JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASSRIGID-FOLDABLE CYLINDERS AND CELLSTomohiro TACHI and Koryo MIURA
Assistant Professor, The University of Tokyo, 3-8-1 Komaba, Meguro-Ku, Tokyo 153-8902, Japan, tachi@idea.c.u-tokyo.ac.jp
Professor Emeritus, The University of Tokyo, Japan, miurak@gakushikai.jp
Editor’s Note: Manuscript submitted 1 January 2012; revisions received 4 August and 16 October 2012; accepted 28
October. This paper is open for written discussion, which should be submitted to the IASS Secretariat no later than June 2013.

ABSTRACTIn this paper, we present newly explored families of rigid-foldable cylinders and the cellular structures
constructed from these cylinders; the families include zonogon extrusion cells, bi-directionally flat-foldable cells,
and a novel type of cells, i.e., woven cylinder cells. We show the geometry of such structures to demonstrate theirvalidity, their parametric design method, and their kinetic behaviors. These types of structures exhibit
continuous rigid-foldability as well as flat-foldability in one or two directions; further, they have different kinetic
properties that are potentially applicable for different purposes. The newly proposed woven cylinder cellular
structure is a bi-directionally flat-foldable one-DOF rigid-foldable structure and has a distinctive geometricproperty: structural stiffness against compression in one of three directions.
Keywords: origami, deployable structure, rigid-foldable structure, cellular structure, honeycomb.

1. INTRODUCTION
Collapsible cylindrical surfaces are useful invarious designs, as such deployable structures canform a close-to-watertight surface that encloses acertain desired volume, and the surfaces can becompactly folded into a two-dimensional state.
Several flat-foldable cylindrical deployablestructures have been proposed thus far, e.g., by
Hoberman [1], Guest and Pellegrino [2], Sogame
and Furuya [3], Nojima [4], Kuribayashi et al.[5],and Wu et al. [6]. However, the deployment ofthese existing tubes rely on the in-plane elastic
deformation or the traveling of hinges (Figure 1),
and the structures cannot be applied to large-scale
structures or robust and reusable mechanisms,
including kinetic architecture, radiation shields forspace structures, and bellows. This is because thestructures do not have a geometrically valid
continuous transformation path from one state toanother.
In order to solve this geometric problem, we
focused on rigid-foldable structures, which arepolyhedral surfaces that realize continuous
deployment mechanisms from one state to anotherif their facets and foldlines are substituted with rigid
panels and hinges, respectively. One example of arigid-foldable structure is a developable double
corrugation surface or Miura-ori [7] (Figure 2).

Variations of such rigid-foldable structures that are
homeomorphic to disks can be designed by only
considering local (i.e., per vertex) fold angle
conditions. However, in order to design cylindricalmechanisms, we need to solve global closure
conditions, for which consideration of thesymmetry is essential. Figure 3 shows a rigidfoldable disk that cannot form a valid rigid-foldable
tube.
We have proposed several types of continuously
foldable designs of cylindrical structures—thus,
rigid-foldable
cylinders—witha
one-DOFmechanism [8][9]. We have found that some ofthese cylinders tessellate a 3D space without
destroying continuous transformability. This leads
to the concept of families of rigid-foldable
collapsible cellular structures. Rigid-foldable
cellular structures have not been extensively
investigated from theoretical and geometrical
viewpoints; however, individual examples such as acellular structure by Hoberman [10] and an origami
model by Yenn [11] are known.
In this paper, we review the geometry of thesefamilies of collapsible cylindrical and cellular
structures. We show the validity of the structures,
their generative design method, and their kinetic
behaviors. We then propose a novel type ofstructures called woven cylinder structures. Woven

217

Vol. 53 (2012) No. 4 December n. 174

cylinder structures are bi-directional flat-foldable
one-DOF rigid-foldable structures similar to otherrigid-foldable cells. In addition, they have thedistinctive geometric property of being stiff against
compression in one of three directions.
In Section 2, we show the basic transformable tube
structure and then generalize it for the zonogon
extrusion type (Section 3, [8]) and bi-directionallyflat-foldable types (Section 4, [9,12]). In Section 5,
we show the novel cellular structure type
constructed from the basic tube structure. In Section
6, we compare the kinetic and static behaviors ofthese structures and propose possible applicationsof the cylindrical and cellular structures.

eight parallelogram panels. The module isconstructed by mirroring a single Miura-ori vertex
with four parallelograms with respect to the zxplane (Figure 2). Observe the boundary curves ofthe Miura-ori vertex (shown in dotted lines). Since
the vertex is constructed from congruent
parallelograms, the curves are planar and areparallel to each other throughout the transformation.
The end loops of this module (shown in thick lines)
are always each planar and parallel to each other.
Therefore, the module can be packed in the z
direction to form a tube, as shown in Figure 5.
Furthermore, the unit tube can be assembled totessellate 3D space without gaps because of its
translational symmetry. This symmetric behavior iscompatible with the folding motion; thus, thetessellated structure can also validly transform.

Figure 1. Conventional bellows. Note that the creases
traverse the surface as the structure folds.
Figure 4. Tubular module.

Figure 2. Miura-ori is a one-DOF rigid-folding
mechanism homeomorphic to a disk.

Figure 5. Constructed tube.

2.2 Kinetic Behavior
Figure 3. A tube constructed from a rigid-foldable disk
does not generally transform.

2. BASIC FORM
2.1 Construction
In order to construct valid variations of rigidfoldable cylinders, we start from a very simple
rigid-foldable tubular module: a polyhedron with218

As shown in Figure 4, we denote the dihedral
angles between adjacent parallelograms by ξ and Ξ.
We denote the half of the angles between adjacentedges by θ and Θ, where θ and Θ vary from 0 to θ0.
Here, θ0 is an intrinsic parameter defining the shapeof the parallelogram; thus, it is constant withrespect to the folding motion. These four angle
parameters ξ, Ξ, θ, and Θ are related by thefollowing three equations to form a one-DOF
mechanism.

JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES: J. IASStan  tan0 cos 2 , sin   sin0 sin ,2 sin  sin sin
0

2

(1)

Therefore, the dimensions of the module can be
written as the functions of a single angle parameter
ξ.
x0

,
 x  x0 cos 22 1  tan  0 cos
2



1

y0
cos
cos 0
2
sin 


,
(2)
 y  y0sin  022 
1  tan  0 cos2 z  z 0 sin 2Equation 2 shows that the dimensions in the x, y,
and z directions are monotonous functions withrespect to the change in the folding angle. Further,
the structure is bi-directionally flat-foldable in boththe y and z directions. The folding motion in thecase
2 arctan √2⁄2 is shown in Figure 6.
Since the tubes are packed without rotation, thekinetic behavior of the cellular structure is
proportional to the original module with eightparallelograms and thus also satisfies Equation 2.

A basic cylinder with eight parallelograms is
generated by a zigzagged extrusion of a section
parallelogram (Figure 5). The section curve can begeneralized to an arbitrary zonogon while
maintaining the rigid-foldability. The construction
procedure (Figure 7) is as follows.
1.

Draw an arbitrary closed polyline with twofold rotational symmetry, i.e., a zonogon, on
the section plane (the xy plane). We call thiszonogon a “section zonogon.”

2.

Make an arbitrary vector on the zx plane,
which forms an angle of ξ0/2 < 90° with thesection plane. We call this vector the“extrusion vector” and denote it by l0 =(ℓcosξ0/2, 0, −ℓsinξ0/2), where the vector has alength of ℓ.

3.

Extrude the section zonogon by the extrusion
vector to form a closed parallelogram strip.

4.

Construct a zigzagged cylinder by sequentially
mirroring the strip by plane, including thetranslated section zonogon.

Figure 7. Zigzagged extrusion of a zonogon produces avalid rigid-foldable cylinder.

3.2 Validity
The validity of the transformation can be confirmed
as follows. We pick up the extrusion of the half ofthe section zonogon along the extrusion vector
forming an open parallelogram strip.

Figure 6. Folding motion of the eight-parallelogram
module of2√2⁄2 .

3. ZONOGON EXTRUSION TYPE
3.1 Geometric Construction(1) When we fold the strip along its edges, it isensured that the edges, i.e., the extruded
vertices, remain parallel after folding. We
represent these parallel edges by a rotatedextrusion vector l = (ℓcosξ/2, 0, −ℓsinξ/2 ).
(2) Since adjacent strips are mirror images of eachother, the section zonogon continues to lie on
the section plane throughout the transformation
process.
Constraints (1) and (2) imply that each facet219

Vol. 53 (2012) No. 4 December n. 174

determines its orientation from the extruded vector
and its original orientation. This results in a halfcylinder with a one-DOF mechanism, whose twoends are translationally symmetric to each other.
The other half strip is folded in the same manner.
Since its deformed section zonogon is congruent tothat of the first half, the two strips are compatible
with each other and form a single rigid-foldable
tube.

flat in the y direction. On the other hand, thestructure always collapses in the z direction when
0. Although there is a difference in the details
depending on the shape of the zonogon, as a whole,the dimensions in the x, y, and z directions aremonotonous functions with respect to the change inthe folding angle. Figure 9 shows the folding
motion for a hexagonal cylinder as an example.

3.3 Kinetic Behavior
Assume that a segment of the section zonogon
forms angle θ ( 90°90°) with the x axis.
This is represented as a vector r = (rcosθ, rsinθ, 0),
where r is the length of the segment (Figure 6). The
vector r is rotated such that the angle between theedge and the extrusion vector is constant. Therefore,
we obtain the constraint l·r = constant, which
results in the following relation (Figure 8):
cos

cos

cos

2 cos

,

(3)

2

Figure 8. Folding motion of each facet. Note that vector r
is determined independent of connectivity.

|
where. The valid sign of θ is chosen sothat a continuous motion is produced, i.e., sign
0. If
0, the mechanism has asign ifsingularity at
0 so that both orientations
are allowed.
The folding motion of a facet is represented asfollows.
cos
cos

2

Figure 9. Folding motion of an example cylinder from ahexagonal section polygon. Note that the structure is only
flat-foldable in the z direction.

,

2

1

cos
cos

2 cos

,

(4)

2

sin .
2
Here, the domain ofis given by cos

cos cos , where equality holds when y = 0. Fora global structure with multiple edges of different θ0,
the shallowest edge, i.e., the edge with the highest
value of cos , limits the domain of global variable. Therefore, the structure does not generally fold220

3.4 Cellular Structures
The fact that the orientation of each facet isdetermined independent of adjacent faces implies
that if multiple cylinders are constructed fromzonogons sharing a vertex and edges, then these
cylinders transform without producing any gaps.
Hence, a planar tessellation of zonogons can beused as the section to produce a rigid-foldable
cellular structure. The tessellation does notnecessarily have periodic symmetry. Figures 10 and
11 show examples of a cellular structure using
Penrose tiling and a multi-layered structure
following a given curve, respectively.

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

as denoted in Figure 12. We can easily generalize
the rectangular module so that it has 5, 7, or 2n + 1
columns by repeating the same tile structure.

Figure 10. Example of a rigid-foldable cellular structure
constructed from a zonogon tiling (Penrose tiling).

Figure 11. Example of a rigid-foldable cellular structure
constructed from a given curve

4. BI-DIRECTIONALLY FLAT-FOLDABLE
TYPE
4.1 Construction
The basic tube is not a single developable surface
but a compound of two developable sheets sharing
congruent boundary curves. By carefully
constructing such a bi-directionally flat-foldable
cylinder, we can construct valid mechanisms. The
following is an overview of the construction steps.
1.

2.

3.

This surface transforms while preserving theparallelism of the fold lines and the planarity andcongruency of the end polylines. This enables us to
set a local coordinate that works throughout thetransformation. As shown in Figure 13, we canextract two pairs of planes that do not rotate byfolding, specifically, the 3- and 0-planes parallel tothe xy plane and the A- and D-planes parallel to theyz plane.
Then, we produce a mirror image of the modulewith respect to its D-plane. This connects two units
to form a half cylinder. The boundary polygons at
the A-plane and its mirror A′ plane aretranslationally symmetric to each other and have acommon axis of 180° rotational symmetry (A-axis)(Figure 14).
We can then construct the other compatible halfcylinder by rotating the original half cylinder by
180° around the A-axis. This forms a valid cylinderof a partially doubly covered surface. If we trim the
doubly covered part, we obtain a star-like rigidfoldable cylinder (Figure 15). We can then extend
the cylinder in the z direction. Figure 16 shows thecrease pattern of two sheets comprising the cylinder
for α = 45°.
lA3Construct a one-DOF mechanism based on aquadrilateral array, such that each of two pairsof ends (in the z and x directions) comprises
translationally symmetric boundary curves inany folded state.
Construct a structure compatible with the resultof step 1. so that both surfaces share the same
boundary curves, then connect the two surfaces.

We show an example design of a bi-directionallyflat-foldable cylindrical structure. First, consider amodular rectangle composed of nine facets withfour identical Miura-ori vertices as its crease pattern,

B3

l
C3

D3

=

d

A2

d

A1

d

A0

=

B2

α

B1
B0

C2
C1
C0

D2
D1
D0

Figure 12. Crease pattern of nine-facet module.
3-planeTrim the doubly covered surface (if it exists) toconstruct a singly covered valid cylinder.

4.2 TMP (Tachi–Miura Polyhedron)m

D-planeD-axisA-axiszyx0-planeA-planeFigure 13. Coordinate system and planes.

221Vol. 53 (2012) No. 4 December n. 174
D-planeA-planeA-axisFigure 14. Symmetry operation to construct a cylinder byconstructing the modules.

Figure 17. Cylinder of α = 45° at β = 89°, 67.5°, 45°,22.5°, and 1° (from left to right).

Figure 15. Trimming out the singly covered star-shaped
polyhedron.

Figure 18. Cylinder of α = 54° at β = 89°, 67.5°, 45°,22.5°, and 1° (from left to right).

Figure 16. Crease pattern of the cylinder (α = 45°).

Figure 19. Folding motions of cylinders of α = 45° (left)
and 54° (right).

4.3 Kinetic Behavior

4.4 Cellular Structure

Patterns with different α produce different folding
motions. Figures 17 and 18 show the transformationof cylinders when α = 45° and 54°, respectively,
with the common parameters l = 1, m = 2, d = 1, N
= 9. For any parameter, the x and z dimensions aremonotonous functions of the folding angle, such as
for a basic tube. However, the y dimension holds a
maximum value at cos
cot between folds(only when α > 45°). Figure 19 compares cylindersof α = 45°and 54°.

The example cylinder is composed of elements
having translational symmetry to each other. Thus,
the cylinder periodically tessellates the threedimensional space without producing any gap, asshown in Figure 20. This tessellation property canbe preserved for any folded state; hence, thistessellation has a synchronized one-DOF rigidfolding motion inherent from the cylinder. This
volumetric polyhedral complex can smoothly folditself into two flat states following thetransformation of a single cylinder.

4

2

1
4

1
sin
sin

222

tan

tan cos
tan
2
2

cos
.

cos
2 ,
2

,
2
(5)

The general key technique for constructing atessellatable rigid-foldable cylinder is to construct a
module whose boundaries are translationallysymmetric, such as the nine-panel module in Figure
13 and its compatible variation. Once we construct
such a module, we can then apply symmetry
operations (reflection and translation) based on therhombus grid, as shown in Figure 21, to construct avalid cellular mechanism.

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

4.5 Generalization
In order to construct such a module, we use joint
structures
composedoffour-valencebidirectionally flat-foldable vertices classified into
fold and elbow types (Figure 22). Such a joint
vertex converts the shared dihedral folding angle ξ
to a sectional folding angle at each joint.
2 arctan cos tan
,
2
2

(6)

2 arccot cos cot.
22Here, we aim to construct a one-DOF mechanism
by connecting such joint mechanisms so that its
ends are always parallel, i.e., ∑0 or. Sucha mechanism can be constructed by combining thefollowing identity equations.
,
,
,

,

The ordering of vertices does not alter the total
angle; thus, we can easily construct compatible
variations of valid half strips, an example of whichis shown in Figure 23.
A cellular structure proposed by Hoberman [3] canbe constructed using this sequence. Therefore, it is abi-directionally flat-foldable type structure and thus
a rigid-foldable structure.
Moreover, bi-directionally flat-foldable structurescomprising planar quadrilateral panels can befurther generalized into non-symmetric freeform
designs by using sufficient conditions for rigidfoldability [12].

≡ 0,
,

≡ 0,

,

≡ .

z

(7)

y

x

Figure 20. Tessellation of a cylinder to produce a oneDOF bi-directionally flat-foldable cellular structure.

Figure 21. Pair of cylinders produced from one module (left) and repeated cylinders using a rhombus grid (right).

5. WOVEN CYLINDER CELLS
5.1 Geometric Construction

Figure 22. Fold and elbow.

In previous sections, we considered cellular
structures whose component cylinders are allaligned in the z direction. In this section, we
propose a novel cellular structure where thecomponent cylinders are “woven” in the y–z
directions (Figure 24). The structure can be formedby the following procedure.
1.

Figure 23. Combining vertices to form an angular identity
equation.

Align the cells side by side (in the y directionof Figure 4) to form a cellular layer (Figure 25).
This compound layer comprises top andbottom corrugated surfaces, each of which istransformable: egg-box type parallelogram
meshes. Here, note that the top and bottom
surfaces are congruent to each other with aphase shift and essentially have fourfold
symmetry.

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Vol. 53 (2012) No. 4 December n. 174

Figure 24. Woven-type cellular structure.

2.

Stack the layers to tessellate a 3D space. Here,
we have two choices for the orientation of eachlayer to be stacked because of the fourfold
symmetry of the egg-box type mesh.

3.

By alternating the orientation of the tubes foreach layer, we get a “woven” type cellular
structure (Figure 26 left). Note that if wealways choose parallel layers, we obtain thebasic cellular structure (Figure 26 right).

Figure 25. We can construct an egg-box surface byaligning cells side by side.

Figure 26. Woven structure (left) and basic cellular
structure (right).

5.2 Kinetic Behavior
The motion of a woven cylinder assembly can be224

understood by observing its smallest motif
composed of two basic modules (two identicalrhombic modules or a pair of modules withcorresponding proportion) (Figures 27 and 28). This
polyhedral shape is known in the origami field as
flip-flop, a kinetic origami model (from one piece ofrectangular paper) originally designed by Yenn [3].
The original flip-flop uses a 60° rhombus, but
arbitrary parallelograms can form such a one-DOF
mechanism. The convex hull of the flip-flop is adodecahedron composed of parallelograms, which
tessellates the 3D space without gaps in any folded
state, in the way rhombic dodecahedra tessellate
space. The whole woven cylinder structure isconstructed with only translational copies of themodule; thus, the transformation mode simply
follows that of a flip-flop.
The x dimension of a flip-flop is the sum of the x
dimensions of its two component modules (x and x′).
These two modules are in opposite folding states,i.e., parameter ξ in one module is equal toparameter Ξ in another module or x′(ξ) ≡ x(Ξ).
Figure 29 shows the change in the x, y, and z
dimensions of a flip-flop using a rhombus with
2 arctan √2⁄2 , i.e., the silver rhombus.
A flip-flop—and thus a woven cylinder structure—
always has the minimum x value in an intermediate
folded state. This means that the structure stabilizes
with x directional compression and is stiff in thisdirection. For
2 arctan √2⁄2 , the structurestabilizes at the state where the module becomes arhombic dodecahedron (ξ = 120° and y = z) that
holds the maximum volume.

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

because it is against Connelly's bellows theorem. In
reality, a small and local distortion at the ends ofcylinders occurs with elastic material; the distortionis caused by constraining shear deformation of theboundaries.

Figure 27. Construction of woven cylinder module: flipflop by Yenn.

Figure 28. Folding motion of a flip-flop. Note the change
in x (height). The middle state forms a rhombic
dodecahedron.

Because the distortion is local, it seems plausible touse the bellows for practical purposes; one example
is a fluid actuator. Figure 30 shows an illustrated
image of the actuator. The actuator has a longstroke (close to 100% of its length) because it does
not require a piston.
The applications of the cellular structures include
collapsible temporary shelters constructed from stiff
panels. The virtue of the cell structure in thiscontext is that each cell acts as a volumetricbuilding block that supports its self-weight andforce while insulating sound and heat. For example,
we can build thick walls that collapse to very thinstates (Figure 32).
We can further interpret the cellular structures as
anisotropic materials from a macroscopic viewpoint.
In other words, we use the intersection between
some given volumetric form and the fine cellular
structure filling infinite 3D space (example shownin Figure 32). This intersection follows the sameone-DOF transformation mechanism as the infinite
tessellation because of the redundant geometricconstraints that work in cylindrical and cellular
structures.

Figure 29. Change in the dimensions of a flip-flop or awoven cylinder cell (compare with Figure 6).

6. POTENTIAL APPLICATIONS
Since the cylindrical and cellular structures we haveshown are purely geometric and thus free of scale,the field of potential application ranges frommicrostructures to architecture. Here, we suggest a
few potential orientations, but the applications are
open-ended.
By placing planes at the ends of a TMP-typecylinder, we can construct a closed and flexible
polyhedron, i.e., bellows, holding a certain volume.
By folding this bellows, we can observe that it
continuously reduces its volume to zero. In theory,
this type of transformable polyhedron does not exist

Figure 30. Actuator from rigid-foldable cylinder (left andmiddle) compared with a conventional actuator.

225

Vol. 53 (2012) No. 4 December n. 174

study on cylindrical deployable spacestructures, IUTAM-IASS Symposium on
Deployable
Structures:
TheoryandApplications, 2000, pp. 383-392.
[4]

Nojima, T., Origami modeling of functional
structures based on organic patterns,
Presentation Manuscript at VIPSI Tokyo,
2007.

[5]

Kuribayashi, K., Tsuchiya, K., You, Z.,
Tomus, D., Umemoto, M., Ito, T., andSasaki, M., Self-deployable origami stentgrafts as a biomedical application of Ni-rich
TiNi shape memory alloy foil, Materials
Science and Engineering A, Vol. 419, 2006,
pp. 131-137.

[6]

Wu, Z., Hagiwara, I., and Tao, X.,
Optimisation of crush characteristics of thecylindrical origami structure, International
Journal of Vehicle Design, Vol. 43, 2007, pp.
66-81.

[7]

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

[8]

Tachi, T., One-DOF cylindrical deployablestructures with rigid quadrilateral panels,
Proc. IASS Symposium 2009, Eds. Domingo,
A., and Lazaro, C., 2009, pp. 2295-2306.

[9]

Miura, K., and Tachi, T., Synthesis of rigidfoldable cylindrical polyhedra, J. ISISSymmetry, Special Issues for the FestivalCongress Gmuend, Austria, 2010, pp. 204213.

Figure 31. Assembly of collapsible building blocks.

Figure 32. Collapsible sphere from woven cylinders.

7. CONCLUSION
We introduced novel families of rigid-foldable
cylinders and three types of cellular structures
based on cylinders: i.e., zonogon extrusion cells, bidirectionally flat-foldable cells, and woven cylindercells. We showed their geometric validity as oneDOF structures, their generalized design method,
and their kinetic properties. We suggested theorientations of potential applications of thesestructures.
8. ACKNOWLEDGEMENTS
This study is supported by KAKENHI (22800009),
a Grant-in-Aid for Research Activity Start-up by
the Japan Society for the Promotion of Science.
Further, this study is supported by the JST PRESTO
program.
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