CUB journal 199.ai

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2/4/19

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JOURNAL OF
THE INTERNATIONAL ASSOCIATION
Vol. 60 (2019) No. 1

FOR SHELL AND SPATIALSTRUCTURES

Prof. D. h-C Eng. E. TORROJA, founder

TRANSFORMABLES: DEPLOYABLE STRUCTURES AND
RAPIDLY ASSEMBLED SYSTEMS

N. DE TEMMERMAN
March n. 199

international association
for shell and spatial structures

PORTADA Y CONTRA.indd 1

Vol. 60 (2019) No. 1
March n. 199
ISSN: 1028-365X

28/12/04 07:06:09

Journal
JournalVOL. 60 (2019) No. 1

n. 199 Marchcontents

Announcements
IASS Symposium Announcement 2019
IASS Symposium Announcement 2020
Upcoming Events

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4
5

Special Issue on Transformables: Deployable Structures and
Rapidly Assembled Systems
N. De Temmerman

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Introduction to Structural Origami
T. Tachi7

Computational Design of Bistable Deployable Scissor Structures:
Trends and Challenges
L. I. W. Arnouts, T. J. Massart, N. De Temmerman and P. Z. Berke

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Development of a Deployable Geodesic Full Sphere
K. Kawaguchi, T. Sato, X. Yang and N. Seo

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A Critical Review on Classification and Terminology of Scissor Structures
F. Maden, Y. Akgün, G. Kiper, Ş. Gür, M. Yar and K. Korkmaz

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Rapidly Assembled Reciprocal Systems with Bending-Active Components:
The ReciPlyDome Project
S. Brancart, O. Popovic Larsen, L. De Laet and N. De Temmerman

65

Reconfigurable Modular Bar Structure
M. C. Phocas, M. Matheou, A. Müller and E. G. Christoforou

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Computational Design and Experimental Behaviour of
Deployable Mass Timber Arches
K. B. Plasencia Alava, L. K. McCann, G. Hodge, K. Baber and J. M. Gattas

90

PrefaceTechnical Papers

COVER: Figures from paper by L. I. W. Arnouts, T. J. Massart, N. De Temmerman
and P. Z. Berke
IASS Secretariat: CEDEX-Laboratorio Central de Estructuras y Materiales
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JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASSINTRODUCTION TO STRUCTURAL ORIGAMI
Tomohiro Tachi
Associate Professor, University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo, Japan, tachi@idea.c.u-tokyo.ac.jp
Editor’s Note: Manuscript submitted 25 February 2019; accepted 20 March. This paper is open for written discussion, whichshould be submitted to the IASS Secretariat no later than September 2019.
DOI: https://doi.org/10.20898/j.iass.2019.199.004

ABSTRACTOrigami, the art of folding sheets of paper, has been appreciated as a source of inspiration for structural design.
Structural origami is a field that is rapidly developing through a collaboration between engineering, design, art,and computation. This paper introduces aspects of structural origami, to share recent exciting developments inthis field.
Keywords: Origami, Structural Origami, Structural Morphology, Folded-plate, Origami Meta-materials
1.

INTRODUCTION

Origami, an art form that consists of folding a sheet(or sometimes multiple sheets) of paper into different
forms without stretching and often without cutting
the sheet, is attracting the interests of not only artists,
but also researchers in diverse fields. Folded shapesof origami have been a great source of inspiration forstructural designs. The study of structural origami,
therefore, aims to explore the relationship between
structural behavior and the forms that emerge fromsheet folding. (Note that the term “origami”
comprises of the Japanese word “oru” meaning “tofold” and “kami” meaning “paper,” but in structuralcontext, “paper” does not necessarily mean material
made with cellulose, but rather means a thin surface
with significantly high in-plane stiffness comparedto out-of-plane stiffness.)
1.1.

Background

About a half-century of studies have supported thedevelopment of a vibrant academic community, inwhich artists, mathematicians, scientists, andengineers collaborate to tackle exciting problems
that originate from the folding of sheets. One of theearliest meetings on this topic was the International
Association for Shell Structures (IASS) Symposium
on Folded Plates and Prismatic Structures held in
Vienna,1970,
whichcollected
researchpresentations on static, modular, or transformablefolded plate systems and shells. Although they were
not particularly called “origami,” the collected works
explored the concepts we now call 'structuralorigami.' Two of the highlights from the proceedings
are the first international appearance of a new figure
composed by the repetition of a “feather pattern” [1]by Koryo Miura, which we now call Miura-ori(Figure 1 left) and a kinematic folded plate system
by Resch and Christiansen [2], which we now often
call the Resch Pattern (Figure 1 right). These
patterns are still repeatedly used in scientific origami.
The early pioneers explored both the high stiffness
and flexibility of sheet folding through repetitive
patterns. Such patterns are called origami
tessellations.

Figure 1: Miura-ori (Left) and a Resch Pattern (Right)

A wider interdisciplinary community of scientificorigami had started to use the terminology “origami,”
when the First International Meeting of Origami
Science and Technology was held in Ferrara, Italy in
1989. The latest of these quadrennial meetings, the
7th International Meeting on Origami Science
Mathematics and Education (7OSME), was held in
Oxford, United Kingdom in 2018. The concept oforigami is rapidly expanding to include a diverse
range of scientific areas, such as material sciences,
biology, robotics, and micro electro mechanicalsystems.

Copyright © 2019 by Tomohiro Tachi.
Published by the International Association for Shell and Spatial Structures (IASS) with permission.

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Vol. 60 (2019) No. 1 March n. 199

Since the 1990s, computational origami has
accelerated interdisciplinary collaboration by
providing geometric models, universal algorithms,
and accessible software tools. In the IASS, theStructural Morphology Group (SMG or WG 15) has
been the platform that supports advances in origami
in a structural context. Since 2011, SMG has been
holding a study group, named Origami, where
researchers exploit the geometry of folding throughcomputation and fabrication to find newopportunities for structural engineering. Theinterdisciplinarity is a key characteristic of this field.
New viewpoints and interpretations given fromdifferent fields make the topics vibrant.
1.2.

Fun with Structural Origami

Because of the fluid nature of this field of study, we
try to keep the definition of origami flexible.
Structural origami deals with any interesting
structural study related to surface folding. This paper
attempts to share what makes structural origami
exciting, rather than provide a systematic review.
Therefore, this introduction to structural origami isunstructured.
Section 2 describes the geometric meaning offolding: an out-of-plane deformation accompanying
sharp creases. The Origamizer algorithm represents
the universality that can be achieved by folding. We
mention how natural systems use a combination offolding and growth of surfaces. The nonlinear elastic
behaviors of origami go beyond stiffness or
flexibility and achieve tunable systems and origamibased mechanical metamaterials. Section 3 briefly
shows some approaches for modeling and computing
such elastic behavior, or origami. An extreme but
scalable model of origami is rigid origami dealt in
Section 4; the kinematic study or rigid origami gives
design tools for novel transformable structures
including overconstrained mechanisms and tubularand cellular structures. In Section 5, we explore theconcept of self-folding, a new way of making things
that emerge from flat surfaces. The intrinsic
difficulty originating from self-folding and its
potential use for reprogrammable systems are
explained. Section 6 briefly mentions curved folding,which remains unexplored in depth. Section 7
concludes the discussion by introducing ourcollaborative activities.

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2.

GEOMETRY OF FOLDING

2.1.

Why Do We Fold?

Paper is a material with a significantly high in-plane
stiffness, proportional to the thickness 𝑡𝑡, relative toout-of-plane stiffness, proportional to 𝑡𝑡 3 , where 𝑡𝑡 isvery small. Theoretically, we consider that thematerial thickness approaches zero in the limit, sothat the surface is completely non-stretching. Such a
deformation can be described by an intrinsicallyisometric mapping of a 2D surface into a 3D
Euclidean space. The most important natural
behavior of paper is folding, defined as out-of-plane
deformation that accompanies the formation ofcreases, i.e., sets of non 𝐶𝐶 1 points along curves.

Paper without creases could be rather boring. To
illustrate this, consider a smooth 𝐶𝐶 2 intrinsicallyisometric mapping, which is well studied in the fieldof classical differential geometry. Theorema
Egregium says that the Gaussian curvature, i.e., theproduct of principal curvatures, is intrinsic, i.e.,unchanged by intrinsically isometric mapping.

The theorem particularly means that the forms thatemerge out of a planar sheet, called a developablesurface, must have zero Gaussian curvature,
meaning that the surface must be singly curved, i.e.,
locally a cylinder, a cone, or a tangent surface. In
other words, doubly curved surfaces like a dome ora saddle require stretching or shrinking of a planar
sheet (Figure 2 top). On the other hand, folded planar
sheets of paper could form synclastic or anticlastic
surfaces as shown in Figure 2, at the bottom. Here,the trick is that the non-smooth creases allow for thesurface to wrinkle and thereby, to virtually shrink.

Figure 2: Top: smooth developable surfaces can form
only singly curved surfaces. Bottom: creased surfaces
allow for doubly-curved developables (pleated hypar,
Yoshimura-pattern, Resch’s pattern)JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

We may experience such a property of origami, i.e.,
virtual shrink by creasing, through an easy
experiment; take a roll of wax-paper, grab two ends,
try to make them shrink. The sheet exhibits selforganized buckling patterns to accommodate thecompression without the material itself shrinking(Figure 3). The buckling patterns of paper arecharacterized by sharp features, i.e., creases, and in
the case of a cylinder, a diamond-pattern arises [1].
The shell buckling behavior stands in contrast to thebuckling of the one-dimensional elastic rod, forming
a smooth elastica curve. Bushnell and Bushnell [3]collect examples and literature of shell buckling
coupled with beautiful patterns of creases.

Figure 4: Crease pattern (left) computed by Origamizer
and the folded model (right)

The main idea for the Origamizer is to separately
place faces of the polyhedral surfaces leaving some
gaps between them. The gaps are folded such thatthey reconvene, allowing multiple vertices on thepaper to fold to a common point, and to allow a pairof edges to fold to a common edge (Figure 5). Sucha crease pattern is generated by folding along theVoronoi diagram generated by the placed faces
under a certain metric.

Figure 3: Buckling patterns of thin sheets arecharacterized by sharp creases

2.2.

Universality of Folding

We have seen that a certain pattern of creases yields
a double-curved surface. The inverse of this is calledorigami design, where we ask to obtain the pattern ofcreases such that the paper transforms into aprescribed form. Computational methods andalgorithms are often useful to solve the design
problem systematically. The study of origami design
was first deeply explored in the 1990s by origami
artists and scientists, which brought about thedevelopment of so-called super-complex origami.
Now, origami design can be extended into 3D
surfaces. Here, the ultimate question is: “Is anything
foldable from a 2D sheet of paper?” This problem isnumerically solved positively by Origamizer [4],software created to generate a crease pattern thatfolds into an almost arbitrarily given polyhedralsurfaces. A constructive proof by Demaine andTachi [5] showed the universality of origami, i.e.,that arbitrarily given polyhedral surfaces can befolded water-tightly.

Figure 5: Folding up to close the gap using a crease
pattern based on the Voronoi diagram

We may interpret this design procedure as theplacement of artificial wrinkles on the surface.
Origami tessellations are a good source of such
wrinkles. A freeform surface can be generated out ofsheet material by populating an origami tessellation
pattern on the surface and solving the developability
constraints through optimization [6] (Figure 6).

Figure 6: Freeform origami tessellations by generating
Resch-type tessellations on a given surface. Computed
using Freeform Origami

9Vol. 60 (2019) No. 1 March n. 199

2.3.

3D to 3D Folding

The wrinkling idea allows for the approximation ofany mapping of a surface to another surface in 3D, if
the mapping is contractive, meaning that all
distances either shrink, or are kept isometric. This istheoretically underpinned by the Nash-Kuiper
theorem [7]. Therefore, we may extend origami
beyond developable surfaces. For example, we may
start from a curved surface and approximate a flat
surface by folding; this is the opposite of making 3D
curvature from a flat surface. In this way, we succeed
in separating the intrinsic metric of the surface (how
it is measured along the surface) and the extrinsic
shape (how they look from the outside). Figure 7
illustrates this separation.

Figure 7: 3D to 3D folding allows for isolation of the
intrinsic and extrinsic curvatures. Vertical direction
corresponds to folding, while horizontal direction requires
growing or shrinking of the material. This contrasts with asmooth (𝐶𝐶 2 ) surface along the diagonals, where intrinsic
metric and extrinsic curvature correspond

2.4.

Growing vs. Folding

The wrinkling technique is also seen in biologicalsystems when biological shells and membranes grow.
The surfaces are made by (1) intrinsic deformation
where actual material grows through cell divisionand (2) isometric deformation, i.e., folding, which
does not include any intrinsic metric change. The
former takes a longer time than the latter process.
Clever usage of the coupling of growth with folding
is seen in the way a beetle grows their new, bigger
shell inside of the older shell. They slowly grow theirshells without extrinsic metric change (horizontal
motion in Figure 7) with “furrows,” and then deploy
them isometrically [8].

103.

STIFFNESS AND FLEXIBILITY

3.1.

Elastic Behaviors

The extreme difference in the in-plane and out-ofplane stiffnesses of paper leads to varieties of stiffand flexible structures when folded. Stiff origami
patterns form folded-plate systems [1] or lightweight
sandwich cores [9][10]. Flexible origami patterns
may lead to deployable space structures [11], stents[12], or energy absorption devices [13]. Moreover,because of the geometric nonlinearity, some origami
structures exhibit multistability [14][15]. Because
the behavior depends on the shapes, one may tune
structural stability with mono-stable, zero-stiffness,or bi-stable properties [16].
Also, origami tessellations viewed as homogenizedmaterials have unusual behavior; so they are often
called mechanical metamaterials. In particular, theauxetic property, i.e., property with a negative
Poisson ratio, has been explored. Shenk and Guest[10] showed that certain origami tessellations have
Poisson ratios of opposite signs for in-plane and outof-plane deformations; Miura-ori is auxetic against
in-plane deformation, i.e., the surface shrinks in both
directions, but the Poisson ratio against out-of-plane
deformation is positive, i.e., the surface forms ananticlastic surface. An eggbox surface has the
opposite property, with positive in-plane andnegative out-of-plane Poisson ratios.
In most cases, the exciting elastic behaviors oforigami can be well captured via a very simple
elastic model of bar-and-hinge when the fold lines
are composed only of straight lines. Note that curved
creases behave quite differently and present a
number of open questions, as we discuss in Section
6.
3.2.

Bar and Hinge Model

An important principle of folding from differential
geometry is that a straight crease stays straight afterfolding. More precisely, folding along a straightcrease on a developable surface by any amountexcept 0 or π will keep the crease straight also in its
folded form [17]. In other words, the surfaces needto stretch in order to deform the crease line. So, astraight crease behaves similarly to a stiff bar.
Accordingly, a face surrounded by three straightcreases is kept stiff and planar. Resch andChristiansen [2] showed the first computationalmodel of kinematic origami, which uses a single

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

triangular finite element for each triangular face,coupled with angular springs representing the hinge
stiffness at the creases.
Faces surrounded by more than three creases needmore care because they bend in a complex manneraccompanying a small amount of stretch [18]. Thus,
the behavior is dependent of the thickness of thematerial. If the surface is extremely thin and thus thein-plane stiffness governs the behavior, a facet gets
triangulated with a finite number of sharper creases.
When the material gets thicker, the triangulation line
starts to avoid sharp creases and gets rounded bycompromising the stretch of the material.
A computational model for acquiring such coupledbehavior of panel bending has been explored throughthe bar-and-hinge model using a central node (e.g.,five nodes and eight bars (N5B8) for a quadrilateral)[19]. The software system MERLIN utilizes thesebar-and-hinge models to capture the geometricallynonlinear behaviors of elastic origami [20]. The barand-hinge model is well contrasted to the diagonalbar model (e.g., four nodes and six bars (N4B6) fora quadrilateral), which is a straightforward strategy
that fails. The diagonal-bar model fails because it
produces singular stiffness matrices when panels areplanar, but the panel becomes extremely stiff as itgets bent (Figure 8). Please, do not use diagonal bars.

Figure 8: The N5B8 with hinge stiffness can model
consistent bending stiffness. The N4B6 model withdiagonal bars can be extremely stiff in its bent state as it
forms a tetrahedron

3.3.

from the most flexible to the stiffest. After six trivial
rigid body zero-stiffness modes (translation androtation in 3D), the seventh mode corresponds to themost flexible deformation mode. For a kinematic
structure, one may expect the seventh mode to be thedesired transformable mode; however, surprisingly,
they show that even the Miura-ori sheet, known to bea one-degree-of-freedom (DOF) kinematic origami
structure, is more flexible against torsion thanfolding with some natural set of stiffness parameters.
The important fact is that the deformation modes andthe eigenvalues relative to each other do not dependon the scale or on material properties, such as themodulus or the density. This is because theproperties will only change the eigenvalues
proportionally. Therefore, the properties analyzed inthis way are intrinsic purely to the shape of thestructure.
A further extreme idea also coming from eigenvalue
analyses is that of mode separation. Whenconsidering an ideal transformable structure, we
desire high stiffness to produce a light-weight andload-bearing structure on the one hand, and highflexibility to enable transformation, on the otherhand. Origami structures used to fail in achieving
this contradicting requirement of being stiff yet
flexible at the same time. The glide-reflection
coupling of origami tubes achieves this best-of-bothworld nature [22]. More precisely, the structure
exhibits mode separation between the desiredfolding mode and the undesired warping mode; thedesirable deployment mode in the seventh mode is
400 times more flexible than the undesirable
deformation in the eighth mode, under plausible
assumptions concerning the material thickness andhinge stiffness. Again, this relative property only
depends on the shape of the structures and how they
are geometrically coupled.

How to Measure Stiffness?

In the study of structural origami, it often makes
more sense not to consider stiffness for a single
loading case, but to understand the “stiffness of thestructure” if such a measure exists. A very good
intuition comes from the eigenvalue analysis.
Eigenvalue analysis provides the natural vibration
modes of the structure; a high-frequency vibration
mode corresponds to high stiffness, and a lowfrequency mode corresponds to low stiffness.
Schenk and Guest [21] use eigen decomposition of
the stiffness matrix to sort the deformation modes

Figure 9: (a) Miura-ori sheets, (b) origami tube [23], (c)parallel alignment [23], and (d) glide-reflection coupling
[22]

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Vol. 60 (2019) No. 1 March n. 199

consisting of closed loops around interior vertices;
this constrains the configuration.

Figure 10: A glide-reflection coupling of tubes [22]actuated from one end. The stiffness of the 8th mode helps
to achieve the uniform, steady deployment

These computational models of origami help inunderstanding the elastic behavior of existing
structures, but they are a bit indirect for designingstructures. In contrast, much inspiration for new
origami structures comes from the kinematic theoryof rigid origami that we explain in the next section.
For example, even though the behavior of the glidereflection coupling of tubes is well-explained bymode separation, the actual design originated fromthe theory of rigid origami.
4.

RIGID ORIGAMI

4.1.

Rigid Origami Model

Rigid origami is a kinematic model of origami where
each face is completely rigid and connected byrotational hinges. In other words, the bars andbending hinges have infinite stiffness. We areparticularly interested in the existence of acontinuous folding motion of rigid origami, called
rigid foldability. Rigidly foldable mechanisms arescalable. Because the mechanisms do not rely on thematerial flexibility, they can be realized with panelsof arbitrary thicknesses and rotational hinges ofdifferent types. Numerous thickness accommodation
techniques have thus been explored to utilize rigid
origami in large-scale structures [24][25][26].
Theoretical results and design techniques for rigid
origami are based on a model parameterized by thefold angles. The fold angle of a crease is the signed
dihedral angle starting with zero at the unfolded state(usually, a positive value is used for valley crease).
The set of fold angles of all creases for a valid folded
state is called a configuration. Conversely, a set ofarbitrary angles does not necessarily represent avalid folded state. This is because most of theinteresting origami form a parallel link mechanism

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Figure 11: Left: serial linkage origami without a loop. We
can assign arbitrary fold angles to all creases. Right:
parallel linkage origami forming a loop around an interiorvertex. The fold angles are geometrically constrained

We may consider a configuration to be a point in an
𝐸𝐸 dimensional space, where the coordinates of thepoint are the fold angles, and 𝐸𝐸 is the number ofcreases. In theory, the set of possible configurations,called configuration space, can be drawn as a set ofthe lower (or equal) dimensional manifolds in the 𝐸𝐸
dimensional space. The dimension of the manifold ata regular point is called the degrees of freedom(DOF). Generically, because each vertex produces
three equations, the DOF of origami (homeomorphic
to a disk) is 𝐸𝐸 − 3𝑉𝑉 , where 𝑉𝑉 is the number ofinterior vertices.
4.2.

Overconstrained Origami

However, this computation of DOF is not true for
special cases. Special cases are the very source ofinteresting behaviors in origami. For example,
Miura-ori is one of such cases. Figure 12 shows thefolding motion of Miura-ori composed of an array ofparallelograms. In a generic case, an 𝑚𝑚 × 𝑛𝑛 array ofquadrilaterals would have 𝐸𝐸 − 3𝑉𝑉 = −(𝑚𝑚 −2)(𝑛𝑛 − 2) + 1. This means that the structure would
be static if the numbers of rows and columns wereboth larger than two. However, because of theperiodicity of the structure, the actual DOF of Miuraori is one. Such a system with redundant constraints
is called an overconstrained mechanism.

Figure 12: Miura-ori is an overconsrained mechanism

Several studies have created overconstrained
mechanisms composed of quadrangular quadrivalentmesh by generalizing the Miura-ori structure. Tachi[27][28] showed that the degeneracy of the

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

constraints in Miura-ori is preserved even without
the periodicity, leading to a freeform generalizationof bidirectionally foldable structures (Figure 13).
The design space of such “exotic” quadrilateral meshis still not fully understood, and novel families haverecently been discovered [29][30].

Figure 13: Freeform generalization of Miura-ori [27]4.3.

Graphical Methods

The constraints of rigid origami around each vertex(assume it is of degree- 𝑘𝑘 ) are expressed as: thecomposition of a sequence of rotations about thecreases 𝑖𝑖 = 0, … , 𝑘𝑘 − 1 incident to the vertex equals
identity. If crease 𝑖𝑖 has fold angle 𝜌𝜌𝑖𝑖 and the crease
direction represented by 𝐋𝐋𝑖𝑖 ,�

𝑖𝑖=0,…,𝑘𝑘−1

𝐑𝐑(𝐋𝐋𝑖𝑖 , 𝜌𝜌𝑖𝑖 ) = 𝐈𝐈,

where crease angle 𝐑𝐑(𝐋𝐋𝑖𝑖 , 𝜌𝜌𝑖𝑖 ) is the rotation aroundcrease 𝑖𝑖. Taking the derivative of these constraints
will give the constraints on the infinitesimal folding
motion 𝜌𝜌𝚤𝚤̇ , or the speed of folding, which isrepresented as,�

𝑖𝑖=0,…,𝑘𝑘−1

4.4.

Cellular Composition

Further novel structural properties are sought out by
going beyond a single sheet. Cellular structures
periodically tessellating a volume can be formed by
combining sheets. Here, the important constraint fortransformable performance is to avoid closed
polyhedral packing of a space, e.g., honeycomb
sandwich panels. Keeping an open cell is necessary
because Bellows Theorem states that a closed
polyhedron does not rigidly fold while changing its
volume [35].
A series of transformable cellular structures can bemade, or can be understood, as the stacking of
tubular structures. One-DOF rigidly foldable tubes
can be made by carefully stacking two folded sheets[23][36]. By choice of the symmetry of the tubes andthe stacking, we may obtain different kinds ofcellular structures, e.g., cellular structures bytranslation [23], interleaving [36], and glidereflection [22]. Stacking of Miura-ori sheets makes acellular material auxetic in all three dimensions [37]while interleaving structures make a highly
anisotropic structure stable in one direction andflexible in two other directions [38] (Figure 15).

𝐋𝐋𝑖𝑖 𝜌𝜌𝚤𝚤̇ = 𝟎𝟎.

This equation can be interpreted as the equilibriumof forces around the vertex when 𝜌𝜌𝚤𝚤̇ is seen as theaxial force applied to the crease. This leads to agraphical connection between origami and selfequilibrium structures. Figure 14 shows the same
polyhedron seen as an infinitesimally (but not
finitely) foldable structure called a shaky polyhedron
and as the self-equilibrium tensegrity structure [31].
The kinematic shell by Mitchel et al. [32]demonstrates the graphical connection between
statics and kinematic structures. Recent studies
further explore graphical methods related to secondorder infinitesimal folding motion [33][34].

Figure 14: Shaky polyhedron (left) and tensegrity
structure (right)

Figure 15: Interleaved tubular structure [38].

Another powerful strategy for creating transformablecellular materials is to replace each face of apolyhedron by a cylinder generated by extruding theperimeter of the face; then the cylinder can sheer
while keeping the planarity of the face. The
technique is known in the origami community assnapology, introduced by Heinz Strobl. Applying
this replacement technique to a periodic polyhedral
packing results in families of multi-DOF cellular
structures [39][40].
5.

SELF-FOLDABILITY

5.1.

Folding-based Fabrication

In biological systems, forms are fabricated based on
self-organization; here, the mechanism of making
things is intrinsic to the material. This contrasts with

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Vol. 60 (2019) No. 1 March n. 199

normal artifacts, where the materials and the tools for
manufacturing are separately prepared. Such an
intrinsic way of making inspires us to develop
techniques called self-folding, where each crease
folds by itself reacting to external energy. Our goal
is to find a proper patterning creases that canselectively actuate hinges such that the entire
structure folds up into the desired shape.
Several methods realize self-folding throughgradational material shrinking and expansion
patterned on a crease, resulting in a controlled
bending moment. The crease can be patterned
through three layers of materials, where some of thelayers are designed to shrink or swell (Figure 16). Na
et al. [41] use swelling polymer gels to induce thefolding and unfolding of origami tessellations in
microstructures. Felton et al. [42] use the one-time
fabrication method of a robot using heat shrink
plastic sheets.

Figure 16: Swelling the middle layer[41](left) or shrinking
the outer layer[42](right) leads to self-folding

By combining the self-folding techniques and theuniversality of origami, we may dream of a future
where we can fabricate desired functional shapes byjust printing a pattern and heating it in an oven. Of
course, there are several issues that need to be
overcome to achieve this goal. An obvious
improvement is to have more precise control over themagnitude and the timing of the applied bending
moment. However, there is a typical issue intrinsic
to origami that cannot be solved even if the applied
force is fully controllable, namely, the bifurcation
problem.
5.2.

Bifurcation

When folding a sheet of paper manually, it is moredifficult to fold a 3D shape from a 2D sheet of paper
than it is to unfold a 3D shape into a 2D form.
Folding requires a proper “sequence of folding,”
otherwise the paper folds into other things or modes.
Once the structure takes a path into a different mode,the only way to correct is to go back to a flat shape(Figure 17).

14

Figure 17: A mistake in the folding (right) can be fixed(left) only through returning to the entirely unfolded state(middle)This irreversibility is also true even for one-DOF
rigidly foldable origami with fully controlled selffolding actuation. To illustrate the behavior
precisely, Figure 18 shows the configuration spaceof a degree-4 vertex. Because a degree-4 vertex is a
one-DOF structure, the configuration space is drawn
as a set of curves in 4D space. Specifically, twocurves meet at one point; the meeting point is the flat
unfolded state. At the flat configuration, theconfiguration space bifurcates into two paths, eachof which corresponds to a different folding mode.

Figure 18: The configuration space of a degree-4 vertex(right) and its corresponding folding modes (left)

Self-foldability [43] is the problem of judging if there
is a pattern of applied force that consistently pushes
the configuration into the correct branch so that it
folds to a specified folded state. In the case of thedegree-4 vertex, there is an assignment of bending
moment that ensures the potential energy decreases
only along the correct path.
Now consider a multi-vertex system. For eachvertex, there is a binary decision of choosing a mode.
Therefore, we may potentially get an exponential
number of folding modes through the combination ofeach bit, while some of the mode assignments are
restricted in an overconstrained mechanism.
Combinatoric analyses show that the number of
combinations can rapidly explode [44]. (Thiscombinatoric behavior also makes the rigid
foldability problem of a given origami patterncomputationally hard, specifically, NP-hard [41].)
Self-folding fails when the dimension of the control
is less than the number of modes. However, theJOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

dimension of the control increases only, at most,linearly with the number creases, while the numberof modes can increase exponentially. So, there is a
general tendency that the more complex the pattern,
the more likely the self-folding fails. The probabilityof success will approach zero just by making thepattern complex.
Kinematic bifurcation, as explained above, is anannoying problem on one hand. However, it can bepotentially exciting, because a single sheet can foldinto different separate mechanisms. For example, we
may obtain re-programmable metamaterials that canswitch between stiff and flexible, auxetic andnormal, or multi-DOF and one-DOF. The author andthe collaborators are now working on several
approaches to tame this bifurcation (Figure 19).

Figure 19: A structure that folds into a cube and an
octahedron [46]6.

CURVED FOLDSThe ideas explained above come from an assumptionthat a creased line is a straight line. If we use a curved
crease, the surface exhibits the coupled behavior of
crease folding and surface bending. Curved folding
is an extremely rich source of designs. Artists anddesigners have explored varieties of forms using
curved folding, and there have been attempts atcomputational analysis and designs [47].
However, these theories are still under development,
when compared to straight-line origami. The typicaldifficulty in the geometry of curved folding is infinding a proper model that can capture both thediscrete and continuous nature of creases andsurfaces. An approach using differential geometry
was recently updated to represent the constraints
between curved creases [48]. This succeeds inreconstructing some of David Huffman’s curved
folding sculptures. However, the majority of existing
curved folded forms are not yet mathematicallydescribed.
The elastic behaviors of curve-folded surfaces areeven more exciting but are even less understood.
Future studies that bridge the geometric study ofcurved folding and the theory of active bending
could aid in understanding such systems.

7. CONCLUSION: LET’S WORK TOGETHER
The structural origami group (Rupert Maleczek andthe author) initiated annual collaborative workshops,
called “Structural Origami Gatherings,” wherearchitects, engineers, mathematicians, and computerscientists collaboratively solved shared questions on
structural origami. (This style is borrowed fromcomputer science community. Erik Demaine calls
the style supercollaboration.) The problems range
from kinematic design, self-folding actuation control,
elastic behavior analysis, structural systems forfreeform surfaces, bending active curved folding
systems, etc.
We believe that structural origami is not a subtopic
closed within engineering, but rather, it looks like a
cloud of academic interests, floating over art,architecture, engineering, science, informatics, andmath. It turns out that collaboration is helpful
especially because it provides new, unexpected
perspectives on the problem, leading to a dozen newquestions.
This collaborative nature is not specific to origami
but is universal in the field of shells and spatial
structures. We believe that SMG in IASS works as
the platform for cultivating interdisciplinary
discussions and collaborations between specialists
from different fields such as engineers, architects,
artists, mathematicians, biologists, using the sharedlanguage of “forms.” The SMG Workshop during
the IASS Symposium in Boston, in 2018, was started
as a way to bridge and reunite structural studies onorigami, scissors, bending active structures, andgraphical statics.
ACKNOWLEDGMENTS
This work is supported by KAKENHIs 16H06106
and 18H03868.
REFERENCES[1]

K. Miura, “Proposition of pseudo-cylindricalconcave polyhedral shells,” in Proc. IASS
Symposium on Folded Plates and Prismatic
Structures, 1970.

[2]

R. D. Resch and H. Christiansen, "The design
and analysis of kinematic folded platesystems," in Proceedings of IASS Symposium
on Folded Plates and Prismatic Structures,
1970.

15

Vol. 60 (2019) No. 1 March n. 199[3]

D. Bushnell and W.
shellbuckling.com, 2015.

[4]

T. Tachi, “Origamizing polyhedral surfaces,”
IEEE Transactions on Visualization andComputer Graphics, Vol.16, pp.298-311,
2010. (DOI: 10.1109/TVCG.2009.67)

[5]

D.

Bushnell,E. D. Demaine and T. Tachi, “Origamizer: A
Practical Algorithm for Folding Any
Polyhedron,” in Proceedings of the 33rd
International Symposium on Computational
Geometry (SoCG 2017), Brisbane, Australia,
July 2017, pp. 34:1–34:15.

[6]

T. Tachi, “Designing Freeform Origami
Tessellations by Generalizing Resch's
Patterns,” Journal of Mechanical Design, Vol.
135, 2013. (DOI: 10.1115/1.4025389)[7]

N. H. Kuiper, “On C1-isometric imbeddings.
I, II,” Nederl. Akad. Wetensch. Proc. Ser. A,
Vol. 58, pp. 545–556, 683–689, 1955

[8]

K. Matsuda, H. Gotoh, Y. Tajika, T. Sushida,
H. Aonuma, T. Niimi, M. Akiyama, Y. Inoueand S. Kondo, “Complex furrows in a 2D
epithelial sheet code the 3D structure of abeetle horn,” Scientific Reports, Vol. 7,13939, 2017.
(DOI: 10.1038/s41598-017-14170-w)[9]E. G. Rapp, “Self-supporting structural unit
having a series of repetitious geometricalmodules,” United States Patent No. 2,963,128,
1960.

[10] K. Miura, “New structural form of sandwich
core,” Journal of AIRCRAFT, Vol. 12, pp.
473–441, 1975. (DOI: 10.2514/3.44468)[11] K. Miura, "Method of packaging anddeployment of large membranes in space," in31st Congress of the International
Astronautical Federation, 1980, IAF 80-A31.
[12] K. Kuribayashi, K. Tsuchiya, Z. You, D.
Tomus, M. Umemoto, T. Ito and M. Sasaki,“Self-deployable origami stent grafts as abiomedical application of Ni-rich TiNi shapememory alloy foil,” Materials Science andEngineering: A, Vol. 419, pp. 131-137, 2006.
(DOI: 10.1016/j.msea.2005.12.016)

16[13] Z. Wu, I. Hagiwara, and X. Tao,
“Optimisation of crush characteristics of thecylindrical origami structure,” International
Journal of Vehicle Design, Vol.43, pp. 66–81,
2007. (DOI: 10.1504/IJVD.2007.012296)
S. D. Guest and S. Pellegrino, “The folding of
triangulated cylinders, Part I: Geometricconsiderations,” ASME Journal of Applied
Mechanics, Vol. 61, pp. 773–777, 1994. (DOI:
10.1115/1.2901553)[14] J. L. Silverberg, J-H. Na, A. A. Evans, B. Liu,
T. C. Hull, C. D. Santangelo, R. J. Lang, R. C.
Hayward, and I. Cohen, “Origami structureswith a critical transition to bistability arisingfrom hidden degrees of freedom,” Nature
Materials, Vol. 14, pp. 389–393, 2015.
(DOI: 10.1038/nmat4232)[15] H. Yasuda, T. Tachi, M. Lee, and J. Yang,“Origami-based tunable truss structures for
non-volatile mechanical memory operation,”
nature communications, Vol. 8, 2017.
(DOI: 10.1038/s41467-017-00670-w)[16] E. D. Demaine, M. L. Demaine, V. Hart, G. N.
Price, and T. Tachi, “(Non)Existence ofPleated Folds: How Paper Folds Between
Creases,” Graphs and Combinatorics, vol. 27,
pp. 377-397, 2011.
(DOI: 10.1007/s00373-011-1025-2)[17] B. A. DiDonna and T. A. Witten, “Anomalous
Strength of Membranes with Elastic Ridges,”
Phys. Rev. Lett. Vol. 87, p. 206105, 2001.
(DOI: 10.1103/PhysRevLett.87.206105)[18] E. T. Filipov, K. Liu, T. Tachi, M. Schenk, andG. H. Paulino, "Bar and hinge models for
scalable analysis of origami," International
Journal of Solids and Structures, Vol.124,
pp.26-45, 2017.
(DOI: 10.1016/j.ijsolstr.2017.05.028)[19] K. Liu and G. H. Paulino, "Nonlinear
mechanics of non-rigid origami: an efficientcomputational approach." Proceedings of theRoyal Society A: Mathematical, Physical andEngineering Sciences, Vol. 473, p. 20170348,
2017. (DOI: 10.1098/rspa.2017.0348)

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASS

[20] M. Schenk and S. D. Guest, “Origami
Folding: A Structural Engineering Approach,”in Origami 5: Fifth International Meeting of
OrigamiScience,
Mathematics,
andEducation, CRC Press, 2011, pp. 291-304.
(DOI: 10.1201/b10971-27)[21] E. T. Filipov, T. Tachi, and G. H. Paulino,“Origami tubes assembled into stiff, yetreconfigurable structures and metamaterials,”
PNAS, Vol. 112, pp. 12321-12326, 2015.
(DOI: 10.1073/pnas.1509465112)[22] T. Tachi, “One-DOF cylindrical deployablestructures with rigid quadrilateral panels,” inProceedings of Symposium of the IASS 2009,
Valencia, Spain, 2009.
[23] C. Hoberman, “Reversibly expandable threedimensional structure,” United States Patent
No. 4,780,344, 1988
[24] T. Tachi, “Rigid-foldable thick origami,” inOrigami5: Fifth International Meeting of
OrigamiScience,
Mathematics,
andEducation, 2011, pp. 253–263. (DOI:
10.1201/b10971-24)[25] Y. Chen, R. Peng, and Z. You, “Origami of
thick panels,” Science, Vol. 349, pp. 396–400,
2015. (DOI: 10.1126/science.aab2870)

figures," Journal for Geometry and Graphics,
vol. 16, pp. 223–234, 2012.
[31] T. Mitchell, A Mazurek, C. Hartz, M. Miki,and W. Baker, “Structural Applications of theGraphic Statics and Static-Kinematic
Dualities: Rigid Origami”, in Proceedings ofthe IASS Symposium 2018, Boston, USA,2018.
[32] N. Watanabe, “(in Japanese) extraction of
foldable mode in a singular state of panelhinge model,” in Proceedings of 31th
Symposium on Aerospace Structure andMaterials, JAXA/ISAS, no. A6000047014,
2015.
[33] E. D. Demaine, M. L. Demaine, D. A.
Huffman, T. C. Hull, D. Koschitz and T.
Tachi, “Zero-Area Reciprocal Diagram ofOrigami,” in Proceedings of the IASS
Symposium 2016, Tokyo, Japan, 2016.
[34] R. Connelly, I. Sabitov and A. Walz: “Thebellows conjecture," Contributions to Algebra
and Geometry, Vol. 38, pp. 1-10, 1997.
[35] T. Tachi and K. Miura, “Rigid-foldable
cylinders and cells,” Journal of the IASS, Vol.
53, pp. 217-226, 2012.

[26] T. Tachi, “Generalization of rigid-foldable
quadrilateral-mesh origami,” Journal of theIASS, Vol. 50, pp. 173-179, 2009.

[36] M. Schenk and S. D. Guest “Geometry ofMiura-folded metamaterials” PNAS, Vol. 110,
pp. 3276-3281, 2013.
(DOI: 10.1073/pnas.1217998110)[27] T. Tachi, “Freeform Rigid-Foldable Structureusing Bidirectionally Flat-Foldable Planar
Quadrilateral Mesh,” in Advances in
Architectural Geometry 2010, 2010, pp. 87102.

[37] K. Cheung, T. Tachi, S. Calisch and K. Miura,“Origami interleaved tube cellular materials,”
Smart Materials and Structures, Vol. 23,
094012, 2014.
(DOI: 10.1088/0964-1726/23/9/094012)[28] P. Dieleman, “Origami metamaterials: design,
symmetries, and combinatorics,” Ph.D.
dissertation, Faculty of Science, Leiden
University, 2018.

[38] J. T. B. Overvelde, T. A. de Jong, Y.
Shevchenko, S. A. Becerra, G. M. Whitesides,
J. C. Weaver, C. Hoberman, and K. Bertoldi,“A three-dimensional actuated origamiinspired transformable metamaterial withmultiple degrees of freedom,” Nature
Communications, vol. 7, 10929, 2016. (DOI:
10.1038/ncomms10929)[29] Z. He and S. D. Guest, “New Rigid-foldable
Developable Quadrilateral Creased Papers,”
in Proceedings of the IASS Symposium 2018,
Boston, USA, 2018.
[30] T. Tachi, "Design of infinitesimally andfinitely flexible origami based on reciprocal[39] J. T. B. Overvelde, J. Weaver, C. Hoberman,
dan K. Bertoldi, “Rational Design ofReconfigurable
Prismatic
Architected17Vol. 60 (2019) No. 1 March n. 199

Materials,” Nature, 541, pp. 347-352, 2017.
(DOI: 10.1038/nature20824)[40] J.-H. Na, A. A. Evans, J. Bae, M. C. Chiappelli,
C. D. Santangelo, R. J. Lang, T. C. Hull, and
R. C. Hayward, “Programming reversibly
self-folding origami with micropatterned
photo-crosslinkable
polymertrilayers,”
Advanced Materials, Vol. 27, pp.79-85, 2015.
(DOI: 10.1002/adma.201403510)[41] S. Felton, M. Tolley, E. Demaine, D. Rus, andR. Wood, “A method for building self-folding
machines,” Science, Vol. 345, pp. 644-646,
2014. (DOI: 10.1126/science.1252610)[42] T. Tachi and T. C. Hull, “Self-Foldability of
Rigid Origami,” ASME Journal of
Mechanisms Robotics, Vol. 9, pp. 021008021008-9, 2017. (DOI: 10.1115/1.4035558)[43] T. C. Hull and T. Tachi, “Double-line rigid
origami,” in Proceedings of 11th Asian Forumon Graphic Science, Tokyo, Japan, 2017.

18[44] H. Akitaya, E. D. Demaine, T. Horiyama, T.
C. Hull, J. S. Ku, T. Tachi, “Rigid Foldability
is NP-Hard,” arXiv, 1812.01160, 2018.
[45] T. Tachi and T. Horiyama, “1-DOF Structure
Folding into Multiple Polyhedra”, 7thInternational Meeting on Origami in Science,
Mathematics and Education, Oxford, UK,2018.
[46] E. D. Demaine, M. L. Demaine, D. Koschitz
and T. Tachi, " A review on curved creases inart, design, and mathematics," Symmetry:
Culture and Science, Vol. 26, pp. 145-161,
2015.
[47] E. D. Demaine, M. L. Demaine, D. A.
Huffman, D. Koschitz and T. Tachi, “Conic
Crease Patterns with Reflecting Rule Lines,”
Origami7 - The proceedings from the seventhmeeting of Origami, Science, Mathematics,
and Education, Vol. 2, pp. 573-590, 2018.