Symmetry: Art and Science
Gmuend, Austria, 2010

SYNTHESIS OF RIGID-FOLDABLECYLINDRICAL POLYHEDRAKORYO MIURA AND TOMOHIRO TACHI

Name: Koryo Miura, Aerospace Scientist (b. Tokyo, Japan, 1930).
Address: 3-9-7 Tsurukawa, Machida, Tokyo 195-0061, Japan.
E-mail: miurak@gakushikai.jp
Fields of interest: Structural Concepts, Geometry, Origami Science.
Awards: Honorary Member, International Association for Shell and Spatial Structures.
Name: Tomohiro Tachi, Origami Scientist (b. Tsukuba, Japan, 1982).
Address: Graduate School of Arts and Sciences, the University of Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo
153-8902
E-mail: tachi@idea.c.u-tokyo.ac.jp; Webpage: www.tsg.ne.jp/TT/
Fields of interest: Computational Origami, Architectural Design.
Awards: Hangai Prize, International Association for Shell and Spatial Structures, 2009.

Abstract: Rigid-foldable structures are foldable surfaces consisting of rigid panels andhinges, thus can be used for wide variety of deployable structures without relying on
flexible materials. In this paper, we present a family of rigid-foldable collapsible
cylindrical polyhedra which is of great interest of structural engineering field. The
symmetry operations in order to synthesize the cylindrical structures and their spacefilling tessellation are shown.

1 INTRODUCTION
One of the basic problem areas on foldable structure is to fold a cylindrical shell in axial
direction to the extreme, i.e., to a flat state, while keeping its axis and internal envelope
like a bellows. There are lots of need for such structures in industry, such as deployablestructures, bellows, and packaging. The present authors are specifically interested in the“bellows-type” foldable structures with a purely geometric folding mechanism whilekeeping substantially its sectional area during the folding/deploying maneuver.

The best way to understand the background of this problem is to consider the case ofpost buckling behavior of a thin cylindrical shell under axial loading. The resulting
geometric form is called Yoshimura-pattern, which consists of repetition of the
diamond pattern. Because the Yoshimura-pattern is obtained through the inextensional
deformation process, it can be considered as the candidate of a foldable cylindricalshell. However, the matter is not simple as expected for this case. For a particulardimension of cylindrical shell, one can define the folded state or the pattern to realize agiven axial length. However, once this geometric parameter is defined, it becomes astable structure that is no more foldable without elastic deformations (Fig. 1). In short,the Yoshimura-pattern is a folded cylinder but not a continuously foldable one.

Fig.1: Yoshimura Patterns. From Left to Right: Original cylinder and Yoshimura
Patterns of radial frequency of 8, 6, 4, and 3. Note that each state is static.
Though the Yoshimura-pattern provides a negative example for the present problem,the “buckling” and “concave polyhedral surface” involved in it are still the major
keywords for the following researches. The typical approach on these studies depends
on the model representing a cylinder with a polyhedron consisting of number ofpolygonal plates members. The most difficult problem is that, in the course of folding
and deploying process, some amount of in plane deformation of polygonal plates areinevitable for geometric compatibility. For example, Guest and Pellegrino (1994)
proposed a variation of cylindrical foldable shell by twisting Yoshimura-pattern toenable a valid state in three-dimensional and also in the flat-folded state. However, theproduced designs are multi-stable structures that cannot be built from rigid or thickmaterial. Therefore, the existence of such in-plane strains severely limits the design areaof the foldable structures.
Here, we consider rigid-foldable geometry, where rigid-folding implies a continuous
folding motion of a polyhedral surface in which its facets are kept congruent. Suchstructures can be used for wide variety of deployable structures built from rigid panelsand hinges without in-plane strains. As it is already shown, not all foldable structures
are rigid foldable. In this respect, a solution of bi-axial rigid folding of a flat plate exists
and is known as Miura-ori (Miura, 1980) (Fig. 2). However, this principle has not
successfully been applied to the case of cylinders until recently, i.e., existing designs offoldable cylindrical structures have relied on the in-plane distortion of panels or thetransition of foldlines.

Fig.2: Miura-ori is a rigid-foldable disk.
Recently, the second author (Tachi, 2009) disclosed a novel concept of rigid-foldable
polyhedral cylindrical structures. The core of the concept is that the basic unit
structure is constructed by joining two pieces: a single vertex origami with fourcongruent parallelograms and its mirror image. In other words, the Miura-ori and its
mirror image are joined. The generalization of the concept results in groups of rigidfoldable cylindrical structures.
During further studies on one of Tachi's tube structure designs, we have accidentally
discovered the characteristic “star-polyhedron” embedded in it. It turned out that, thepolyhedron constitutes the very core of the design (Fig. 3). In this paper, we will
disclose the characteristic of the star-shaped cylindrical polyhedron by producing it bymeans of symmetry operations; the polyhedron can be constructed from pasting
together Miura-ori vertices. Through this study, we will also show that the polyhedronis actually a space filling polyhedron. This leads to a 3D cellular structure that cansimultaneously folds in x, y, and z dimensions.

Fig.3: Star-polyhedron with a synchronized motion.

2 MODULAR STRUTCTURE
First of all, let us examine the basic geometric properties of Miura-ori and its vertex. Avertex of Miura-ori (composed of 4 foldlines) exhibits a synchronized motion of folding
angles as shown in Fig. 4. This results in a one-DOF mechanism from unfolded to flatfolded states. We can repeat this vertex structure to construct an array, i.e., Miura-ori.
Within the motion, the parallel lines are kept parallel, and planar polylines are kept
planar. This is the very core of our study.

γααβγγγβγββγtanα cosβ = tanγα = constant: an inner angle of parallelogram
β = variable: 1/2 of dihedral angle
γ = variable: angle between the straight foldline and the base plane
Fig. 4: A folding motion of Miura-ori vertex.
Then we consider a modular rectangle composed by 9 facets with 4 identical Miura-ori
vertices as denoted in Fig. 5. Because of the repeating structure, this surface transforms
preserving the parallelism of foldlines and the planarity and congruency of end
polylines. This enables us to set a local coordinate that works throughout thetransformation (Fig. 6). Specifically, we extract a set of parallel lines {A0B0, ..., A3B3,C0D0, ..., C3D3} and use this direction as the x-axis. Symmetric structure of Miura-ori
vertex forces polyline AiBiCiDi (i = 0, 1, 2, 3) to be planar; here we term the containing
plane i-plane. We define xy-plane as 0-plane. Since we cut parallel lines byperpendicular boundary polylines A0A1A2A3 and D0D1D2D3 (termed A- and D-polylines,repsectively), each of the folded state of polylines lies on a plane perpendicular to xaxis, which is termed A- or D-plane. Here, note that A- and D-polylines are congruent
to each other and each has 180° rotational symmetry around the axis through themidpoint of A1A2 or D1D2 along x-direction. We term these axes A- and D-axes. Themodule transforms so that the surface boundary is tangent to A-, D-, 0-, and 3-planes,
where A- and D-planes are kept parallel to yz-plane, and 0- and 3-planes to xy-plane.
In a design context, we can generalize the rectangular module so that it has 5, 7, or
2n+1 columns by adding extra 2, 4, or 2n zigzagged foldlines similar to B- and Cpolylines. This is because the property of the boundary (e.g., A- and D-planes) in thiscase is equivalent to that in the 3 columns case. Hence our method described in thispaper is essentially a general approach to merge 4 pieces of Miura-ori to form acylindrical surface.

lA3ml

B3C3

D3=dA2dA1dA0=

B2

αC2

B1
B0

C1C0

D2
D1D0Fig 5: Notation of the module.

3-planeD-planeD-axisA-axiszyx0-plane
A-planeFig 6: Coordinate system.

3 CYLINDRICAL POLYHEDRA
Described co-planarity and congruency of edges enable us to paste modules edge-toedge with symmetric operation without producing any gap between them throughout
the transformation. The following shows the process (Fig. 7 and Fig.8).
1) Produce a mirror image of the module with respect to its D plane. This connects
two units (in any folded state) without gap. In this procedure, the A-axes of theoriginal and mirrored modules coincide since the x-directions of the original andcopied modules are opposite (Fig.7 middle).

2) With respect to the common A-axis, rotate the connected modules by 180°. This
also validly connects four modules in any folded state since A-polyline on theboundary has rotational symmetry with respect to the axis. Here, line AiBi is rotatedto the position of A3-iB3-i (Fig.7 right).
3) The rotation makes the connected part doubly covered by facets (Fig. 8(a)).
However the difference between the lengths of AiBi and A3-iB3-i produces themismatch. If we remove the doubly covered part, we obtain the singly covered starshaped polyhedron as shown in Fig. 8 (b).
4) Since 0- and 3-planes are kept planar by the process 1) and 2), we can also extend
this geometry to the z-axial direction by mirror reflecting with respect to 3-plane.
We can also cut out one or two of the three-rows of the module in order to adjust
the corrugation frequency (Fig. 8 (c)).
Once the cylindrical polyhedron is built, we can naturally extract a two-row module andits glide reflectance (Fig. 8 (d)). In fact, the rotation in 2) is essentially equivalent to theglide reflection with respect to a plane through A-axis and parallel to zx-plane with thehalf wavelength translation in z-direction.
D-planeA-planeA-axisFig 7: Symmetry operation to construct a cylinder by composing the modules.

(a)

(b)

(c)

(d)

Fig. 8: Trimming out the singly covered star-shaped polyhedron and extending thecylinder by repeating mirror reflection to z-direction.

The pattern of the cylinder can be parameterized by the dimension of the module
represented by , l, m, and d and the number of repetition N. On the other hand, thetransformation of the shape is represented by single variable  (0  90°) representing
the half of dihedral angle of any edge between adjacent facets in the same column suchas A1B1 or B2C2 (all of them are equal). Patterns with different  produce different
folding motions. Fig. 9, 10 and 11 show the transformation of cylinders in case of
=30°, 45°, and 54°, respectively, using common l=1, m=2, d=1, N=9.

Fig. 9: The cylinder of =30° at  = 89°, 67.5°, 45°, 22.5°, 1° (from left to right).

Fig. 10: The cylinder of =45° at  = 89°, 67.5°, 45°, 22.5°, 1° (from left to right).

Fig. 11: The cylinder of =54° at  = 89°, 67.5°, 45°, 22.5°, 1° (from left to right).

Because the proposed cylindrical polyhedra are composed of rigid facets, we can attach
solid panels to the facets while producing kinetic motion without distorting them. By
applying the thickening method introduce by (Tachi, 2010), we could obtain a thickpanels structure that follows the kinetic motion of proposed cylindrical polyhedron asshown in Fig. 12. This indicates that our structure can be utilized in different scales anddesign contexts.

Fig. 12: Cylindrical polyhedra using thick panels. (=45°)

4 SPACE FILLING AND 3D CELLULAR STRUCTURE
The proposed cylinders can periodically tessellate the three-dimensional space without
producing any gap as shown in Fig. 13. In other words, the cylinders consists a 3D
cellular structure. This tessellation property can be preserved for any folded state; hence
this tessellation has a synchronized one-DOF rigid-folding motion inherent from thecylinder. This volumetric polyhedral complex can smoothly fold itself to two flat states.
The validity of the structure can be similarly described by applying symmetry
operations to the modules we used for synthesizing cylinders.

Fig. 13: 3D cellular structure in folding motion.

In the construction of a cylinder, we used D-plane for the reflection and A-axis for therotation. However, because of the symmetric structure of the module, A and B cansubstitute C and D, and vice versa via 180° rotation with respect to the axis through thecenter and along the z-axis. Therefore, a pair of cylinders is produced from a single
module as shown in Fig. 14 left. Constructing a cylinder produces four congruent
modules. We recursively construct cylinders from already built modules. This
procedure forms a regular cellular structure that fills the three-dimensional space. In
order to understand this regularity, consider the contact points of A- and D- polylines
with 0-plane or the xy-plane. The points are located on the vertices of rhombus whose
diagonals are along x- and y-axes. Therefore constructing a cylinder from a module
adds a rhombus pasted edge-to-edge to the existing ones, and thus A- and D- polylines
are essentially located at the nodes of rhombus grid on xy-plane (Fig. 14 right).

Fig. 14: Left: A pair of cylinders produced from one module.
Right: Rhombus grid on whose vertices A- and D- polylines are located.
Now we obtained a foldable cellular structure of single-walled (non-manifold)polygonal mesh. Then, we show that the star-shaped cylindrical polyhedra fill the spacewith every part of the surface touching an adjacent polyhedron. This means that thepolyhedra can completely doubly cover the cellular polygonal mesh. We firstdecompose the cylinder into 6 parts, comprising of three pairs of congruent corrugatedsurfaces P-P, Q-Q, and R-R (Fig. 15). Then we consider 6 cases of translation, each ofwhich moves one part of the unit to its congruent counterpart, e.g., P to P or Q to Q.
Under such transformations, cylindrical unit moves to adjacent six units. Therefore
every part of the cylinder is shared by and only by the corresponding adjacent cylinder
as shown in Fig. 16. Here, note that the orientations of congruent counterparts (e.g., Pand P) are opposite. This makes adjacent cylinders “touch” each other. Therefore every
surface is doubly covered, and thus the cylindrical polyhedron can tessellate the threedimensional space.

5 CONCLUDING REMARKSFirst, we have accidentally discovered the characteristic “star-polyhedra” following thegeneral concept proposed by Tachi (2009). The symmetry operations in order tosynthesize the cylindrical structures are shown. The family of the polyhedra can be used
for wide variety of deployable structures and other mechanisms.

Second, during the detailed study on the polyhedra, we have unexpectedly discoveredthat it has the space filling property in spite of its star-shape. The resulting volumetricpolyhedral complex can smoothly fold itself to two flat states. This novel concept will
surely attract the interests of potential users.
Through this study, we were able to present a novel concept of polyhedra having
unprecedented properties.

P'Q'
R

R'
P

Q

Fig. 15: Cylinder decomposed into 6 parts.
4
5

P'
Q'

P'

R

Q'

Q

R'

3
R

0R'
P

6QP

21Fig. 16: Correspondence of parts to adjacent cylinders. Parts {P, P, Q, Q, R, R} ofcylinder 0 is shared by adjacent cylinders {1, 4, 2, 5, 3, 6}, respectively.

References
Guest, S. D. and Pellegrino, S. (1994) The folding of triangulated cylinders, Part 1: Geometric Considerations,
In: Journal of Applied Mechanics, Vol. 61, 773-777.
Miura, K. (1980) Method of packaging and deployment of large membranes in space, In: Proceedings of 31st
Congress of International Astronautics Federation (IAF-80-A31), Tokyo, Japan, 1980, 1-10.
Tachi, T. (2009) One-DOF Cylindrical Deployable Structures with Rigid Quadrilateral Panels, In:
Proceedings of IASS Symposium, Valencia, September, 2009, 2295-2305
Tachi, T. (2010) Rigid-foldable Thick Origami, To be presented at: 5th International Conference on Origami
in Science, Mathematics, and Education (5OSME), Singapore, July 2010.