Freeform Rigid-Foldable Structure using
Bidirectionally Flat-Foldable Planar Quadrilateral
Mesh
Tomohiro Tachi
The University of Tokyo

Abstract. This paper presents a computational design method to obtain collapsible
variations of rigid-foldable surfaces, i.e., continuously and ﬁnitely transformablepolyhedral surfaces, homeomorphic to disks and cylinders. Two novel techniques
are proposed to design such surfaces: a technique for obtaining a freeform variationof a rigid-foldable and bidirectionally ﬂat-foldable disk surface, which is a hybridof generalized Miura-ori and eggbox patterns, and a technique to generalize thegeometry of cylindrical surface using bidirectionally ﬂat-foldable planar quadrilateral mesh by introducing additional constraints to keep the topology maintained
throughout the continuous transformation. Proposed methods produce freeform
variations of rigid-foldable structures that have not been realized thus far. Sucha structure forms a one-DOF mechanism with two possible ﬂat states. This enables
the designs of deployable structures useful for packaging the boundary of architectural spaces, space structures, and so on.

1

Introduction

A polyhedral surface composed of rigid facets connected by rotational edges forms
a kinetic mechanism: a rigid folding mechanism. Several collapsible structures areproposed using this type of kinetic mechanism, such as Miura-ori [Miura 1980] andeggbox patterns [Brunner 1965] (Figure 1). Such a rigid-foldable and ﬂat-foldable
surface is useful as a deployable and transformable structures in an architecturalcontext because of the following advantages.
1. The existence of a collapsed state enables compact packaging of the structure.
2. The synchronized complex folding motion produced by constrained rotational
hinges can be controlled with simple manipulation.
3. The transformation mechanism that does not rely on the ﬂexibility of materials can be made out of thick rigid panels and hinges.
4. The watertightness of the surface maintained throughout the transformationis potentially suitable for the envelope of a space, a partition, and the facadeof a building.

T. Tachi

Figure 1: Folding motions of Miura-ori (Top) and eggbox pattern (Bottom).

In order to apply such kinetic surfaces to design purposes, it is required that we
can control the three-dimensional shape of the surface and its behavior so that they
are consistent with the design requirements. Design requirements in architecturalcontext include environmental conditions, functional requirements, conﬁguration ofsurrounding buildings and structures, and user’s preference. However, since therigid-foldability of known surfaces strongly relies on the symmetry of the pattern,
designing such structures could not be done in an ad-hoc approach; the geometricconstraints must be generally investigated to solve an inverse problem of obtaining
a pattern from the resulting form and behavior. The objective of our study is thus tofreely design kinetic forms using design methods based on computational geometry.
As the ﬁrst step to achieve this goal, the author proposed a freeform design methodof kinetic structures based on origami (developable surface) to provide design variations of Miura-ori through an interactive design system in which a user can deform the surface freely while sustaining the rigid-foldability of the surface ([Tachi
2009a]). This method yields variations of rigid-foldable origami models that havenever been achieved otherwise, however, the method was not general enough fordesign applications, especially of architecture, since the developability condition
used is not always an essential condition when we construct structures from multiple parts. In particular, this developability condition disables the design of rigidfoldable collapsible cylinders, realization of which can contribute the designs ofsurrounding of a volume of a space, collapsible containers, compound surface structures, and so on. This limitation mainly comes from the fact that a cylindrical surface
cannot have a globally developed state under the conventional deﬁnition.
This paper provides a novel method that enables a freeform variation of such rigidfoldable structure, not restricted by the developability of the surface or the disk
topology. We will solve this design problem by extending and combining the ideasof generalized Miura-ori [Tachi 2009a] (or ﬂat-foldable 4-valent-vertex origami)

Rigid-foldable Flat-foldable Surface

and generalized eggbox pattern (or discrete Voss surface) [Schief et al. 2007] to produce bidirectionally ﬂat-foldable planar quadrilateral mesh. Such a mesh has nicegeneralized characteristics: it is bidirectionally ﬂat-foldable and produces one-DOF
mechanism. Required design condition is loose compared to purely developablesurface and it allows asymmetric and cylindrical variations. We will show a novel
design method to obtain rigid-foldable collapsible cylinders by deforming symmetric rigid-foldable cylinders of [Tachi 2009b] into asymmetric ones.

2

Geometry of Rigid-Foldable Quadrilateral Mesh Disk

This section shows the geometry of 4-valency rigid-foldable mesh with two ﬂatstates that combines ﬂat-foldable origami surface and discrete Voss surface. A 4valency mesh or a quadrilateral-mesh surface in general does not enable a continuous rigid-folding motion because an overconstrained system is constructed. This isbecause the conﬁguration of a rigid-foldable structure homeomorphic to a disk canbe represented by the folding angles of edges, while these variables are constrainedat each vertex by 3 equations as it is used in the simulation of rigid origami [Tachi
2009c]; if we are to construct a quadrilateral mesh, then the number of constraints
exceeds the number of variables only by making 3 × 3 array. However, Miura-ori[Miura 1980] and eggbox patterns [Brunner 1965] are known to produce singular
one-DOF mechanisms (Figure 1) because of their redundant constraints. Miura-ori
is a ﬂat-foldable origami surface composed of parallelograms used for packagingof large membranes in the space. The surface produces a synchronized kinetic motion of expanding in x and y directions at the same time from a collapsed state to acompletely developed state (Figure 1 Top). Eggbox pattern is a polyhedral surface
similarly composed of parallelograms but is not a developable surface. This has two
ﬂat-folded states and produces a different one-DOF kinetic behavior from that ofMiura-ori; the surface expands in x direction when it collapses in y direction (Figure 1 Bottom). While Miura-ori is a surface with foldlines folded at the same time,
eggbox pattern is a surface that has two groups of foldlines collapsed at the sametime.
Since Miura-ori and eggbox patterns are surfaces composed of congruent parallelograms, the redundancy of constraints required for the mechanism seem to originate
in its global repeating symmetry. However, the condition to produce such redundancy actually is a result of its intrinsic symmetry, i.e., the angles coincidence ateach vertex, which also contribute to bi-directional ﬂat-foldability of the surfaces(Miura-ori is developable and ﬂat-foldable. eggbox pattern is ﬂat-foldable in two
directions). The intrinsic symmetries of Miura-ori and eggbox pattern, yield generalized forms of rigid-foldable origami surface [Tachi 2009a] and discrete Voss surface [Schief et al. 2007], respectively. In addition, these surfaces can be merged into
one generalized hybrid surface, i.e., bidirectionally ﬂat-foldable planar quadrilateralmesh, based on the fact that the behaviors of the two surfaces can be represented using a common form.

T. Tachi

Figure 2: A corrugated vault used as a transformable architecture that connects two
existing buildings.

2.1

Generalized Miura-ori

The rigid-foldability condition of generalized Miura-ori is presented by [Tachi
2009a]: consider a planar quad mesh surface of disk topology whose vertices are
incident to 4 foldlines; when the surface is developable and ﬂat-foldable, ﬁnite rigidfoldability of the surface is equivalent to the existence of one valid semi-folded state.
This sufﬁcient condition for rigid-foldability yields rigid-foldable generalization ofMiura-ori. The condition also applies to any other degree-4 vertex based patterns
such as assymetric vault pattern shown in Figure 2.
The behavior of the surface is represented by the conﬁguration of fold angles. A vertex with 4 foldlines, and thus with 4 sector angles θi (i = 0, 1, 2, 3), form a one-DOF
mechanism (Figure 3 Top). In the case of developable and ﬂat-foldable surface,condition of origami (i.e., developability) and ﬂat-foldability given by ∑3i=0 θi = 2π
and∑3i=0 (−1)i θi = 0, respectively, force the sector angles to satisfy
θ0 = π − θ2and

θ1 = π − θ 3 .

(1)

According to [Tachi 2009a], the fold angles ρi and ρ j incident to the vertex are
related as follows:ρAi, j tan 2j (i − j = 1 or 3 mod 4)
ρi
tan =
(2)
ρj
2
(i − j = 2 mod 4)± tan 2

Rigid-foldable Flat-foldable Surface

where the latter represents that pairs of opposite foldlines have an equal absolute
folding angles, and Ai, j is the coefﬁcient between these two equivalent pairs determined by θ0 , · · · , θ3 , i.e., intrinsic measure in the crease pattern independent fromthe folding angles. Speciﬁcally, if |ρ0 | = |ρ2 | > |ρ1 | = |ρ3 |,
1 + cos (θ0 − θ1 )|A0,1 | =.
1 + cos (θ0 + θ1 )Note that this relationship is a special case of the generalized form of origami vertex
presented by [Huffman 1976].
This gives an explicit conﬁguration of folding angles of all foldlines globally connected via degree-4 vertices: ρi (t)
ρi (t0 ) tan 2t
tan
,
(3)
= tan
2
2
tan t20
where t (0 ≤ t ≤ π) is the parameter that deﬁnes the folding amount(t =0 andρi (t0 )t = π indicate developed and ﬂat-folded states respectively) and tan 2
is anarbitrary semi-folded state (0 < t0 < π). Generalized Miura-ori thus shows a kinetic
motion of expanding in x and y directions at the same time.
1

0

θ0

θ3

θ1

θ2

ρ1
3

ρ2

2
θ0

1

θ1

3

θ3
θ2
2

2.2

ρ0

ρ3

0

ρ1

ρ3
ρ0
ρ2

Figure 3: The folding
motions of vertices oforigami (Top) and discrete Voss surface (Bottom). Note that set ofplanes used is identical.

Generalized Eggbox Pattern

Eggbox pattern can be generalized as a discrete Voss surface; the rigid-foldability of
discrete Voss surface is proved by [Schief et al. 2007]. Even though the term discrete
Voss surface implies a mesh pattern that represents a smooth surface, corrugatedsurfaces such as eggbox pattern is also included in the family, since it is deﬁned
by the intrinsic angular condition that the opposite angles are equal. Every interior

T. Tachi

vertex on a discrete Voss surface is incident to 4 edges (or foldlines) and sector
angles θi (i = 0, 1, 2, 3) that satisfy the following:
θ0 = θ2andθ1 = θ3 .

(4)Note the similarity to (1): in fact, the set of planes that form an origami vertex andeggbox vertex are identical (Figure 3). According to [Schief et al. 2007], the folding
angles ρi and ρ j are similarly written as the following form:ρAi, j cot 2j (i − j = 1 or 3 mod 4)
ρi
tan =
ρ
2
(i − j = 2 mod 4).
tan 2jIf we measure the folding angles of one of the opposite pairs by their complementary
angles ρi = π − ρi , this becomesρρiAi, j tan 2j (i − j = 1 or 3 mod 4)
(5)
tan =
ρ
2
(i − j = 2 mod 4).
tan 2jTherefore the kinematics of Miura-ori and discrete Voss surface are essentially identical as represented by Equation (3).
2.3

Hybrid Surface: Bidirectionally Flat-foldable Planar Quadrilateral Mesh

Here, we introduce a new type of functional patterns, i.e., BDFFPQ mesh (deﬁned
later), based on combining the geometry of origami and discrete-Voss surface We
introduce the concept of complementary foldlines, which are edges whose folding
angles are measured by their complementary angles, thus a complementary foldline
folds from ±π to 0 when an ordinary foldline folds from 0 to ±π. By using complementary foldlines (CFLs) along with foldlines (FLs), we can produce a 4-valency
network that consistently joins the vertices of origami and discrete Voss surfaces, sothat one-DOF kinetic motion exits.
In order to understand this behavior, we deﬁne extended “developed state” and “ﬂatfolded state” as follows:
Developed state: A ﬂat state in which every edge has rotational angle of 0 (thusfoldlines are unfolded and complementary foldlines are folded).
Flat-folded state: A ﬂat state in which every edge has rotational angle of ±π (thesign shows mountain or valley).
Here, notice that complementary foldlines and foldlines are interchangeable by calling a developed state a ﬂat-folded state and vice versa.
Here we deﬁne bidirectionally ﬂat-foldable planar quadrilateral mesh (BDFFPQmesh) a 4-valency network mesh surface that follows the following conditions.
• The surface is polyhedral, i.e., composed of planar facets, and every edge iseither FL or CFL.

Rigid-foldable Flat-foldable Surface• The surface has developed and ﬂat-folded states, where the intersection of theoverlapped facets is ignored.
• Each interior vertex’s incident edges are either 4 FLs, 4 CFLs, or 2 FLs +2CFLs in which a pair of CFLs or FLs is an opposite pair.
Rigid foldability of such a surface is represented as follows when the surface ishomeomorphic to a disk.
Theorem 1 If and only if BDFFPQ mesh homeomorphic to a disk with more than
one interior vertex has one intermediate folded state, the surface is ﬁnitely rigidfoldable.
Proof: Rigid-foldability of a polyhedral disk surface can be represented by theexistence of a continuous solution in the conﬁguration space. From the developability and ﬂat-foldability condition, it follows that subnetwork composed of only
FLs (CFLs) must divide the surface completely into one or multiple regions each ofwhich is assigned with either +1 or −1 such that the signs of adjacent regions areopposite. The sign indicates the orientation of facet in the ﬂat-folded (developed)
state. Then the developability and ﬂat-foldability of the network for each vertex isrepresented as follows:∑3i=0 σdev (i)θi = 0 for 4CFL or 2FL + 2CFL vertex(6)developability:
for 4FL vertex,∑3i=0 θi = 2π
∑3i=0 σff (i)θi = 0 for 4FL or 2FL + 2CFL vertex
ﬂat-foldability:(7)
for 4CFL vertex,∑3i=0 θi = 2π
where σdev (i) and σff (i) indicate the assigned signs of the facet incident to the sector
angle i in the developed and ﬂat-folded states respectively.
A 2FL+2CFL vertex satisﬁes ∑3i=0 σ(i)θi = 0 for both σ in developed and ﬂatfolded states ({σ(0), σ(1), σ(2), σ(3)} = {1, 1, −1, −1} and {1, −1, −1, 1}); therefore θ0 = θ2 and θ1 = θ3 . Hence, the vertex is a discrete Voss vertex and the folding
motion follows (5). A 4FL or 4 CFL vertex is essentially a vertex of Miura-ori andthe folding angles of incident edges follow the kinetic motion represented by (3).
Since we represent the folding angles of complementary foldlines by their complementary angles, these relations can be represented in the following single form:
tan

ρj
ρi
= Ai, j tan ,22where Ai, j = 1/Ai, j for 4CFL vertices. Since all the foldlines are connected to eachother, the transformation follows (3).
 Therefore if and only if there exists one inρi (t0 )termediate conﬁguration tan 2 , there exists a continuous valid conﬁgurationstan ρi2(t) .
2

T. Tachi

3

Design Variations of Rigid Foldable Surface

For obtaining design variations, we adopt the perturbation based approach used in[Tachi 2009a]. We ﬁrst obtain a valid existing BDFFPQ mesh in a ﬂat state; thiscan be done quite easily, e.g., by using a square grid and assigning mountain, valley,
complementary mountain, and complementary valley to the edges. Here, we canroughly design the kinetic behavior since an origami vertex tries to expand the surface in two directions at the same time, while a discrete Voss vertex tries to expand
in one direction while collapsing in the other direction.
Then we deform the pattern while sustaining the developability and ﬂat-foldability
conditions. The conﬁguration of the mesh surface is represented by the coordinatesof vertices x forming a triangular mesh whose edges are either triangulation edges
or complementary or ordinary foldlines. This triangular mesh satisﬁes the planarity
condition for each facet and ﬂat-foldability (7) and developability (6) conditions foreach interior vertex. Thus an inﬁnitesimal deformation of mesh conﬁguration mustsatisfy:
⎤
⎡ dev
c (x)
(8)
c (x) = ⎣ cff (x) ⎦ = 0,
cplanar (x)where cdev (x) is a N Vint -vector (N Vint is the number of interior vertices) whose element is∑3i=0 σdev (i)θi for 4CFL or 2FL + 2CFL vertex
for 4FLvertex,
2π − ∑3i=0 θicff (x) is a N Vint -vector whose element is∑3i=0 σff (i)θi for 4FL or 2FL + 2CFL vertex
2π − ∑3i=0 θi for 4CFL vertex,
and cplanar (x) is a vector with dimension of the number of triangulating edges, andits element is its corresponding folding angle ρi .
Since the number of variables exceeds the number of constraints for a quadrilateralmesh structure with multiple boundary edges, the Jacobian matrix of the constraint∂c∂x is a rectangular matrix whose number of columns exceeds the number of rows.
An inﬁnitesimal solution space of this constraint can be calculated as:    
∂c + ∂c(9)Δx = I −
Δx0 ,
∂x
∂x
+

∂c
where ∂xis the pseudo-inverse (Moore-Penrose generalized inverse) of the Jacobian matrix, and Δx0 represents an arbitrary vector. Equation (9) calculates the valid
perturbation closest to Δx0 by orthogonal projection to the solution space; therefore,

Rigid-foldable Flat-foldable Surface

the user input such as the drag motion of vertices through GUI can be used as Δx0
in the implementation system of the method. A larger deformation of the shape canbe achieved by accumulating small deformation using Euler integration while eliminating the residual c by the Newton-Raphson method. Figure 4 shows an example
process of design in a design system that can interactively solve Equation 9 whiledisplaying the 3D conﬁguration, ﬂat-folded state, and developed state of the surface.
Figure 4 shows an example process of design in a design system that can interactively solve Equation 9 while displaying the 3D conﬁguration, ﬂat-folded state, anddeveloped state of the surface.

 
 

Figure 4: The design process of hybrid pattern obtained by deforming a square grid.
Red and dark blue lines indicate mountain and valley foldlines, and pink and light
blue indicate complementary mountain and valley foldlines. Each screen shows
3D conﬁguration (left) developed state (upper-right) and ﬂat-foldable shape (lowerright).

The proposed method succeeded in generalizing eggbox pattern and hybrid
BDFFPQ mesh. Figure 5 shows an example of a generalized eggbox pattern thathas “globally” positive curvature surface, and Figure 6 is an example complex
foldable structure based on hybrid surfaces.

4

Rigid Foldable Cylinder

In an architectural context, collapsible cylindrical surfaces are a signiﬁcant design
target since a cylinder qualitatively surrounds a volume of space. Here, note thata cylindrical surface is one of the best possible solutions in this direction since a

T. Tachi

Figure 5: The folding motion of a generalized eggbox pattern.

Figure 6: The folding motion of hybrid BDFFPQ mesh.

topologically closed polyhedron, which is ideal in the sense of enclosure of a space,is actually known to forbid a rigid folding motion with volume change as the bellows
conjecture [Connelly et al. 1997] states. Although several designs are proposed toenable collapsible and non-rigid-foldable cylinders such as triangulated cylinders[Guest and Pellegrino 1994] and single curved corrugated surface [Hoberman 1993],
these exiting designs relied on the ﬂexibility of each facet for the folding motion;
they were not applicable to smooth kinetic mechanisms that work with stiff materials
and mechanical hinges often required for architectural-scale structures.
[Tachi 2009b] gives a family of cylindrical surfaces and compounds of cylinders
that are rigid-foldable and ﬂat-foldable, by solving the angular identity equations via
constructing a symmetric modular cylinder based on rotational symmetry and thenarray copying the obtained module in an axial direction. This successfully produces

Rigid-foldable Flat-foldable Surface

xi+1

yi+1

zi+1
θi

Ti,i+1(ρi)

ρi

yi

3 2
4 1
5
0

zi
xi

Figure 7: A closed strip of facets (Left). A loop can continuously shrink or expand
using conditions around vertices (Right).

a folding motion that does not separate the structure into an open disk. However,the derived surface can only have a constant section along an axis in this approach,and the design variations are limited. Producing freely designed cylindrical surface
is the goal of the work described in this section.
The biggest issue in the design of cylindrical surface is that the local condition
around each interior vertex is not enough because loops surrounding a hole cangeometrically separate by folding; and therefore cylindrical BDFFPQ mesh is notproved to be rigid-foldable in general. In general, it is required that for any closed
strip consisting of k facets, the strip does not separate by the folding motion of
hinges. We assign a local coordinate to each facet i and denote the local afﬁnetransformation from i-th to i + 1-th coordinates by 4 × 4 matrix Ti,i+1 ({ρi }) (Figure
7 Left). This transformation is regular and the inverse can be described as T−1
i, j =
T j,i . The condition thus is represented as, there exists a vector function ρi (t) (0 ≤t ≤ π) that satisﬁes the following for any loop of facet strip:
T0,1 T1,2 · · · Ti,i+1 · · · Tk,0 ≡ I.

(10)

If the facet fan around each vertex is ensured to be rigid-foldable, loop condition
around the vertex modiﬁes the condition along a loop to another homologous loop.
For example, in Figure 7 right, the transformation from facet 0 to 5 can be simpliﬁed
as:(11)
T0,1 T1,2 T2,3 T3,4 T4,5 ≡ T0,1 T1,4 T4,5 ,using the condition around an interior vertex: T1,2 T2,3 T3,4 T4,1 ≡ I. Therefore, therigid-foldability of a disk with a hole (or n holes) can be represented by the combination of the local rigid-foldability around each interior vertex and the condition
around the hole (or the n-holes).
Since BDFFPQ mesh satisﬁes local rigid-foldability conditions around vertices, theproblem of obtaining a rigid-foldable cylindrical BDFFPQ mesh turns into a problem of obtaining a state with one rigid-foldable loop connected to the surface. We
transform an existing rigid-foldable cylinder while ensuring the rigid-foldability of
one loop satisﬁed. In order to do this, we use an isotropic type of rigid-foldable
cylinders from [Tachi 2009b] as an initial state, which are symmetrically repeating
structures constructed using the combination of degree-4 vertices termed folds andT. Tachi

elbows producing a valid loop motion, where a fold and an elbow are symmetriccases of origami vertex and discrete Voss vertex, respectively, in our interpretation.
Therefore the cylinder is a symmetric type of rigid-foldable cylindrical BDFFPQ.
Since the exact rigid-foldability condition around a loop is still not fully investigated, we guarantee the rigid-foldability of the whole model by using a sufﬁcientcondition: one loop around the hole is unchanged from the original symmetric cylinder. We pick up a strip loop along one of the boundaries that exactly produces a oneDOF motion and rigidify this part using a rigid bar model, while we deform otherparts under the local conditions of developability, ﬂat-foldability, and planarity ofpolygons. The developability and ﬂat-foldability keep the sector angles incident tovertices between rigid strip and ﬂexible BDFFPQ strips ﬁxed and enable a one-DOF
motion.
We obtained asymmetric form variations from one symmetric cylindrical structureusing the design system that solves BDFFPQ mesh and extra constraints. The design process is shown in Figure 8, and Figure 9 shows the derived crease pattern ofthe design. The pattern shows the location of foldlines in the developed state (notethat the lines are drawn on multiple layers since we use extended deﬁnition of developed state). Here, we can notice sets of parallel lines remaining in the generalized
pattern. This implies the limitation of our method since this globally symmetric behavior forbids cylinders to change their overall radius; this may restrict the design
applications of the cylinders. Since this behavior presumably originates in the ﬁxed
boundary of the surface, investigating the exact rigid-foldability condition around aloop and loosening the design constraints can contribute to more ﬂexible designs ofrigid-foldable cylinders, which still remains to be a future work.

5

Materialization

Because of the developability of the surface, the structure applied for a small scale
object can be manufactured from sheets of material such as paper and plastics. The
pattern can be ﬁrst perforated by cutting plotter, laser cutter, or other CNC 2-axis
machines on sheets of thin and hard materials, then pasted together, and folded
to form a three-dimensional shape. Careful folding control is needed only at the
beginning of the fold; once it is semi-folded, the surface folds automatically since
the pattern produces a one-DOF motion. Figure 10 shows a folded model of a rigidfoldable cylinder designed using the proposed method.
For a larger scale architectural structure such as shades, retractable roofs, and transformable partitions, the following thickening method can be used to produce a robust kinetic structure. Each facet is ﬁrst substituted by two layers of constant thickness panels whose contact plane is the ideal geometric surface. In order to avoid theintersection of panels by the folding motion of 0 ≤ ρi ≤ π − δi for each foldline, theboundary of the panel in the valley side of the surface is relocated on an offset ofthe foldline by t cot 2δ , where t is the thickness of the panel (Figure 11). This yields

Rigid-foldable Flat-foldable Surface

 
 

Figure 8: A design process of rigid-foldable cylinder. The initial state is a cylinder
by [Tachi 2009b]. The surface is deformed keeping the condition of BDFFPQ meshwhile a loop strip (two strips in this redundant case) is maintained to be rigid, as
indicated by green rigid bars.

Rigid Part
Figure 9: A crease pattern of generalized cylinder.

designs of kinetic structure composed of rigid panels using their edges as rotational
hinges as shown in Figures 12 and 13. The hinges can be constructed mechanicallyor using a sheet or cloth of negligible thickness between panels; in the latter method,
the cloth also becomes a watertight covering.

T. Tachi

Figure 10: Folded model of a rigid-foldable cylinder. Left: in the developed state,
Middle:Intermediate state, Right: ﬂat-folded state.

δtt cot(δ/2)

Figure 11: Thickening using two layers of constant thickness panels.

Figure 12: Cylindrical rigid-foldable BDFFPQ mesh consisting of thick panels.

6

Conclusion

In this paper, we presented a design method of rigid-foldable ﬂat-foldable disks
and cylinders by introducing bidirectionally ﬂat-foldable planar quadrilateral(BDFFPQ) meshes and their generalization method. The concepts of BDFFPQ
mesh, complementary foldlines, and extended deﬁnitions of developed and ﬂatfolded states successfully unify and generalize the ﬂat-foldable origami and discrete
Voss surfaces. This generalization enables the perturbation based method that can

Rigid-foldable Flat-foldable Surface

Figure 13: Folding motion of cylindrical structure using thick panels.

freely control the shape of the surface in a direct way; the method yields various
designs of rigid-foldable and bidirectionally ﬂat-foldable polyhedral disk. We also
investigated the rigid-foldability of a cylindrical surface and its sufﬁcient condition
to design a rigid-foldable cylinder. A design method by modifying a cylindrical
BDFFPQ mesh with one ﬁxed loop copied from a known rigid-foldable cylindricalstructure is introduced. The method enables us to obtain a novel generalized formof rigid-foldable ﬂat-foldable surfaces that can be implemented for architecturaldesign purposes.
It should be noted that we used sufﬁcient condition of the rigid-foldability in thispaper and did not provide the exact condition, which still remains to be an unsolved
problem as it is seen in [Stachel 2010]. The sufﬁcient condition for enabling cylindrical topology also made the deformation less ﬂexible, the effect of which can beobserved in the globally parallel edges. Deriving a more general condition of therigid-foldability of cylindrical surface and thus enabling a less constrained design is
one of the future works of this study.

T. Tachi

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Authors’ address:
Tomohiro Tachi (tachi@idea.c.u-tokyo.ac.jp): The University of Tokyo 3-8-1,
Komaba, Meguro-Ku, Tokyo, 153-8902, Japan.