstrated
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FIGURE 11
Lattice kirigami. (a) The basic principle of lattice kirigami: a wedge is removed from a honeycomb lattice (top), the edges are brought together and the paper
is folded along prescribed fold lines (known as a “climb”move). (b) Another basic move, the “glide”, in which the gaps are closed through folding and sliding
along the slit connecting the two excised triangles. (c) A “5–7”disclination dipole, characterized by one vertex surrounded by five hexagon centers and one
vertex surrounded by seven hexagon centers (represented with the solid circles). d) A “2–4”disclination dipole: one vertex has two neighboring hexagon
corners, the other vertex has four neighboring corners (solid squares). (e) Excising an entire hexagon results in a “sixon”, which could be folded into different
configurations by popping the plateaus up or down. (f) Folding a pluripotent “sixon sheet”, i.e. a tessellation of sixons on a triangular lattice, enables step-wise
approximations of curved surfaces (e.g. left bottom pane). All scale bars are 2 cm.
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lattice kirigami on other Bravais lattices and arbitrarily complex
lattices with a basis [123]. However, the most extensive general-
izations came in the form of area-preserving kirigami, in which
only slits are made and no material is removed, and additive kir-
igami, in which new material is actually inserted in the slits, rem-
iniscent of natural growth of cells. Furthermore, Castle et al.
showed that complex cuts could be decomposed into the general
basic kirigami operations they presented [123]. The researchers
envision that the generalized lattice kirigami framework provides
more opportunities to create arbitrary shapes from initially flat
sheets than the original method, due to the increased freedom
in distributing local sources of Gaussian curvature along the
sheet. However, a drawback is that the generalizations are not
yet suitable with inverse design algorithms, which inhibits the
use of such kirigami techniques in practical applications [123].
Lattice kirigami has not received the same attention as tradi-
tional origami by the scientific community, the great steps
undertaken by the abovementioned researchers notwithstand-
ing. However, it is clear that lattice kirigami offers an exciting
and promising paradigm toward 3D structures. By strategically
removing material or creating incisions, this technique could
alleviate some of the traditional origami issues such as interlock-
ing folds or the cumbersome tucking of excess material (which is
non-existent in kirigami), thereby offering higher design free-
dom and simplicity [62,122,124].
Kirigami-engineered elasticity
The second kirigami technique we consider here involves cutting
the paper at many locations without folding it afterward. The basic
idea is that specific cut patterns imbue flat sheets with a high “ap-
parent”elasticity, or stretchability, which does not arise from
stretching the actual material but rather from the geometric
changes enabled by the cuts, which is why we term this technique
kirigami-engineered elasticity [39]. Owing to the high stretchability
and the scale-independent nature, this technique has recently
been proposed as an interesting avenue toward stretchable elec-
tronic devices [38,39], small-scale force sensors [113],macro-scale
sun-shading [125], and solar-tracking photovoltaics [114].How-
ever, the kirigami-engineered elasticity technique is also useful
for wrapping flat sheets on intrinsically curved surfaces since the
cuts allow the sheet to locally stretch in-plane, thereby permitting
the required metric distortions to conform to the curved surfaces.
We distinguish two approaches toward kirigami-engineering
elasticity in the available literature, one involving out-of-plane
buckling of cut struts and one involving in-plane rotation of
polygonal units, as shown in Fig. 12 for standard paper models.
The former approach involving the out-of-plane buckling was
first used by Shyu et al. to create highly stretchable nanocompos-
ite sheets with predictable deformation mechanics [39]. The
researchers enriched the nanocomposite sheets with a cut pat-
tern consisting of straight slits in a rectangular arrangement such
as the one shown in Fig. 12a. Upon tensile loading perpendicular
to the slits, the struts formed by the cutting operation buckle out-
of-plane, allowing the overall sheet to reach an ultimate strain of
almost two orders of magnitude higher than that of the pristine
material (from 4%to 370%)[39]. Around the same time, Blees
et al. demonstrated that the same technique is applicable to gra-
phene since this one-atom-thick material behaves similar to
paper in terms of the Föppl–von Kármán number, a measure
for the ratio of in-plane stiffness to out-of-plane bending stiffness
[113]. Although Fig. 12a shows the struts buckling all in the same
direction, this is not necessarily the case and struts might ran-
domly buckle upward or downward, resulting in unpredictable
and non-uniform stretching behavior. In order to control and
program the tilting of the struts in the desired direction, Tang
et al. recently introduced “kiri-kirigami”in which additional
notches are etched into the material between the cuts [125].
Those notches are geometrical imperfections in the context of
buckling and serve as cues to guide the tilting in the desired
direction. By implementing the appropriate notch pattern, the
tilting orientation of the struts could be programmed before-
hand, and could be varied throughout the same kirigami sheet
[125]. All of the abovementioned works used the kirigami tech-
nique solely for imparting greater elasticity to flat sheets. How-
ever, the out-of-plane buckling that this kirigami technique
entails could also be used to efficiently create textured metama-
terials from flat sheets, as was recently demonstrated by Rafsan-
jani and Bertoldi [126]. These researchers perforated thin sheets
with a square tiling of orthogonal cuts with a varying orientation
with respect to the loading direction. Uniaxial tensile loading
results in out-of-plane buckling of the square units, and this
could be made permanent by increasing the load beyond the
plastic limit of the hinges between the squares. The results are
textured “metasheets”which are flat-foldable and could show
similar deformation characteristics as Miura-ori sheets, such as
negative Gaussian curvature upon non-planar bending [126].
The second method to achieve kirigami-engineered elasticity
involves in-plane rotation rather than out-of-plane buckling. In
this approach, the imposed cut pattern divides the sheet into
(typically square or triangular) units connected through small
“hinges”(Fig. 12b). Upon stretching the sheet, the units rotate
(almost) freely around the hinges, resulting in a deformation that
is driven mostly by the rigid unit rotation instead of stretching of
the units themselves. Cho et al. used this principle and devel-
oped a “fractal”kirigami technique in which the sheet is hierar-
chically subdivided into ever smaller units that could rotate and
contribute to the overall extension of the sheet (Fig. 12b)
[40,127]. While an increased level of hierarchy (i.e. more subdivi-
sions) could increase the expandability, there is a limit which is
dictated by the allowable rotations of the units. Cho et al.
showed that the maximum expandability may be increased by
alternating the cut motif between levels, allowing larger rotation
angles for the individual units [40]. The researchers demon-
strated the fractal kirigami technique at different length scales
and achieved an areal expandability of up to 800%. Furthermore,
they showed that the kirigami sheets could conform to an object
of non-zero Gaussian curvature (a sphere in this case) through
non-uniform stretching of the pattern [40], as also shown in
Fig. 12b. Further research into fractal kirigami was aimed at
understanding the complex mechanics of the hinges, prone to
stress concentrations, as well as the influence of material proper-
ties [128]. Based on experiments and numerical simulations,
Tang et al. proposed dog bone-shaped cuts and hinge widths that
vary with the hierarchy level in order to increase the strength
and ultimate expandability of the fractal cut patterns (even when
applied to brittle materials) [128]. Despite the impressive
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expandability that could be achieved with these standard cut
patterns, a drawback impeding the adoption in real applications
was the lack of compressibility. Therefore, Tang and Yin recently
proposed to extend the standard cut pattern, consisting of only
slits with actual cut-outs [129]. By introducing circular pores in
the original square units, sheet compressibility could be obtained
through buckling of the pore walls, while stretchability was still
guaranteed by the straight cuts [129].
Comparing the two kirigami-engineered elasticity approaches
discussed above, it seems that the fractal cut method is currently
more suitable to conform to intrinsically curved surfaces, as it
allows biaxial stretching and compression. A drawback of both
approaches is that a full coverage of the target surface cannot
be achieved, as the stretching is enabled through significant
“opening up”of the material. Nonetheless, both methods are
expected to receive considerable attention in future research,
not only in the field of stretchable electronics but also as a path-
way toward mechanical metamaterials. For example, the fractal
kirigami patterns are very similar to earlier work on rotation-
based auxetic mechanical metamaterials [130,131].
Discussion and conclusions
We reviewed current origami and kirigami techniques that could
be used to approximate or conform to intrinsically curved sur-
faces. Starting off with some concepts from differential geome-
try, we highlighted the inherent difficulty of transforming flat
sheets into intrinsically curved surfaces. Moreover, we explained
the geometry of origami, which involves isometric deformations
of developable surfaces and therefore retains the intrinsic flatness
of the starting material. While scientific research into origami
and kirigami is still in its infancy, we could nonetheless identify
several promising techniques for the transformation of flat sheets
into curved geometries.
Approximations of intrinsically curved surfaces
The origami and kirigami techniques that we have reviewed
could essentially approximate intrinsically curved surfaces in
two different ways. One approach is to use origami and kirigami
to transform ordinary flat sheets into “metasheets”with signifi-
cantly altered properties, which are then deformed into the desired
geometry. In a second approach, the prescribed fold and cut pat-
terns directly correspond to the final 3D shape and no additional
deformation is required after folding. The kirigami-engineered elas-
ticity techniques [39,40,113,125,128,129] are examples of the first
approach, while the techniques with tucking molecules [59,81,88]
and curved-crease couplets [97,98,101] are examples of the second.
Some techniques could be classified in both categories. For exam-
ple, origami tessellations could be employed to texture sheets so
RESEARCH: Review
FIGURE 12
Kirigami-engineered elasticity. (a) A parallel arrangement of slits (left) causes the small struts in between to buckle out of the plane upon tensile loading
(middle), thereby providing the sheet with higher elasticity (Scale bar is 2 cm). Right: a closer view showing the struts buckling in the same direction (Scale bar
is 1 cm). (b) Fractal cut kirigami. Left: a three-level cut pattern with motif alternation, according to Ref. [40]. The small amount of material between adjacent
cuts serves as hinge between the rigid square units. Middle: the cut pattern applied to a standard piece of paper (Scale bar is 2 cm). Right: the cut sheet of
paper shows a high degree of expandability and could conform to a spherical geometry through rotation of the square units.
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that they may be deformed into an intrinsically curved geometry
[28,68,72,106], but (generalized) tessellations have also been cal-
culated to fit a target surface once folded [73–77]. The lattice kir-
igami technique may be considered an example of the second
approach, as the target shape is programmed into the flat sheet
using appropriate cuts and folds. However, a pluripotent version
of lattice kirigami has been also proposed [122], in which the
same kirigami pattern could be folded to fit multiple curved
surfaces.
Due to the developability constraint, no origami or kirigami
technique can exactly fitaflat sheet onto an intrinsically curved
smooth surface. Such surfaces could be approximated in a “glo-
bal”sense, but locally the folded sheets remain intrinsically flat.
Some techniques, such as lattice kirigami [62,122,123] or the
origamizing technique [59,88] do imbue the sheets with Gaus-
sian curvature, but this curvature remains concentrated in single
points surrounded by developable patches, i.e. “non-Euclidean
vertices”[10]. Even when a sheet of paper is crumpled, the
majority of the paper remains developable and non-zero Gaus-
sian curvature only arises at specific points due to local stretching
of the paper [55,132]. However, owing to the different underly-
ing mechanisms, some techniques will result in a “smoother”
approximation of the target surface than others. This is an
important factor to consider in applications where the surface
topography plays a role, e.g. in fluid flow over an object. Surfaces
approximated using origami tessellations or periodic pleating
exhibit a textured surface topography. For the periodic pleating
technique, this texture is in the form of sharp, parallel ridges.
In the case of origami tessellations, the texture is determined
by the specific unit cell geometry, with the square waterbomb
or Ron Resch patterns resulting in a smoother surface than the
Miura-ori pattern, for example. The axisymmetric geometries cre-
ated by the curved-crease couplets could result in a relatively
smooth surface due to the bent faces, although the frustration
between the curved and straight creases might entail additional
creases that disturb this smoothness. The smoothest approxima-
tion of the target surface is likely obtained with the origamizing
technique, as the calculated crease pattern (almost) exactly folds
into a polygon mesh of this surface. Naturally, a finer mesh
results in a smoother representation, yet also entails a more com-
plex folding process. The lattice kirigami technique results in a
stepped surface approximation, consisting of many small units
that simulate the convex and concave curvatures of the sheet.
Interestingly, both the origamizing and lattice kirigami tech-
niques bear strong similarities with computer graphics tech-
niques used to represent 3D objects, either using
polygonization of the surface or by means of a “voxel”(volume
pixel) representation. Finally, the fractal kirigami technique
allows conforming initially flat sheets, made from relatively rigid
materials, to surfaces of non-zero Gaussian curvature through
non-uniform opening of the cut pattern [40]. However, it was
also mentioned that this opening of the perforations inhibits a
full coverage of the target surface, which might be a drawback
in certain applications.
Practical considerations
There are several practical challenges that need to be overcome in
order to accelerate the adoption of origami and kirigami as a
shape-shifting technique for development of advanced materials.
First and foremost is the challenge of the folding itself, which is a
labor-intensive manual process in traditional origami. Although
all the macroscale paper models that we have presented in this
review could be folded by hand, such manual folding becomes
increasingly complex at much smaller or much larger scales
and for more advanced materials [14,133]. As a consequence,
self-folding (i.e. “hands-free”[11]) techniques are required. A
wide range of such techniques have been developed during
recent years, in particular aimed at smaller length scales [9–11,
13–16,105,134–142]. While these techniques, for example, differ
in terms of materials used, speed, actuation method, and suitable
length scales, the underlying principle is typically the same: the
self-folding behavior is programmed into the flat starting materi-
als, most often in the form of “active hinges”which are triggered
by an external stimulus to activate folding. In many cases, stim-
ulus-responsive polymeric materials have been employed for this
purpose. Examples are hydrogels that swell or de-swell upon a
change in aqueous environment [11,141,142], or shape memory
polymers (SMP) that shrink when heated above the glass transi-
tion temperature [9,105,137–140]. Especially the use of ther-
mallyresponsive SMP for self-folding has attracted the interest
of many researchers due to its simplicity and versatility in terms
of actuation method [138], for example via uniform oven heat-
ing [122,137], localized joule heating [9], or localized heating
by light or microwave absorption [138,139]. The shape memory
effect is not limited to polymers but is also present in certain
metallic alloys (giving rise to shape memory alloys (SMA)), mak-
ing these materials also suitable for thermally activated self-
folding origami [134]. In addition to these more common tech-
niques, many other actuation methods for self-folding have been
developed such as the folding of rigid panels driven by surface
tension [143–145] or cell traction forces [136] to create nano-
and microscale origami, and mechanically driven origami/kiri-
gami [146,147] approaches in which folding is achieved through
controlled buckling at specified locations. The reader interested
in more detailed information on self-folding techniques and
the associated materials is referred to other excellent reviews on
these topics [11,14,16,133,148,149].
Despite many recent developments, self-folding remains a
challenging task, in part, because of the need for sequential fold-
ing and control over the direction of folding and fold angles
[9,11,150]. While many of these challenges have been addressed
in recent years [9,15,138,150–153], self-folding origami demon-
strations have often been restricted to single folds or basic poly-
hedral shapes [16,138–140,144], while demonstrations of more
complicated patterns such as origami tessellations are not yet
so common [137,152]. Given these inherent complexities, it is
not surprising that some of the reviewed origami and kirigami
techniques are better suited for self-folding than others. Com-
pared to straight creases, self-folding of curved-crease origami is
expected to be more challenging despite some recent demonstra-
tions [153]. The facet bending, which is inherent in this type of
non-rigid origami, would necessitate larger hinge actuation
forces than for purely rigid origami with straight creases. More-
over, the possibility of arriving at a “locked”state during folding
may further complicate the automated folding of curved-crease
origami [93]. Comparing the tucking molecules origami
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technique and the lattice kirigami technique, it has been argued
that the latter is more applicable to self-folding [62,122,123]. The
tucking molecules approach requires excessive material to be
tucked away behind the outer surface, which is a cumbersome
process involving many small (crimp) folds and high folding
angles. This is in sharp contrast with the lattice kirigami tech-
nique, where no excessive material needs to be tucked away
and a simple, repetitive folding pattern is used [122]. Indeed,
self-folding of basic lattice kirigami units (millimeter and cen-
timeter scale) has been recently demonstrated using localized
heating [153] or controlled compressive buckling [146,154], but
more complex 3D geometries have not yet been reported.
Regarding the techniques that employ the “metasheet”
approach, such as the origami tessellations and the kirigami-
engineered elasticity, one could argue that these are less suited
for self-folding as the sheets need to be actively deformed into
the desired shape. In order for these sheets to self-fold into the
target geometry, local control over the fold angles or gap opening
would be required. Preliminary results show that the standard
kirigami cut pattern, consisting of parallel straight cuts, could
be actuated using thermally activated local shrinkage [125], but
more investigations are needed for metasheets to automatically
fit curved surfaces through remotely actuated opening or closing
of folds and cuts.
In addition to the self-folding process, another practical con-
sideration is related to locking the origami and kirigami struc-
tures in their curved folded geometries. This has, for example,
been achieved by annealing titanium-rich origami structures at
high temperatures [155]. Another approach is using sequential
self-folding to include self-locking mechanisms [149,156]. Alter-
natively, the locking mechanisms might be inherent to the used
origami or kirigami technique. For example, the origamizing
technique with tucking molecules uses crimp folds to keep the
tucks closed and maintain the desired shape on the outside sur-
face [59]. On the contrary, the standard lattice kirigami cannot
benefit from such excess material to lock the folds, although
recent generalizations of lattice kirigami enable this to a certain
extent by retaining some material for use as “fastening tabs”
[123].
In addition to aspects such as self-folding and locking,
another prominent challenge is related to the medium to which
origami and kirigami are applied. Origami and kirigami are often
considered to be scale-independent processes on zero-thickness
surfaces. The thin paper sheets that have traditionally been used
in these art forms are not too far from zero-thickness surfaces
[157]. However, in engineering and scientific applications, the
thickness of the flat starting materials cannot be ignored, espe-
cially not for applications where the ratio of the sheet thickness
to other sheet dimensions is substantial. An important conse-
quence of finite sheet thickness is that appropriate hinge design
is required to enable obstruction-free folding and flat-foldability
[158]. In recent years, several hinge design approaches to
account for finite material thickness have been proposed for
rigid-foldable origami [158–160]. In addition to hindering flat-
foldability, the material thickness also affects the fold regions
themselves, i.e. when the materials are actually folded and the
folds are not replaced by hinges. Origami design methods typi-
cally assume perfectly sharp folds applied to the (zero-
thickness) sheets, meaning that all folding is concentrated along
a single line of infinitesimal width (such a sharp fold is consid-
ered to be “G
0
continuous”)[133,161]. While such idealized
sharp folds could be approximated to some extent in a thin paper
sheet, a fold in a finite thickness sheet will never be perfectly
sharp but will rather be defined by some bent region with a cer-
tain radius of curvature, especially when thicker materials or
materials that cannot withstand high bending strains are used
[133,161]. Peraza et al., inspired by Tachi’s origamizing
approach, recently developed an origami design method based
on “smooth”folds [161], which are bent surface regions of finite
width. Such “smooth”folds form the connection between the
rigid origami faces and are characterized by higher order geomet-
ric continuity than the sharp folds. Smooth folds are not only
relevant for folding of thicker materials but also relevant for
self-folding techniques based on “active hinges”[161]. Swelling
or shrinking at these hinge locations also results in finite regions
of bending rather than perfectly sharp folds [162]. This distinc-
tion between folds in idealized origami (zero-thickness sheet)
and folds in origami on real materials (non-zero-thickness
sheet) essentially revolves around the subtle difference between
bending and folding [11,157]. Lauff et al. described bending
(i.e. smooth fold) as “distributed curvature”while folding is
“localized curvature”. However, Liu et al. correctly stated that
an overlap between bending and folding exists as it is difficult
to draw a clear boundary between localized and distributed cur-
vature [11]. Note that in this context, the term “curvature”
refers to single or extrinsic curvature. Bending and folding
both result in zero Gaussian (intrinsic) curvature, see Fig. 3b
[157].
In light of the techniques reviewed in this paper, origami with
finite material thickness is expected to be particularly challeng-
ing in the tucking molecules approach, due to many small folds
(which are considered to be sharp folds in the origamizer soft-
ware) and the requirement for flatfolding to tuck away excess
material. As mentioned before, improvements to the standard
tucking molecules approach have been proposed to make the
technique more apt to real applications with finite thickness
materials [81,89,161]. The approach based on the curved-crease
couplets is limited by the fact that very high folding angles (close
to p) are required, which could be difficult to realize with thicker
materials. The techniques based on origami tessellations, con-
centric pleating, and lattice kirigami are expected to be better
suited for origami with finite thickness sheets as they are
generally characterized by simple fold patterns and have
already been (self-) folded from materials other than paper
[122,152,153,158,162]. Finally, the kirigami-engineered elastic-
ity techniques do not require folding, hence they do not suffer
from challenges with tight folds or flat-foldability. Nevertheless,
the sheet thickness does influence the load at which out-of-plane
buckling of the struts occurs (see Fig. 12a), thereby making it also
an important design parameter [39,125].
Besides the thickness, another aspect that is often neglected in
“idealized”origami is the mechanical properties of the material,
which may also hinder practical application of the origami and
kirigami techniques [162]. For example, permanently folding a
finite thickness sheet entails a complex stress state involving
plasticity and some degrees of stretching, aspects which are
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strongly linked to the mechanical properties of the sheet
[55,104]. Furthermore, non-rigid origami also involves facet
bending [90], which is strongly tied to the bending stiffness of
the sheets that are used. As for the kirigami-engineered elasticity
techniques, it has been already mentioned that these techniques
are characterized by high stress concentrations, both for the out-
of-plane buckling and in-plane rotation approaches
[39,40,125,128]. Consequently, implementation of these tech-
niques to real materials will require certain levels of understand-
ing regarding the local material behavior under these high stress
states [128]. Some techniques might therefore be more suitable
than others for a given application depending on the chosen
material.
As is clear from the preceding discussion, scientific and engi-
neering origami/kirigami are not purely “scale-independent”
processes that could be treated solely from a geometrical perspec-
tive. For example, self-folding techniques that are suitable for
micro-scale origami are not necessarily suitable for
architectural-scale origami (e.g. surface tension or cell traction
forces). As a final remark, we note that traditional paper seems
to remain an excellent medium for origami and kirigami, consid-
ering its balance of relative thickness, bending and tearing resis-
tance, and the ability to withstand relatively sharp creases.
However, this does not necessarily mean that other materials
(on different scales) are less well suited as these materials might
behave very similar to paper when used for origami or kirigami
[113].
Outlook
Approximating intrinsically curved surfaces using origami and
kirigami is relevant for many applications that can benefit from
the specific advantages offered by these techniques: the ability
to obtain complex geometries from (nearly) non-stretchable flat
sheets and the ability to apply this on virtually any length scale.
As such, the folding-and-cutting paradigm could enable the
development of flexible electronics [38,40], shape-morphing
materials [122], nano- and microscale devices [14,163], architec-
tural structures [84,159], or any other complex geometry involv-
ing intrinsic curvature. The applications of origami and kirigami
are not limited to static designs, but could also be of a more
dynamic nature. In fact, certain fold patterns involving facet
bending or curved creases may form energetic barriers between
different folding states, giving rise to bi-stability and fast snap-
ping motions that could be leveraged for switchable or tunable
devices [103,164].
Interesting opportunities for origami and kirigami may be also
found in the rapidly expanding field of biomedical engineering
[36], with such examples as patterned micro-containers for con-
trolled drug delivery [35], origami stent grafts [34], or self-folding
tetherless micro-grippers [165]. A particularly interesting bio-
application is the folding of 3D tissue scaffolds from flat sheets
enriched with cell-regulating surface topographies. Self-folding
of patterned scaffolds with simple geometries has already been
demonstrated [166], but more complex curved geometries would
be needed in order to better stimulate and guide tissue regenera-
tion [167]. For example, it has been hypothesized that promising
bone-mimicking scaffolds could be based on triply periodic min-
imal surfaces (TPMS) [168,169]. These are area-minimizing 3D
surfaces with zero mean curvature (H) everywhere, correspond-
ing to negative (or zero) Gaussian curvature everywhere
(H¼
1
2
ðj
1
þj
2
Þ¼0 and thus j
1
¼j
2
)[42]. These intrinsically
curved minimal surfaces are ubiquitous in biological systems
[41,170–172] and are nature’s best attempt at dealing with the
frustration of embedding constant negatively curved surfaces in
Euclidean 3-space (Hilbert’s theorem) [173,174]. Current TPMS
scaffolds are created with additive manufacturing [4,169], mean-
ing that the surface topographies needed to enhance tissue
regeneration cannot be included. However, the origami and kir-
igami techniques we have reviewed here might enable trans-
forming patterned flat sheets into intrinsically curved scaffolds
through appropriate cutting and folding.
In conclusion, we have reviewed recent work on origami and
kirigami to identify the techniques that enable shape shifting of
flat sheets into complex geometries. By introducing aspects from
differential geometry, in particular the Gaussian curvature, we
have illustrated the fundamental difference between flat sheets
and intrinsically curved surfaces, which can explain gift-
wrapping of spheres to wavy edges in plant leaves. While in-
plane distortions could imbue the flat sheets with intrinsic curva-
ture, we have shown that origami and kirigami offer alternative
approaches to approximate curved surfaces with (almost) no
stretching of the underlying material. Despite originating from
centuries-old art forms, the techniques we have reviewed here
are promising for many applications across a broad range of
length scales. It could therefore be expected that the relatively
recent interest in “scientific”origami and kirigami will only keep
on growing in the near future.
Acknowledgments
This research has received funding from the European
Research Council under the ERC grant agreement n°[677575].
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