Origami tubes assembled into stiff, yet reconfigurable
structures and metamaterials
Evgueni T. Filipova, Tomohiro Tachib, and Glaucio H. Paulinoa,c,1
a
Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801; bGraduate School of Arts andSciences, University of Tokyo, Tokyo 153-8902, Japan; and cSchool of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta,
GA 30332

Thin sheets have long been known to experience an increase instiffness when they are bent, buckled, or assembled into smaller
interlocking structures. We introduce a unique orientation forcoupling rigidly foldable origami tubes in a “zipper” fashion that
substantially increases the system stiffness and permits only one
flexible deformation mode through which the structure can deploy.
The flexible deployment of the tubular structures is permitted
by localized bending of the origami along prescribed fold lines. Allother deformation modes, such as global bending and twisting of
the structural system, are substantially stiffer because the tubular
assemblages are overconstrained and the thin sheets become engaged in tension and compression. The zipper-coupled tubes yieldan unusually large eigenvalue bandgap that represents the uniquedifference in stiffness between deformation modes. Furthermore,
we couple compatible origami tubes into a variety of cellular assemblages that can enhance mechanical characteristics and geometric versatility, leading to a potential design paradigm for structuresand metamaterials that can be deployed, stiffened, and tuned. The
enhanced mechanical properties, versatility, and adaptivity of thesethin sheet systems can provide practical solutions of varying geometric scales in science and engineering.

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stiff deployable structures origami tubes rigid origami
thin sheet assemblages reconfigurable metamaterials

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ntroducing folds into a thin sheet can restrict its boundaries,
cause self-interaction, and reduce the effective length forbending and buckling of the material (1–4). These phenomena
make thin sheets practical for stiff and lightweight corrugatedassemblies (5, 6); however, such systems tend to be static, i.e.,functional in only one configuration. For creating dynamic structures, origami has emerged as a practical method in which continuous thin sheet panels (facets) are interconnected by prescribedfold lines (creases). Existing origami patterns and assemblages caneasily be deployed; however, they tend to be flexible and need to bebraced or locked into a fixed configuration for a high stiffness-toweight ratio to be achieved (7–10). The zipper-coupled system isdifferent because it is stiff throughout its deployment without
having to be locked into a particular configuration.
Origami principles have broad and varied applications, fromsolar arrays (11) and building façades (12) to robotics (13),mechanisms in stent grafts (14), and DNA-sized boxes (15). The
materials and methods used for fabricating, actuating, and assembling these systems can vary greatly with length scale. On the
microscale, metallic and polymer films or, more often, layered
composites consisting of stiff and flexible materials can be folded
by inducing current, heat, or a chemical reaction (16, 17). Largescale origami structures can be constructed from thickened panels
connected by hinges and can be actuated with mechanical forces(11, 18, 19). The kinematic motion, functionality, and mechanicalproperties of the origami are governed largely by the folding pattern geometry. For example, rigid origami systems are defined as
those having a kinematic deformation mode in which movement isconcentrated along the fold lines, whereas the panels remain flat
(20, 21). Among various rigid folding patterns, the Miura-ori has
attracted attention for its folding characteristics (22, 23), elastic

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stiffness properties beyond rigid folding (24, 25), geometric versatility (26, 27), and intrinsic material-like characteristics (28, 29).
The zipper-coupled tubes introduced here are derived from
the Miura-ori pattern and can undergo the same type of rigid
kinematic deployment. All other deformations are restrained asthey require stretching and shear of the thin sheets. Thus, thestructure is light and retains a high stiffness throughout its deployment. It has only one flexible degree of freedom and can be
actuated by applying a force at any point (Fig. 1 and Movie S1).
To explore unique mechanical properties of the zipper tubes, we
introduce concepts of eigenvalue bandgaps and cantilever analyses
to the field of origami engineering. Zipper assemblages can be
fabricated with a variety of materials and methods. We envisionapplications of these assemblages will range in size from microscale metamaterials that harness the novel mechanical properties
to large-scale deployable systems in engineering and architecture(Movies S2–S4).
This paper is organized as follows. First, the Miura-ori pattern isintroduced, and the geometries of three fundamental coupling orientations are discussed. Next, we demonstrate how the system
stiffness changes as we assemble two sheets into a tube and then two
tubes into the unique zipper-coupled tubes. The fundamental coupling orientations are then studied as deployable cantilevers that can
carry perpendicular loads. Next, we discuss cellular assemblages,
geometric variations, and practical applications that can be createdfrom coupled tubes, and we conclude with some final remarks.
Geometric Definitions
A Miura-ori cell consists of four equivalent panels, defined by aheight a, width c, and vertex angle α (Fig. 2A). For our analyticalinvestigation, we use a cell with a = c = 1 and α = 55° as the basis
for all structures unless otherwise noted. A sheet is created byrepeating this cell Nð=5Þ times in the X direction (Fig. 2B). We
SignificanceOrigami, the ancient art of folding paper, has recently emerged
as a method for creating deployable and reconfigurable engineering systems. These systems tend to be flexible because thethin sheets bend and twist easily. We introduce a new methodof assembling origami into coupled tubes that can increase theorigami stiffness by two orders of magnitude. The new assemblages can deploy through a single flexible motion, butthey are substantially stiffer for any other type of bending or
twisting movement. This versatility can be used for deployablestructures in robotics, aerospace, and architecture. On a smallerscale, assembling thin sheets into these tubular assemblages
can create metamaterials that can be deployed, stiffened,
and tuned.
Author contributions: E.T.F., T.T., and G.H.P. designed research, performed research, contributed new reagents/analytic tools, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1

To whom correspondence should be addressed. Email: paulino@gatech.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1509465112/-/DCSupplemental.

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Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved August 7, 2015 (received for review May 14, 2015)

Fig. 1. Deployment and retraction sequence of a zipper-coupled tube system. This origami has only one flexible motion through which it can deform,
and thus it is deployed by actuating only on the right end. See Movie S1 forcomplete simulation.

define the configuration of this sheet as a percentage of extension(a percentage of the maximum extended length =2Nc). At 0%
extension the sheet lies folded on the Y − Z plane and at 100%extension it lays flat on the X − Y plane. The single tube structure (30) is created by combining the single sheet with its mirrored counterpart (Fig. 2C and Movie S5). Previously, Miura-ori
sheets and tubes have been coupled (or stacked) to create several
different folding assemblages (7, 9, 30, 31). For the zipper orientation of tube coupling, one tube is rotated and connected in a
zig-zag (zipper) manner (Fig. 2D and Movie S1). We explore themechanical properties of this assemblage and compare it with analigned-coupled system where tubes are only translated (30, 31)
and an internally coupled system consisting of two compatible
tubes (7) (Fig. 2 E and F). Symmetry in these assemblages permits them to be rigid foldable, meaning that they can fold withno bending of the panels, but only the zipper and aligned systems
can reach a flat state when deployed to a 100% extension. The
range of extension for internally coupled systems is limited because the internal tube reaches a flat configuration and locks thesystem in place. Geometric parameters of the compatible internal tube aI , cI , and αI , are discussed in SI Text, section S5.
Eigenvalue Analyses
To model the stiffness and mass of origami, we use a bar andhinge approach that provides insight into the structural behaviorof the origami assemblages. It captures three fundamental
physical behaviors: bending along fold lines, bending of initiallyflat panels, and stretching and shearing of panels (SI Text, section S1 and Fig. S1) (24). The simplicity of the bar and hingemodel allows for straightforward implementation and versatility
where tubes can be coupled and analyzed as more complex assemblages (see SI Text, sections S2 and S5 and Figs. S2 and S3).
The model is made scalable and incorporates thickness (t = 0.01),
Young’s modulus (E = 106), Poisson’s ratio (ν = 1=4), and density(ρ = 1) of the material (32). A parameter, RFP = 1=10, relates foldline bending to panel bending stiffness and can vary greatly,
depending on the materials, the fabrication technique, and theactuation process (33–36). Using a sensitivity analysis in SI Text,
section S3 and Fig. S4, we show that the value of RFP and otherparameters do not affect the qualitative results and conclusions
presented in this paper. A finite-element (FE) model, where
each panel is discretized with shell elements, is used to verify the
results and to study local deformations and stress concentrationsin the origami (SI Text, section S4; Fig. 3; and Figs. S5–S7). The
bar and hinge model cannot capture all localized effects, but it issignificantly faster and provides a sufficiently high level of accuracy for global origami behaviors.
Stiffness (K) and mass matrices (M) are used to construct a
linear dynamics system of equations Kvi = λi Mvi (no sum), where
λi is the ith eigenvalue and vi is the corresponding eigenmode of
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the structure. Each eigenmode represents a structural deformation mode of the system, and its corresponding eigenvalue (inunits of 1=s2) represents an excitation frequency that would result in the given deformation. The eigenvalue is proportional to
the total energy in the system (kinetic and elastic strain energy),
meaning that a deformation mode with higher energy will require a higher excitation frequency. The structure is analyzed
with no boundary constraints, and thus the first six eigenmodes
correspond to rigid body motion of the structure in 3D space(three displacement and three rotational modes). We omit thesemodes from our study, as they require zero energy. The seventh
eigenmode and those following represent deformations requiring
incremental energy input. The eigenvalues of a single Miura-ori
sheet, a single tube, and a zipper-coupled tube are plotted vs.
percentage of extension (Fig. 3 A–C), and representative deformation modes are shown for each structure at 70% extension(Fig. 3 D–F). Eigenvalues scale linearly with mass and stiffness
and therefore comparisons remain scale independent and highlight the influence of coupling on the global system behaviors.
The single sheet can bend and twist globally in several directions
with little energy input, resulting in low eigenvalues. The distribution of energy for the seventh and eighth deformation modes(Fig. 3G and Fig. S7) illustrates that bending occurs in thecentral panels and folds whereas the remainder of the structure
remains unstressed. As expected, the bending energy in thepanels is highest at the vertices, where the curvature approaches
infinity (1, 37). The ninth mode of the sheet and the seventh
mode of the tube represent a rigid folding motion where bending
is primarily concentrated in the folds and the panels remain
essentially flat throughout the deformation (Fig. 3 D and E).
For practical purposes, we prefer that λ7 corresponds to rigid
folding and that subsequent eigenvalues are higher, creating a
bandgap β = λ8 − λ7. A structure with a large bandgap has a

A c

D

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Y
ZY

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ZY
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Fig. 2. Geometry and assembly of origami tube structures. (A) Miura-ori
pattern defined with a = c = 1, α = 55° and folded into a three-dimensional
Miura-ori cell. This cell is used as the basis for all structures unless otherwise
noted. (B) A Miura-ori sheet in four configurations defined as a percentageof extension. (C) Two mirror-image sheets shown at 70% extension andcombined into a rigid, flat foldable tube. (D) Zipper-coupled tubes. A tube istranslated and rotated in the Y − Z plane until the opposing faces of the two
tubes align. For clarity, a different shade is used for one of the zipper tubes,
but both tubes are identical to the definitions in A–C. (E) Aligned-coupledtubes. A tube is translated in the Y − Z plane and coupled with the tube in
the initial configuration. (F) Internally coupled tubes, inspired by ref. 7. External tube is the same as in C, and the internal tube is defined to reach a flat
configuration when the external tube is at 80% extension.

Filipov et al.

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λ8
λ8

λ7
λ7

Bar and hinge model:
FE model:

A

λ9
λ9

λ10
λ10

Bandgap β = (λ8 - λ7)

BC

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λ7=1.37

Mode 8Z100

λ7=2.78λ8=1.43λ9=2.03X

λ8=11.5λ9=226.8

λ8=1194

λ9= 1647

YX
Energy distribution (%)

S

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Mode #S - Mode 80

7 8 9Mode #

flexible mode that allows for easy deployment but is stiff for allother types of loading. Mode switching may occur, meaning thatthe deformed shape corresponding to a given eigenvalue candiffer drastically, depending on extension. For example, theseventh and eighth modes switch for the single Miura-ori sheet
around 70% extension (Fig. 3A). The single and zipper-coupledtubes display a continuous bandgap and do not experienceswitching of the rigid folding mode (seventh). The eighth modeof the single tube constitutes a “squeezing” deformation, where
one end of the structure is folded and the other end is unfolded(Fig. 3E and Movie S5). This mode requires bending of folds andpanels at the ends of the structure; however, the panels do notstretch or shear, and thus λ8 is only slightly higher than λ7. The
aligned and internally coupled assemblages are also prone tosqueezing and the bandgap is not increased (SI Text, section S5
and Fig. S8).
However, when coupling the tubes into a zipper assemblage
we observe that the structure has a substantially increased
bandgap, where λ8 is often two orders of magnitude greater than
λ7. The large increase in λ8 occurs because all deformation modes
except for rigid folding (λ7) require stretching and shearing of thethin sheets and thus a substantially higher amount of energy.
Deforming the zipper-coupled tubes in a global bending mode(eighth or ninth) requires stretching the structure’s panels (Fig. 3
F and I). When actuated at one end, a single, aligned, or internally coupled tube squeezes (Movie S5 and Fig. S8), whereas
zipper-coupled tubes expand and contract in a consistent rigid
folding motion (Fig. 1 and Movie S1). The zipper-coupled tubes
are not prone to the squeezing-type deformation, as it would
require differential movement between the tubes and stretching
the thin sheet (SI Text, section S2 and Fig. S3).
Filipov et al.

Fig. 3. Stiffness characteristics of a sheet, a single
tube, and a zipper-coupled tube. (A–C) Eigenvalue
vs. configuration (percentage of extension) for the
sheet, the single tube, and zipper-coupled tubes,
respectively. The large bandgap from the zippercoupled tubes is also verified with an FE model.
(D–F) Seventh, eighth, and ninth deformation eigenmodes when structures are at 70% extension(undeformed outline in red). (G–I) Energy distributions for the sheet, the single tube, and zipper
tubes, respectively. The energy distribution is presented as a bar graph indicating the percentage ofenergy in fold bending (F), panel bending (B), andpanel stretching and shearing (S). Representative
element energies are shown for modes 7 and 8 ofeach structure. For detailed energy presentation see
SI Text, section S4 and Fig. S7.

Structural Cantilever Analyses
The zipper, aligned, and internally coupled tube systems can beapplicable as deployable cantilever structures when restrained on
one end (Fig. 4). At the supported end all nodes are fully fixed(X, Y, and Z displacements), whereas a total load of 1 is distributed on the other end of the structure (Fig. S9 D–F). Whenused as cantilevers, the tubes exhibit behavior similar to that ofan I-beam, wherein the second moment of area (or area momentof inertia) is increased by distributing material away from thecentroid. The aligned and internally coupled tube systems often
experience squeezing-type deformations when loaded on one
end, whereas the zipper-coupled tubes experience more uniform
bending deformations (Fig. 4A). The stiffness of the structures iscalculated for the three Cartesian directions for different extensions (Fig. 4 B–D). The loads are applied for each configuration, and the resultant displacements (ΔX , ΔY , and ΔZ) arecalculated from the linear function F = KΔ, where F is a force
vector. A quantitative stiffness of the system is then calculated as
K = F=δ = 1=δ based on the maximum system displacement (δ).
The zipper-coupled tubes tend to be stiffer than the other two
systems and have a greater stiffness when closer to full extension.
The internally coupled tubes tend to be stiff for Z-direction
loading and when locked into a fixed configuration (i.e., 80%
extension). The results for stiffness presented here show thegeneral system behavior and are in consistent units of force perlength (e.g., newtons per millimeter). A realistic length scale andelastic modulus can be substituted to find quantitative results forthe cantilevers.
The stiffness for different directions of loading, orthogonal to
the X axis, is investigated, by rotating the load in the Y − Z plane.
The stiffness for each structure is calculated at extensions of 40%,PNAS Early Edition | 3 of 6

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0Fixed end

Distributed
load

Fz
Fy

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Fx
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At 40% extension

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102

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FyY

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KYZ

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20

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ZipperAlignedInternal

−1

−2

10 0

25

50
75 100% extension

ZipperY
AlignedFig. 4. Zipper, aligned, and internally coupled tubes used as cantilevers. (A) Initial (red line) and deformed geometry of the structures at 70% extension
when the left end is fixed and a uniform load is applied on the right end. The deformed shapes are scaled so the maximum displacement for each case is equal
to the panel height (a = 1) and do not necessarily represent stiffness. (B–D) The stiffness (force/length) of the cantilevers in the three Cartesian directions. Theinternally coupled tube cannot extend beyond 80% extension of the external tube. (E–G) The stiffness for loads in the Y − Z plane represented as a radial plot
at extensions of 40%, 70%, and 95%, respectively. Stiffness is shown as distance from the origin.

70%, and 95% and presented as a radial plot corresponding to thedirection of loading (Fig. 4 E–G). The structure is analyzed withthe same constraint and load distributions, and only the directionof the load vector (equating to 1) is rotated in the Y − Z plane.
The aligned and internally coupled tube systems demonstrateanisotropic behavior, wherein only one loading direction (in the
Y − Z plane) displays high stiffness. On the other hand, the zippercoupled tubes tend to be stiff for all directions of loading, as
depicted by the more circular profile on the plots (Fig. 4 E–G).
The fixed end of the cantilevers may be constrained in a different fashion, so that a mechanism is installed to fold and deploy the entire structure (Fig. S9 G–I). The mechanism would
control the rigid folding mode of the system, thus permitting easy
deployment (especially for zipper-coupled tubes—see actuationin Movie S1). When the mechanism is contracted, the structure
will deploy, and when it is extended, the structure will fold. Whenthe length of the mechanism is fixed, the cantilever will behavemuch as if the support is fully fixed.
Cellular Assemblages
Cellular origami can permit self-assembly of engineered hierarchical materials (5, 38), whose mechanical properties depend on
the microstructure geometry. The structural stiffness and energy
absorption properties of cellular origami can be optimized tocomplement and improve naturally occurring materials (39).
Zipper-coupled tubes can be integrated with aligned or internal
coupling to create layered foldable assemblages (Fig. 5 andMovie S3). Structures that incorporate zipper coupling inherit
the large bandgap, while also retaining properties from the othercoupling techniques (e.g., space filling from aligned or locking
from internal coupling). The zipper/aligned metamaterial studiedin Fig. 5 B–E has the same parameters as before, except the foldto panel stiffness ratio is set to RFP = 1, to simulate an assemblage constructed by additive manufacturing (Fig. 5 F and G).
Origami metamaterials created with 3D printing do not fold liketraditional origami, but possess novel characteristics such as the
single flexible mode of zipper coupling.
We analyze the assemblage in Fig. 5 by applying symmetricuniform forces (summing to 1) on opposing faces of the systemand calculate the compression stiffness as K = F=δ = 1=δ, where δis the mean total displacement in the direction of loading.
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Because of the zipper geometry, the system is primarily flexible in the X direction at lower extensions (0–70%) and in the
Y direction at higher extensions (70–100%). The peak in the
Y-direction stiffness (at 96.4% extension) corresponds to a bistablestate, where the tube cross section is square, and can transition toa different rhombus, depending on the direction of folding (Movie
S3). In addition to the stiffness, the deformation characteristics ofthe material are also anisotropic. The perceived Poisson’s ratio isnegative in the Y direction when compressed in X, whereas it ispositive in the Z direction when compressed in Y (Fig. 5 C and D).
The structure is substantially stiffer in the Z direction, and deformations do not follow the kinematic folding mode.
Geometric Variations
There are numerous other ways in which rigid foldable tubes canbe defined, combined, and coupled. For example, a different
vertex angle α and panel height a may be used with rigid and flat
foldability of the system preserved. Zipper coupling can becontinued in one direction to create an easily deployable slab
structure resistant to out-of-plane loads; potential applications
are architectural canopies (Fig. 6A), bridges (Fig. 6B), solar arrays, or synthetic materials. When the vertex angle is varied forthe coupled zipper tubes, we can create arbitrarily curved systems with high out-of-plane stiffness (Fig. 6A and Movie S4).
The bridge deck uses tubes with a vertex angle of α = 85° toprovide a more even surface, whereas the bridge parapet tubes
have a lower vertex angle to provide stiffness during deployment(Fig. 6B and Movie S2). If the zipper coupling is repeated in
more than one direction, the structure will self-interlock during
the deployment, creating a stiff array of coupled thin sheets (Fig.
6C and Movie S2). The structure does not necessarily need to havea square final cross section (four segments of zipper-coupled tubes),
but can be any radially symmetric shape (SI Text, section S6).
The cross section of the tubes is not restricted to a quadrilateral shape (30, 31); any polygon with translational symmetry
can be used to extrude a rigid, foldable origami tube. The tubes
can have any cross section where each segment of the crosssection has an equal-length counterpart (or counterparts) that is(are) translationally symmetric. The internal panels of the sixsided polygonal tube (blue in Fig. 7A and Movie S2) are defined
to conform with the flat and rigid foldability of the system. The
Filipov et al.

versatility of the systems. Extensions and refinements of thiswork may explore geometric arrangements that improve thestiffness-to-weight ratio, and impact energy dissipation and othermechanical properties. Further study of the hierarchical systemproperties with respect to fabrication, scale, and materials will beneeded to inform potential applications. As origami becomes more
widely used in science and engineering, the coupled tube assemblages will serve as an important component that allows flexible
deployment while simultaneously retaining a high global stiffness.

direction of the folds in these segments can be reversed (e.g.,
mountain to valley), thus changing the geometry. Polygonal tubes
with more sides can be reconfigured in the same way, with many
more variable cross sections. In Fig. 7B we show that finite
thickness may also be incorporated into the construction of thecoupled origami tubes. Using available techniques for thick origami (18, 19), we can create structures of thick panels adjoinedwith physical hinge elements. With these techniques cost-effective materials (e.g., thin wood panels with metal hinges) can beused to create large structures that can be easily deployed, butpossess large global stiffness from the zipper-coupling framework. Finally, the characteristics of origami assemblages can be
modified by coupling only specific portions of a tube. The longtube in Fig. 7C has zipper coupling only in the middle portion to
restrict the global squeezing and bending of the system. The ends
remain uncoupled, allowing for a rigid connection to the outsideedge while still permitting the system to fold and unfold.
Concluding Remarks
This paper introduces an approach for coupling origami tubes ina zipper fashion and explores their unique mechanical characteristics. These assemblages engage the thin sheets in tension,
compression, and shear for any deformation mode that does notfollow the kinematic deployment sequence. The origami tubes
can be stacked and coupled into a variety of cellular assemblages
that can further enhance the mechanical characteristics andFilipov et al.

Fig. 7. Local variations in coupling origami tubes. Different stages of deployment are shown with approximate percentage of extension. (A) Zipper
coupling of a reconfigurable tube with a polygonal cross section. The sixsided polygonal tube has two possible shapes (I and II). (B) Computer visualization of zipper tubes from thick material (thick origami). (C) Actuatorsystem from zipper coupling of two tubes of different length. The edges ofthe long tube are restrained, but squeezing can occur on the uncoupled
sections, allowing the system to fold. For details see Table S1.

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Fig. 5. Cellular origami metamaterial consisting of 36 tubes with α = 75°,
N = 3 arranged as zipper in the horizontal Y direction and as aligned in thevertical Z direction. (A) Kinematic folding sequence of the assemblage(Movie S3). (B) Compression stiffness of the metamaterial in the three Cartesian directions vs. extension. (C–E) Initial (red line) and deformed geometry of the assemblage at 90% extension for uniform compression tests in
the X, Y, and Z directions, respectively. The deformed shapes are scaled so
the mean displacement of the loaded surfaces is equal to the panel height(a = 1) and do not represent stiffness. (F and G) Metamaterial prototypes
constructed with additive manufacturing cannot undergo the full folding
motion in A, but inherit the anisotropic mechanical properties of the cellularzipper assemblages. In F a soft metamaterial from resin with a wall thicknessof 0.09 mm can be deformed by hand, whereas the polyamide assemblage in
G with a wall thickness of 0.8 mm is substantially stiffer (Materials andMethods).

Fig. 6. Variations of cellular assemblages in different stages of deployment.
Approximate percentage of extension is shown. (A) Deployable architectural
canopy with high out-of-plane stiffness for transformable building design.
(B) Bridge structure from zipper-coupling tubes of different geometries. The
structure is flat foldable in two directions and stiff for out-of-plane loading.
(C) Structure that interlocks into a fully conforming shape. The structure’s
edges are each composed of four tubes coupled in a zipper orientation. For
details see Table S1 and Movie S2.

Materials and Methods
Prototypes of the origami were created for demonstrating the mechanicalcharacteristics and for showing the kinematic folding. The origami assemblages in Movies S1, S2, and S5 and Figs. 1, 6, and 7 are created from perforated paper that is folded and adhered together. The Miura-ori sheets
were created from 160-g/m2 paper by perforating along the fold lines withcuts of length 0.5 mm spaced evenly at 1-mm intervals. Because each tube
cannot be developed from a single flat piece of paper, it is assembled byconnecting two Miura-ori sheets (Fig. 2C and Movie S5). One of the sheets isconstructed with perforated tabs at the edge, which can be folded and attached with standard paper adhesive to a mirror-image Miura-ori sheet.
When connecting two tubes into either the zipper or the aligned assemblage, the adjacent facets are adhered together (Fig. 2 D and E and Movie
S1). For internally coupled tubes, the structure is assembled in sequence (Fig.
2F) by connecting individual Miura-ori sheets with the joining tabs. The
numerous variations of the coupling configurations (Figs. 6 and 7) were also
assembled using the same techniques. In Fig. 5 F and G, we show how additive

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6 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1509465112

manufacturing can be used to create cellular metamaterials with characteristics inherited from the zipper tubes. The model in Fig. 5F was created with a
Keyence AGILISTA 3000 3D printer that uses digital light processing (DLP)
technology. The model material is an AR-M2 transparent resin and the
support material is an AR-S1 water-soluble resin. The cube has dimensions of
68 × 62 × 61 mm with wall thickness of 0.09 mm. The model in Fig. 5G was
created with an EOS Formiga P 100 selective laser sintering (SLS) system. The
model uses a PA 2200 (Polyamide 12) material that is sintered together anddoes not require support material. This assemblage has dimensions of 102 ×
93 × 92 mm with wall thickness of 0.8 mm.
ACKNOWLEDGMENTS. This work was partially funded by the National Science
Foundation (NSF) through Grant CMMI 1538830. E.T.F. is grateful for support
from the NSF Graduate Research Fellowship and the Japan Society for the
Promotion of Science Fellowship. The authors also acknowledge support fromthe Japan Science and Technology Agency Presto program and the Raymond
Allen Jones Chair at the Georgia Institute of Technology.

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Natick, MA), pp 87–99.
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panels. Proceedings of the International Association for Shell and Spatial Structures,eds Domingo A, Lazaro C (Universidad Politecnica de Valencia, Valencia, Spain),
pp 2295–2305.
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Structures 53(4):217–226.
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Science, Mathematics and Education, eds Kawasaki T, Uehara R, Tachi T, Maekawa J(University of Tokyo, Tokyo), p 121.
33. Lauff C, et al. (2014) Differentiating bending from folding in origami engineering
using active materials. Proceedings of the ASME 2014 International Design Engineering Technical Conferences & Computers and Information Engineering Conference (ASME, Buffalo, NY), p V05BT08A040.
34. Giampieri A, Perego U, Borsari R (2011) A constitutive model for the mechanical response of the folding of creased paperboard. Int J Solids Struct 48(16–17):2275–2287.
35. Mentrasti L, Cannella F, Pupilli M, Dai JS (2013) Large bending behavior of creased
paperboard. I. Experimental investigations. Int J Solids Struct 50(20–21):3089–3096.
36. Lechenault F, Thiria B, Adda-Bedia M (2014) Mechanical response of a creased sheet.
Phys Rev Lett 112(24):244301.
37. Cerda E, Chaieb S, Melo F, Mahadevan L (1999) Conical dislocations in crumpling. Nature
401(6748):46–49.
38. Fratzl P, Weinkamer R (2007) Natures hierarchical materials. Prog Mater Sci 52(8):1263–1334.
39. Gibson LJ, Ashby MF, Harley BA (2010) Cellular Materials in Nature and Medicine(Cambridge Univ Press, Cambridge, UK).
40. Ahn BY, et al. (2010) Printed origami structures. Adv Mater 22(20):2251–2254.
41. Hibbitt D, Karlsson B, Sorensen P (2010) ABAQUS FEA Version 6.10 (Dassault Systemes,
Vélizy-Villacoublay, France).

Filipov et al.

Supporting Information
Filipov et al. 10.1073/pnas.1509465112
SI Text
S1) Stiffness Calculations
For our analyses and discussion in the main text, we examine thethin sheet origami in a consistent fashion to highlight the behaviors, while staying independent of scale (eigenvalue and static
analyses). The eigenvalues (λ) have units of 1=s2 and represent
the global characteristics of the structure considering both massand stiffness. The static analyses present the global structural
stiffness in the form of force divided by displacement. The twodifferent computational models used herein (bar and hingemodel and finite elements) give similar results and correspondwell with observations of physical models.
For our analyses we use the following parameters unlessotherwise noted. The Miura-ori sheet (Fig. 2B) with unit dimensions a = c = 1, α = 55°, and N = 5 is used to define the tubes
and coupled tube systems used in the analyses of Figs. 3 and 4(see SI Text, section S5 for additional details on internally coupled tubes). A Miura-ori sheet with a = c = 1, α = 75°, and N = 3 isused for the cellular assemblage in Fig. 5. The thickness t = 0.01
is used to simulate a thin sheet that is much thinner than it islong (L=t = 100). A Young’s modulus E = 106 and material density ρ = 1, both arbitrary but within a realistic range, are selectedto scale stiffness and mass linearly and do not affect the global
behavior or characteristics of the origami tubes (Fig. S4 andaccompanying discussion). The elastic modulus is assumed to bein units of force per length squared (e.g., newtons per square
millimeter) and the density as mass per length cubed (e.g., kilograms per cubic millimeter). When appropriate numerical values and length scales are used, the analyses presented here will
provide quantitative approximations for stiffness and eigenvalue
response. Other parameters used in the modeling are discussed in
the text below and are chosen based on existing literature, approximate physical behavior, and type of computation performed(e.g., Poisson’s ratio ν = 1=4).
Here, we describe the simple bar and hinge method for thenumerical modeling of thin sheets in origami systems. A previously established model (24) is used as a basis, and several
improvements are incorporated to make the model scalable andisotropic and to incorporate material characteristics (32). We
make the stiffness dependent on the material’s thickness (t),Young’s modulus (E), and Poisson’s ratio (ν). The stiffness
matrix (K) for the origami structure incorporates stiffness parameters for (i) panels stretching and shearing (K S), (ii) panels
bending (K B), and (iii) bending along prescribed fold lines (K F ).
The global stiffness matrix is constructed as
2

3T 2
C
KS
K = 4 JB 5 4 0
JF
0

0
KB
0

32 3
C
0
0 54 JB 5,
JF
KF

[S1]

where the compatibility matrix (C) and Jacobian matrices (JB andJF ) relate the stiffness of the various components to the nodal
displacements, as discussed in detail below. Each node has 3
degrees of freedom (DOFs) (X, Y, and Z displacement) andthe stiffness matrix is of size (ndof × ndof ), where ndof represents
the total number of DOFs in the system. The mass of each panel
is calculated from the panel volume and material density ρ. A
mass matrix M for the entire structure is constructed by distributing one-quarter of the mass of each panel to each of its connecting nodes.
Filipov et al. www.pnas.org/cgi/content/short/1509465112

S1.1) Panel Stretching and Shearing. In-plane axial and shear
stiffness is simulated using the indeterminate bar frame (Fig.
S1A). A general formulation for bar elements is used with anequilibrium matrix (A) relating internal bar forces (t) to nodal
forces (f) as At = f, a compatibility matrix (C) relating bar nodal
displacements (d) to bar extensions (e) as Cd = e, and a diagonal
matrix (KS) relating bar extensions (e) to local forces (t) as
KS e = t. Using the static-kinematic duality that C = AT , the linear
system for stretching and shear of the panels (i.e., the bars) isrepresented as the first row of Eq. S1. The crossed bar frame(Fig. S1A) has six bars connected at the four corner nodes of theorigami panel. This crossed bar geometry results in the frame
behaving as an isotropic panel. The bar stiffness parameters (i.e.,
components of KS) are defined for each bar as KS = EAB =LB,
where LB is the bar length and AB is the bar area. To achieve theisotropic behavior for the panel, the bar areas are defined as

AX = tH 2 − νW 2,
2Hð1 − ν2 Þ

[S2]

AY = tW 2 − νH 2,
2W ð1 − ν2 Þ

[S3]

AD = t
3=2
ν H2 + W 2,
2HW ð1 − ν2 Þ

[S4]for the horizontal (X), vertical (Y), and diagonal (D) bars, respectively. The height (H) and width (W) of the panel are taken
as an average for skewed panels. To verify the frame model wedefine a panel with W = H = 1, t = 0.01, E = 1,000, and ν = 1=3
and subject it to a tensile (Fig. S1B) and a shear patch test
(Fig. S1C). For tensile loading the system always satisfies thepatch test, but for shear loading, the behavior of the model ishighly dependent on the chosen Poisson’s ratio. From Eq. S4,
when a low ν is used, the diagonal bars have a low area, and theframe demonstrates a low shear stiffness. The converse is also
true, countering the behavior expected in a truly isotropic material. Alternatively, when ν is set to 1=3, the behavior of the frame
model in shear is identical to that of a homogeneous, isotropic
block of material. Each diagonal bar carries a force of F=2 in the
X direction, and the top of the frame laterally displaces in the
direction of loading. When ν = 1=3, the frame displacement
matches the lateral displacement of a solid block with dimensions W × H × t loaded in pure shear, analytically defined as
Δx = FX H=GWt, where FX is the total shear force and G is theshear modulus, defined as G = E=2ð1 + νÞ for a homogeneous,isotropic, linear elastic material. With ν = 1=3, the frame is scale
independent for shear loadings. If the shear patch is remeshed,
then the frame model converges to the same solution as any
generic finite-element approach. However, because only a single
six-bar frame is used to model each panel, the shear stiffness ofthe panel is overestimated (similarly to any finite-element approach). In reality, the shear stiffness of the panels is lower thana pure shear case because, in addition to the shear deformations,
tensile deformations also occur over the width and height of thepanel. Using a lower Poisson’s ratio, we artificially reduce theshear stiffness of the frame. To this end, we choose to useν = 1=4. At this value, the single bar frame exhibits approximately
the same shear deformation as a patch test performed on a meshof 2,500 finite elements.
1 of 13

S1.2) Panel Bending. Out-of-plane bending of the panel is modeled
as an angular constraint between two triangular segments of thepanel (1–2–3 and 1–3–4 in Fig. S1D). For small displacements,
the choice of the diagonal does not affect the kinematics of thesystem (7). Previous findings (1, 4) show that the bending energy
is lower along the shorter length, so we formulate our model assuming that bending occurs along the shorter diagonal (i.e., 1–3
in Fig. S1D). An angular constraint, F, is formulated based on thedihedral bending angle, θ, which can be calculated by using crossand inner products of the vectors a, b, and c from the nodal coordinates of the panel p. This constraint is defined as

F = sinðθðpÞÞ,[S5]

and the corresponding Jacobian for panel bending, JB, is calculated asdθ =1 X ∂Fdpi = JB d,cosðθÞ
∂pi[S6]

where d are the displacements of the panel nodes. The second
row of Eq. S1 incorporates panel bending stiffness where eachelement in the diagonal matrix K B corresponds to the bending
stiffness of an individual panel.
We assume that the in-plane stiffness of the thin sheet (e.g.,paper) is high enough to prevent bending and buckling at theedge connecting two panels (i.e., at the fold line of a thin sheet).
The bending energy of thin sheets increases when the edges of the
sheet are restrained. In this case, tensile forces develop over the
sheet’s surface, and flexural deformations become restricted to asmall area focused at the bending ridge (i.e., the diagonal 1–3 in
Fig. S1 D–F) (1, 2, 4). This phenomenon occurs with large displacements and the elastic energy of the panel bending scales
approximately as kðL2 =tÞ1=3, where k is the bending modulus of
the sheet, defined as k = Et3 =12ð1 − ν2 Þ (1). In our model wedefine the panel stiffness as 1=3
Et3
L2,
KB = CB12ð1 − ν2 Þ t[S7]which incorporates the nonlinear effect of the L2 =t ratio in the
restricted bending of a thin sheet. We use a finite-element model(discussed in detail later) to show thin sheet bending for small
and for large displacement cases. We define a thin sheet with along diagonal L1 = 1 and a short diagonal of L2 = 0.8. The thinpanel is restrained at the edges by thin sheets in a perpendicularorientation (similar to typical origami). The structure is restrained in the vertical direction at three corners and is subjected
to bending with a downward force F at the fourth corner (Fig.
S1E). When a small force is applied (F = 0.01), the structure
remains in the small displacement regime and the thin sheet
experiences curvature along both diagonals (Fig. S1F, Top).
For a larger force (F = 1.0) the structure experiences large displacements, and curvature occurs in the center of the L1 diagonal, creating a bending ridge along the L2 diagonal (Fig. S1F,
Bottom). As discussed in previous literature (1, 4) the large displacement case has a higher stiffness and the rotational hinge
formulation in Eq. S7 would give a realistic result with a factor of
CB = 0.794. Here we restrict ourselves to small displacements,and so we use CB = 0.441 for all analyses. The stiffness coefficient
CB has not been investigated in detail yet. A more comprehensive formulation would incorporate a nonlinear scaling associated with the dihedral bending angle θ into Eq. S7 that would
scale stiffness between small and large displacement cases. For
example, ref. 1 suggests that the total bending energy (and thus
stiffness) of the restricted sheet should scale with θ7=3 vs. θ2 for alinear rotational hinge.
Filipov et al. www.pnas.org/cgi/content/short/1509465112

S1.3) Folds Bending. Folds are modeled in a similar fashion to thebending of panels. Realistic origami behavior does not allow forout-of-plane displacements along fold lines due to the restrictive
nature of the perpendicularly oriented sheets. Thus, it is sufficient
to use a simplified approach: modeling the origami fold as arotational hinge along an edge. A schematic of the fold model
contains a fold spanning nodes 2 and 3 connecting two panels (1–
2–3–4 and 2–5–6–3) (Fig. S1G). The length of the fold is LF , andthe stiffness is expected to scale linearly with this length, because
curvature and bending energy are expected to exist only on theinfinitesimally small width of the fold. The same constraint formulation presented for the panel bending is used to formulate
two independent fold elements from the two vector sets: (i) a, b,
and c and (ii) −a, d, and e. This approach distributes the stiffnessof the fold to all relevant nodes on the two adjacent panel elements. The initial fold angle, θ0, is different for different folds on
the Miura-ori sheet and tube and can be calculated using basic
geometric relations (7, 25, 27) for each chosen configuration(percentage of extension). Here, the angle θ represents a rotation away from the initial folded configuration.
We introduce a parameter RFP to relate the stiffness between
the bending of a fold with length LF = 1 and the bending of apanel with a diagonal of L2 = 1. Structures consisting of thickpanels with near frictionless hinges (18, 19) could have a low RFP(e.g., 10−4); structures where perforation or plastic deformation
is used to facilitate creasing (34, 35) could have a medium RFP(e.g., 0.1); whereas origami created with additive manufacturing(40), or where creasing is near elastic (36), could have a RFP = 1
and perhaps higher if movement at folds is intentionally restricted. The interaction between bending of flat material andfolding at fold lines as well as highly nonlinear folding behaviorsof origami are currently being studied (33–36). Our simplifications assume that only linear elastic deformations occur at thefolds. For the analyses in Figs. 3 and 4 and SI Text, we assume
that the folds are less stiff than the panels, and we use an arbitrary choice of RFP = 1=10 based on visual observations of ourphysical models. For the cellular assemblage in Fig. 5, we use
RFP = 1 to simulate folds that are as stiff as the panels, because
this is likely the case for continuous 3D printed structures. In SI
Text, section S3 we show that RFP influences the magnitude of
eigenvalues associated with fold bending; however, it does not
influence the overall behavior and the qualitative results presented in this paper. For our implementation, the stiffness foreach of the twofold elements (i.e., a, b, c and −a, d, e) is calculated similarly to Eq. S7, as
KF = RFP 1=3
LF
Et3
1
CB
,
2
12ð1 − ν2 Þ t[S8]except here the function scales linearly with the length LF . Morecomprehensive future formulations may incorporate nonlinear
scaling associated with material properties, the angle θ, the angle
θ0, and other fold-related variables (33–36).
S2) Coupling of Zipper Tube Structures
In the computer model, the coupling of the zipper structures isperformed by inserting coupling elements that restrict relative
movement between the adjacent panels of the two tubes. These
coupling elements can be thought of as an adhesive joint between
the adjacent faces of the tube. The relative local coordinates (X′and Z′) are different for the odd and even panels of the structure(Fig. S2). The X′ coupling elements restrain relative movement
between panels along the length of the tube (relative to the local
X′ axes), and the Y ′ coupling elements restrain differential orthogonal movement in the Y ′(=Y) axis. The X′ and Y ′ couplingelements are formulated using a compatibility relation where bar
2 of 13elements are used to restrain relative movement between two
nodes. The stiffness for the coupling elements is defined as
KX′ = CX′

EA
E0.5t
= CX′
,
L
1

[S9]

KY ′ = CY ′

EA
E0.5t
= CY ′
,
L
1

[S10]

and

where E is Young’s modulus, t is the thickness of the thin sheet,and the coupling coefficients (CX′ and CY ′) can be used to vary
the stiffness of the coupling elements. When these coupling coefficients are set to 1, each coupling element has the same axial
stiffness as a 0.5-unit-wide, 1-unit-long, and t-thick piece of panel
material. Finally, Z′ coupling elements are used to prevent nodes
lying on the same plane (on a panel) from moving out-of-plane(in the relative Z′ direction). The Z′ coupling elements are defined in the same fashion as the out-of-plane rotational hinges
for panel and fold bending. Here, we consider a node on one of
the zipper tubes that overlaps a panel on the opposite tube, indicated as a red/white triangle in Fig. S2C. A rotational hinge
element is constructed that restricts out-of-plane movement between each of the overlapping nodes and the three corresponding nodes on the opposing tube. For clarity, only two sample
corresponding node sets are illustrated in Fig. S2C. The vectors
groups (a, b, and c) and (d, e, and f) can be used to define therotational hinge for each set. The stiffness for each of theserotational hinges is defined as 1=3Et3
1
KZ′ = CZ′ CB,
12ð1 − ν2 Þ t[S11]where the parameters are the same as the panel definition in Eq.
S7, and the coupling coefficient CZ′ can be used to vary the stiffness of the coupling elements. With this formulation, the stiffnessof each Z′ coupling element is equal to the bending stiffness of apanel element with a diagonal length, L2 = 1. Because the equation for KZ′ is based on bending of the thin sheet, the value of thisstiffness is substantially lower than that of the X′ and Y ′ couplingelements.
The sensitivity of the model eigenvalues vs. the magnitude ofeach of the coupling coefficients is explored for the zippercoupled tubes deployed to 70% extension (Fig. S2). The rigid
folding mode (λ7) of the coupled tube is not affected by any of
the coupling elements; i.e., it is neither easier nor harder to fold
and unfold the structure due to the coupling. When the coupling
coefficient CX′ is substantially reduced (lower than 10−2), theeighth and subsequent eigenvalues experience a drop in magnitude from ∼1,200 to ∼500. This drop occurs because the tubes
are not restrained in the relative X′ direction, and thus the eighthmode switches to a deformation mode in which the tubes begin
to separate. Due to the presence of the Z′ coupling elements, themagnitude of this eigenvalue remains relatively high even when
CX′ = 10−5 (λ8 = 560 vs. λ8 = 20.7 for a single tube). When decreasing the value of the CY ′ coupling coefficient, there is a moresignificant effect on λ8 and the subsequent eigenvalues. The eigenvalue corresponding to bending reduces to λ8 = 590, when
CY ′ = 10−3, and experiences a mode switch for lower coupling
coefficients. When CY ′ is substantially reduced, the tubes are free
to slide apart in the Y direction. The CZ′ coupling coefficients, on
the other hand, have no influence on any of the eigenvalues. Thisis due to the zig-zag geometry of the tubes, which causes the axial
members to restrain global out-of-plane movement between the
two tubes. When any of the coupling coefficients are increased
Filipov et al. www.pnas.org/cgi/content/short/1509465112

past a value of 1, there is negligible increase in the eighth andsubsequent eigenvalues of the structure.
When changing the value of the vertex angle α or the extended
configuration of the zipper-coupled tubes, there is little change
in the sensitivity of the different coupling elements, and thegeneral trends remain. The tubes are effectively coupled when
the coupling coefficients are about equal to 1. The X′ and Y ′coupling elements have a higher influence on the coupling andon maintaining the relatively large bandgap (β = λ8 − λ7), whereas
the Z′ coupling elements have little influence on the eigenvalues.
To better understand why the zipper coupling results in thelarge bandgap increase, we investigate how the squeezing deformation mode is restrained by the new geometry. We perform
an eigenvalue analysis on two tubes arranged in a zipper-couplingfashion, when the stiffness of all coupling elements is substantiallyreduced (CX′ = CY ′ = CZ′ = 0.0005). In this scenario the tubes cansimultaneously undergo the squeezing deformation mode,
where the tubes fold on the left side and unfold on the right (Fig.
S3 A–C). However, this motion is incompatible for an effectively
coupled zipper system. On the left side, the first vertex of thebottom tube moves downward (point I in Fig. S3A), whereas thefirst vertex on the top tube moves upward (point II in Fig. S3B)
and vice versa on the right side. This transverse motion between
the two tubes can be quantified by tracking the distance between
adjacent panel-edge center points on the two tubes (Fig. S3D). In
an undeformed (or effectively coupled) system the distance between adjacent edge center points is uniform at 0.7 units. The
squeezing of the loosely coupled system results in separation on
the left side (distance increases up to 0.9 units) and a closing on
the right side (distance decrease down to 0.5 units). In an effectively coupled zipper system these in-plane motions are restrained,
and it would be necessary to stretch and shear the thin sheet toachieve a squeezing-type deformation.
S3) Sensitivity of Model and Analysis
In Fig. S4 we show differences in scaling of eigenvalues, for the
tube and zipper-coupled tubes, with respect to different model
parameters. The seventh eigenvalue for both the single tube andthe zipper-coupled tubes corresponds to the rigid folding mode,in which deformation primarily occurs as bending of the prescribed folds (Fig. 3 and Fig. S7). The eighth mode for the single
tube corresponds to squeezing, in which bending occurs in thefold and panel elements. However, the eighth mode for thezipper-coupled tubes requires stretching and shearing of the thinsheet, which require much more energy than bending and resultin drastic differences for the scaling of eigenvalues.
All eigenvalues (λ7 and λ8 for both systems) scale proportionally with Young’s modulus E (i.e., doubling E doubles the
eigenvalue) and inversely proportionally with the material density ρ (Fig. S4 A, B, E, and F). This is expected because E scales
the stiffness proportionally, and ρ scales the mass proportionallyin equation Kvi = λi Mvi. Scaling of other parameters, however,does not necessarily influence all eigenvalues proportionally.
When changing the sheet’s thickness, t, its stretching/shearing
stiffness and the system’s mass scale proportionally. On the other
hand, the bending stiffness for both the panels and the folds
scales as 8=3 based on Eqs. S7 and S8. Therefore, when scaling
the thickness, the eigenvalues scale as 5=3 for the rigid folding
modes (λ7) of both tubes, and the squeezing mode (λ8) of thesingle tube (Fig. S4 C and G). The eighth eigenvalue for thezipper-coupled tubes does not change because both mass andstretching/shear stiffness scale proportionally (both are definedby t). When scaling the fold stiffness ratio, RFP, λ7 scales proportionally for both structures, λ8 scales as 0.8 for the single tube,
and λ8 remains constant for the zipper tubes (Fig. S4 D and H).
The system behaviors that lead to these scaling relations arecomplemented by the energy distributions shown in Fig. S7.
3 of 13

In summary, Young’s Modulus E and material density ρ directly scale all system eigenvalues, whereas the material thickness t and fold stiffness ratio RFP scale eigenvalues influenced bypanel and fold bending. The scaling of different parameters does
not cause mode switching and the order of eigenmodes remains
the same as shown in Fig. 3. Scaling different parameters canchange the quantitative results (e.g., magnitude of λ7); however,it does not influence the qualitative results presented here. The
eighth eigenvalue for the zipper-coupled tubes remains substantially higher than the seventh, ensuring a large bandgap(β = λ8 − λ7) regardless of the parameter values used. The resultspresented in this paper are independent of length scale, making
the zipper-coupled tubes applicable across scales.
S4) Model Verification with Finite-Element Analysis
In this section, we verify and explore the accuracy of the bar andhinge model by comparing the results from the bar and hingemodel to those of a shell model created using the commercial
finite-element (FE) analysis software ABAQUS (41). Details of
the FE implementation are shown in Fig. S5A for a single
Miura-ori cell. For the FE model, we discretize each panel into
D segments in each direction, such that each panel will now bemodeled using D2 shell elements (D = 8 for Fig. S5B and for all
FE analyses herein). General purpose, four-node shell elementswith finite membrane strains are used (S4 elements) and areconnected with one node at each corner. The fully 3D FE
model contains 6 DOFs at each node (3 displacements and 3
rotations). Mass in the model is distributed based on the volume and the density, ρ, of the shell elements. All model parameters are defined to be the same as those for the bar andhinge model.
At the fold lines, collocated (overlapping) nodes are placedwith one node connected to the shell elements of each adjoining
panel. Each set of collocated nodes, shown in Fig. S5A, Inset, isconnected with an element that restricts the nodes to the same
XYZ coordinates throughout the analysis. At the same location,
the fold stiffness is modeled as an elastic rotational spring that
resists rotation between the adjacent panels. The spring is placed
locally on each pair of collocated nodes and resists rotations
about the local a vector shown in Fig. S5A, Inset. The stiffness ofthe fold is based on the dimensions of connected shell elements
and is defined asðKF ÞFE =ðLS1 + LS2 Þ=2
KF ðLF Þ,
LF

[S12]

where LS1 and LS2 are the lengths of the connected shell element
parallel to the fold line. The parameter LF denotes the length ofthe entire fold, and the equation KF is a function of stiffness (i.e.,
Eq. S8) based on the fold length and the same material properties as before. Eq. S12 distributes the stiffness of the fold (ascalculated for the bar and hinge model) based on the tributary
length of the shell elements used. Collocated nodes that are at
the end of a fold (i.e., at a vertex) will have only one adjacentshell element and thus the stiffness will be based only on LS1 =2or on LS2 =2.
Discretized FE models of the sheet, the single tube, and zippercoupled tubes are created with D = 8 and a natural eigenvalue
extraction is performed. The eigenvalue spectra calculated with
the FE model and the bar and hinge model are presented in Fig.
S6. The seventh through ninth eigenmodes for the FE model
match those of the bar and hinge model when considering
the global deformation of the structure (Fig. 3 D–F and Fig. S6
D–F), which we expect because these deformation modes correspond to global behaviors of the structure such as bending andrigid folding. The magnitudes of the eigenvalues follow similar
trends between the two models but do not always match exactly.
Filipov et al. www.pnas.org/cgi/content/short/1509465112

For example, with the single tube, λ7 and λ8 match well because
the fold stiffness between the two models is the same (i.e., Eqs.
S8 and S12), and in the FE model, panel bending occurs globallyon the diagonal (as predicted for Eq. S7). However, the magnitude of the eigenvalues λ9 and λ10 for the tube is lower with the
FE model. The difference in magnitude can be attributed to the
localized deformations and stress concentrations that can be
captured with the FE discretization, but cannot be captured
using the bar and hinge model. For example, in the ninth mode
the Miura-ori cell in the middle of the tube experiences bending,stretching, and shear localized at the vertex. The FE model cancapture localized effects and individual shell elements can deform more than others. On the other hand, in the bar and hingemodel stretching, shear, and bending can be captured globally
only over the entire panel.
We also use the FE model to perform energy analyses for theMiura-ori sheet, the single tube, and the zipper-coupled tubes(Fig. S7). The energies are calculated based on the structuraldeformation for the normalized mode shapes (Fig. S6 D–F). The
percentage of distribution between the different sets of element
deformations (i.e., fold bending, panel bending, and panel
stretching/shear) and the total energy for each mode are shown.
The total energy for eigenvalues of the single sheet is relatively
low, because deformations consist of localized bending in panels
and folds. On the other hand, the eighth and ninth modes of thezipper-coupled tubes require much more energy, because thethin sheets are engaged in stretching and shearing. The rigid
folding modes, ninth for the sheet and seventh for the tube
structures, primarily engage the fold lines in bending as previously expected.
S5) Aligned and Internally Coupled Tubes
The eigenvalue vs. configuration plots and representative eigenmodes at 70% extension are illustrated for the aligned-, internally, and zipper-coupled tubes in Fig. S8. The alignedcoupled tubes are constructed by translating a tube and attaching
it to the initial tube without rotation (Fig. 2E). The tube could bea copy of the original geometry or could have different parameters a and/or α. For internally coupled tubes, the geometries of
the internal and external tubes have to conform to preserve flatand rigid foldability as discussed in ref. 7. Here, we define theinternal tube geometry, using a parameter IExt that describes the
maximum extension of the internal tube with respect to the external tube. For example, if we take an internal tube defined with
IExt = 0.8, then the internal tube will reach 100% extension (lieflat in the X − Y) plane, when the external tube reaches 80% of
its maximum extension. The internal tube geometry (aI , cI , andαI ) is calculated froma cosðαÞaI = qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ,
1 − ðIExt Þ2 sinðαÞ2cI = IExt c,

[S13]

and
αI = sin−1 ðIExt sinðαÞÞ,[S14]where a, c, and α define the external tube geometry. The internally coupled tubes discussed in this paper (Figs. 2F and 4 andFigs. S8 and S9) use IExt = 0.8 and the conforming Miura-ori cellgeometry is therefore aI = 0.759, cI = 0.8, and αI = 40.94°. Computational modeling of the connectivity required to createaligned and internally coupled tubes is much simpler to formulate than that of the zipper-coupled tubes. Nodes on adjacentfaces and edges of each tube are conveniently located at thesame coordinates; therefore, the same DOFs are used for bothtubes, and only one of the colocated nodes is used in the model.
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This modeling approach effectively connects the two systems but
provides more stiffness than the coupling element approach used
for the zipper-coupled tubes.
The eigenvalues λ7 and λ8 for the aligned-coupled tubes (Fig.
S8A) are almost identical to those of the single tube (Fig. 3B),
because coupling tubes in this fashion doubles the system’s mass,and simultaneously adding an identical set of elements to thesystem doubles the stiffness. The seventh and eighth modeshapes are identical to the single tube and are not restricted bythis form of coupling (Fig. 3E and Fig. S8D). The magnitude of
the ninth eigenvalue increases slightly, which can be attributed tothe aligned coupling.
Coupling tubes internally (Fig. 2F) has more of an effect on their
behavior; however, λ7 and λ8 still correspond to the rigid folding
and squeezing modes, respectively, and the bandgap is not substantially increased (Fig. S8 B and E). When approaching an extension of 80% (with respect to the external tube), the internal
tube becomes flat, and the system essentially locks. At these larger
extensions (60–80%), the eigenvalues λ7 and λ8 increase significantly, meaning that the system is substantially stiffer. When theinternal tube is flattened, folding the structure (rigid folding mode)or deforming it in any other mode becomes more difficult. Withother coupling orientations (i.e., zipper or aligned), the eigenvalues

drop rapidly at large extensions. In those cases, the entire structure
becomes essentially flat (in the X − Y plane), and it becomes easy
to bend the system out-of-plane. In contrast to the zipper-coupledtubes, neither the aligned-coupled tubes nor the internally coupledtubes show a substantial increase in the bandgap (β = λ8 − λ7) (Fig.
S8 A–C). However, multiple coupling orientations can be integrated into a single system to take advantage of the different
behaviors. For example, a combined zipper-coupled and internallycoupled system would not experience squeezing and it would lockat a specified configuration.
S6) Self-Interlocking Structure
Each side of the self-interlocking structure (Fig. 6C) can becomposed of any number of zipper-coupled tubes together (ofsame α). The structure can be any radially symmetric shape withn sides, as long as there is no self-intersection. The structure willinterlock when an angle (γ) between the two faces on the Y − Zcross section of the Miura-ori cell is at γ = 360°=n. Thus, thestructure will interlock at an extension ofrﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ.
1 − cosðγ=2Þ2 sinðαÞ2Ext = 100%.
[S15]
sinðγ=2Þ

Fig. S1. Schematics and tests of the bar and hinge model. (A) Schematic of indeterminate bar frame for simulating in-plane panel stretching and shearing.
(B) Tensile patch test of bar frame exhibiting isotropic behavior. (C) Shear patch test of bar frame. (D) Schematic orientation of rotational hinge simulating
panel bending along the diagonal L2. (E) FE model used to study bending of thin sheet restrained at edges by orthogonally placed thin sheets. (F) Curvature
along the diagonals of the FE model for a small displacement (F = 0.01) and a large displacement (F = 1) case. (G) Schematic of the two rotational hinges used tosimulate bending along a prescribed fold line of length LF . (H) The three components of the bar and hinge model shown on the single tube.

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A

B

C

e
f

d

Y’
Z’

b

X’

a

c

Y’
Z’
X’

E

D
10

Eigenvalue

F

3

3

Mode 8 λ8=1136

3

10

10

Mode 8 λ8=590

λMode 8 λ8=1193
2

2

10

10

Mode 8 λ8= 562

10

λ9

2

10

Mode 8 λ8= 10.4

λ

8

λ

7

1

1

1

10

10

−4

10

−2

10

2

1

10

4

10

Coupling coefficient - CX’

10
−4

10

−2

10

1

2

10

−4

4

10

10

Coupling coefficient - CY’

−2

2

1

10

10

4

10

Coupling coefficient - CZ’

Fig. S2. Coupling elements used to connect zipper tubes in the bar and hinge model. (A–C) Configuration of X′, Y′, and Z′ coupling elements, respectively, fora segment of zipper-coupled tubes. The local axes are shown for the leftmost and rightmost panels of A. Only representative coupling elements are shown for
the Z′ coupling. (D–F) The sensitivity of the system eigenvalues vs. magnitude of each of the coupling coefficients CX′, CY ′, and CZ′, respectively, when the othercoupling coefficients remain at a value of 1.

C

A

11

1
6

Y

Y

B

II

Y
X

X

D

Change incenter distance

X

I

0.2
0

7
1

2

3

4

5

8

9 10 116- 0.2

Edge number

Fig. S3. Squeezing deformation of zipper-coupled tubes when the coupling coefficients are reduced to CX′ = CY′ = CZ′ = 0.0005. (A) Top view of only the
deformed bottom tube with an outline of its initial configuration. The x marks indicate the panel-edge center points on the side of the tube that is attached.
(B) Only the deformed top tube. (C) The squeezing deformation of the zipper-coupled assembly with low coupling. (D) Change in the distance between thepanel-edge center points of the two tubes. Initial distance before deformation is 0.7 units for all locations; the squeezing results in separation on the left side
and closing on the right side of the coupled tubes.

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Zipper Tubes

Single Tube
102

B

A

D

C

λ7

101
1

100

-1

1

1 23

10-1
104

F

E

λ8

1
102
1

H

G
-1

≈0

-1

1 23

100
105
106
107
Elastic modulus ( E )

0.1

≈0

≈0.8

0.1
0.001
0.01
1
0.1 0.01
Fold stiffness ratio ( RFP )Thickness ( t )1
10
Density ( ρ )Fig. S4. Eigenvalue sensitivity for a single tube and the zipper-coupled tubes. (A–D) Scaling of λ7 with respect to E, ρ, t, and RFP . (E–H) Scaling of λ8 for the samecases. These relations directly govern the scaling sensitivity of the bandgap.

Shell

A

B
a

LS1

b
LS2

LF

Fold joint0

Fig. S5. FE model used for verification. (A) Miura-ori cell discretized with D = 5 elements for each panel; Inset shows local zero-length connectivity at fold lines.
(B) Discretization of single tube with D = 8.

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Bar and hinge model:
FE model:

λ8
λ8

λ7
λ7

λ9
λ9

λ10
λ10

Bandgap β = (λ8 - λ7)

C

B

A

3

Eigenvalue

10

2

10

1

10

0

10

Mode 7

0

25

100

0

λ7=1.20

D

25

E

50
75
% extension

100

0

25

50
75
% extension100

λ7=2.84F

λ7=2.80

Y

Z

Y
X

Mode 850
75% extensionX
λ8=1.32X

λ8=12.5λ8=649Z
λ9=1.90

Mode 9X

λ9=80.8

λ9= 708

Y
X

Fig. S6. Eigenvalue analyses of a sheet, a single tube, and a zipper-coupled tube performed with the FE model. (A–C) Eigenvalue vs. extension for the threestructures, performed with both the FE and the bar and hinge models. (D–F) Seventh, eighth, and ninth deformation eigenmodes when structures are at 70%
extension (undeformed outline in red).

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Fig. S7. (A–C) Eigenmode energies of the sheet, the single tube, and the zipper-coupled tubes, respectively, using FE analysis. Energy distributions (as apercentage and as a total amount) are presented in stacked bar graphs indicating energy in the different sets of components [i.e., fold bending (F), panel
bending (B), and panel stretching (S)]. The energy within the structure is shown by shading of elements to show the concentrations of energy. The color scale
indicates the magnitude of energy.

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Fig. S8. Eigenvalue comparison for the different coupling assemblies. (A–C) Eigenvalue vs. extension for the aligned-, internally, and zipper-coupled structures, respectively. (D–F) Seventh, eighth, and ninth deformation eigenmodes when structures are at 70% of the maximum extension length (undeformed
outline in red).

Fixed endA
Z

Distributed
load

Y

B

Z

C

Y

Z

X

Y

X

X

Node constrained in X, Y, and Z

D

Applied load = 1/4

Applied load = 1/8

E

F

Z
Z Z
Y

Z Z
Y

Y

Nodes constrained in:GY

X, Y, and Z

X and ZY

Mechanism

X only

I

10%

10%

80%

80%

70%

10%
50%
Z
70%

Z
97%

Z

Y

H

Z
70%

Y

97%

Y

79%

Y

Fig. S9. Support and loading conditions of origami cantilever structures. (A–C) Initial configurations for the zipper-, aligned-, and internally coupled structures, respectively. (D–F) Support conditions (Left) and loading conditions (Right) used in the analyses of the three cantilever systems in Fig. 4. The total load of
1 is distributed by placing one-eighth of the load at each of the eight nodes, whereas double the load (one-quarter) is placed on colocated nodes occurring in
the aligned- and internally coupled systems. (G–I) Support conditions (Left) for the cantilevers, if a mechanism is installed to deploy the system, and cantileverconfigurations (Right) for different levels of system extensions/contractions.

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Table S1. Geometric definitions and characteristics of coupling variations shown in Figs. 6 and 7
Structure

Number and geometries of tubes

Canopy structure (Fig. 6A)

Thirty-two alternating tubes with α varying
between 58° and 84°. All tubes witha = c = 0.3 m and N = 16.

Bridge (Fig. 6B)

Two tubes with α = 55° (yellow) and six with
α = 85° (green). All tubes with a = c = 25 mm
and N = 5.

Self-interlocking
structure (Fig. 6C)

Twelve tubes with a = c = 25 mm, α = 75°,
and N = 5 (red).

Coupled polygonal
tube (Fig. 7A)

Polygonal tube N = 5: cross section with four
panels of a = c = 25 mm, α = 55° (yellow) andtwo panels of a = 19 mm, c = 20 mm, α = 40.94°(blue). One tube has a = c = 25 mm, α = 55°(brown).

Zipper-coupled tubes
with thickness (Fig. 7B)

Two tubes with a = 80 mm, c = 40 mm, α = 75°,
and N = 4. The thickness is t = 5 mm on thethin part and t = 10 mm on the thick partof the panels.

Actuator (Fig. 7C)

One tube of N = 6 and one of N = 4 cells, witha = c = 25 mm, α = 55° (orange). The rigid
boundary is a bounded paperbox (blue).

Coupling orientation and discussion
All tubes are coupled in the zipper orientation.
The vertex angles α are calculated so the overall
cross section follows a smooth planar curve (e.g.,
α = 58°, 84°, 60°, 83°, 62°, 82° for the first six tubes).
This structure covers an area of 8.1 × 9.3 m with a
2.6-m rise when deployed to 97% extension and can folddown to a size of 5.1 × 0.8 × 1.3 m at 5% extension.
Zipper coupling is continued in one direction only, where
each tube is coupled on two opposite faces. The systemremains rigid and flat foldable in two directions. The
structure has a high out-of-plane stiffness.
Zipper coupling is continued in one direction for each sideof the structure. The tubes at the corners have zipper
coupling in orthogonal directions (i.e., on two adjacentfaces of the tube). The system is flat foldable in one
direction and interlocks into a stiff conforming assemblyat an extension of 96.3%.
Zipper coupling between the polygonal and the regularsquare tube can be done on any of the polygonal tube
faces. The internal (blue) panels of the polygonal tube
are defined to reach a flat configuration when external
panels (yellow) are at 80% extension. The structure is flat
foldable only in one direction, but the internal folds(on blue) can change their polarity and can reconfigure
the overall structural shape (Movie S2).
A technique of cutting out material (18) is adopted,
allowing the structure to fold down completely tothe minimal feasible thickness of 2Nt = 80 mm. The
structure cannot extend to 100% extension becauseintersection of the thick material will occur.
Zipper coupling along the middle section of the longer tube,
while its ends are adhered to stiff boundaries. The
zipper-coupled section prevents local and squeezing-type
deformations. At the ends, the longer tube canundergo squeezing, allowing for the rigid attachment.

Movie S1. Zipper-coupled tubes. Shown are assembly from two single tubes, flat folding, eigenmodes, and system actuation from one end.
Movie S1

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Movie S2. Zipper-coupled origami assemblages. Shown are: a bridge structure rigid deployment and loading with 2.5 kg; a self-interlocking structure rigid
deployment and interlocking; and a coupled polygonal tube where rigid motion is used to reconfigure the polygonal tube between two different shapes.
Movie S2

Movie S3.

Folding sequence of a cellular origami metamaterial consisting of zipper and aligned tubes.

Movie S3

Movie S4.

Folding sequence of an architectural canopy structure created from geometrically varied zipper-coupled tubes.

Movie S4

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Movie S5. Miura-ori sheet and a single tube. Shown are flat folding and eigenmodes of the sheet, assembly of a tube from two sheets, eigenmodes of a tube,
and system actuation of the tube from one end.
Movie S5

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