Journal for Geometry and Graphics
Volume VOL (YEAR), No. NO, 1-13.

Design of Infinitesimally and Finitely Flexible
Origami Based on Reciprocal Figures
Tomohiro Tachi
Graduate School of Arts and Sciences, The University of Tokyo
3-8-1 Komaba, Meguro-Ku, Tokyo 153-8902, Japanemail: tachi@idea.c.u-tokyo.ac.jp

Abstract. This paper investigates novel computational design method for inﬁnitesimally and ﬁnitely foldable rigid origami based on solving a ﬁrst-order folding mode, which can be represented by a reciprocal ﬁgure. We derive thesegraphical conditions from a matrix representation of rigid origami, and extend
the conditions to cases when the surface includes holes. We propose an algorithm
to obtain forms that satisfy the conditions and an interactive system to freely
design inﬁnitesimally foldable forms. We show design examples of shaky polyhedron and origami, and ﬁnitely foldable quadrivalent mesh origami.
Key Words: Origami, Rigid origami, Inﬁnitesimal ﬂexibility, Shaky polyhedron,
Reciprocal ﬁgure

1. Introduction
Rigid origami is a polyhedral origami modeled using rigid panels and rotational hinges. The
designs of mechanisms and the structures based on the kinematic property of rigid origami areinteresting topics in the ﬁeld of origami engineering, since they can facilitate the developmentof possible applications of deployable structures and ﬂexible meta-material at diﬀerent scales,e.g., from medical devices inside the human body to architectural and space structures. Froma theoretical viewpoint, rigid origami structures are especially interesting when they acquire
ﬂexibility from redundant mechanical constraints based on their non-trivial singular geometricconﬁguration, such as ﬂexible closed polyhedra and rigid-foldable quadrivalent mesh origami
structures. In designs using rigid origami mechanisms, such redundancies can be used forremoving a part of the structure without disturbing the mechanism or combining multiple
compatible parts to form more complex and robust structures.
Inﬁnitesimal and ﬁnite kinematic properties of polyhedral surfaces have been studied theoretically and have also been associated with reciprocal ﬁgures in 3D, speciﬁcally reciprocalparallel, by Sauer [6] for quadrangle meshes. Schief et. al. [7] studied reciprocal ﬁgures thatc YEAR Heldermann Verlag, Berlin
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T. Tachi: Flexible Origami Based on Reciprocal Figures

also have second-order ﬂexibility. Redundant ﬁnite mechanisms have been studied as well,e.g., ﬂexibility of closed polyhedra [3] and ﬂexibility of quad panel meshes [7, 9, 8]. Despite
these theoretical studies, design methods of such structures are limited. A computational
system for bi-directionally ﬂat-foldable quad mesh origami [9] facilitates interactive formﬁnding of ﬁnitely foldable rigid origami structures with redundant constraints. The drawback ofthis method is that the system relies on the existence of a valid state in a half-folded form.
In this paper, we propose a novel design method for obtaining inﬁnitesimally and ﬁnitely
transformable origami and polyhedral surfaces by developing an interactive design systemthat solves the condition of the existence of an inﬁnitesimal folding mode, which can begraphically represented by a reciprocal ﬁgure.
In Section 2, we derive the inﬁnitesimal ﬂexibility conditions for disk meshes equivalent
to existing results [6, 7], using a matrix representation of transformation. Next we naturally
extend the conditions to the case of a disk with holes by distinguishing conditions around eachhole. In Section 3, we propose a computational method for obtaining a valid conﬁguration
and a reciprocal ﬁgure. We ﬁrst obtain an initial conﬁguration based on optimization anduse the same system to explore possible solutions by projecting perturbation in the solution
space. Using the system, we can obtain design variations of inﬁnitesimally foldable closed
polyhedron, as shown in Section 4. In Section 5, we apply a second-order ﬂexibility condition
represented by reciprocal ﬁgure to a bi-directionally ﬂat-foldable quadrivalent 2D crease pattern. Despite the lack of a proof, we observed that this condition yields ﬁnitely rigid-foldable
origami patterns. One of the results is “curved rigid origami,” which is a discretization of
the curved folding that acts as a one-degree-of-freedom(1-DOF) mechanism.

2. Infinitesimal Folding Mode and Reciprocal Figure
A rigid origami structure comprises rigid panels and hinge joints at the foldlines, and can bemodeled like a robot arm; however it has many closed loops. The constraints of its overallmechanism can be described using matrix equations as in [4, 2], which are redundantly given
for any possible sequence of foldlines forming a closed-loop strip. Here, we will focus only on
the kinematics and ignore self-collision.
In the case of a rigid origami model that is homeomorphic to a disk, it is suﬃcient
to consider only the constraints resulting from the closed faceted fan around each interiorvertex. This is because any loop condition decompose into a contracted loop condition and aninterior vertex condition, which further decomposes into multiple interior vertex conditions
after several iterations (Figure 1). This implies that this condition is suﬃciently applicable
to a simply connected 2-manifold.
Single-Vertex Constraint The constraint along the fan around an interior vertex is described as follows. Let the fold angle of the vertex’s incident foldline i be denoted by ρiand the angle between adjacent foldlines i and i + 1 be denoted by θi,i+1 . Here, note that
there are n incident foldlines incrementally numbered counterclockwise, and their index isnumbered in modulo n. Then, the rotation by foldline i is described by a 3 × 3 matrix χi−1,i ,
which transforms the local coordinate of a facet i − 1, i to that of facet i, i + 1. Let the local
coordinate of facet i − 1, i have its x and z axes along the foldline and the facet’s normal,

T. Tachi: Flexible Origami Based on Reciprocal Figures

3

Figure 1: Contraction of a closed loop. This also implies that a non-simply connected surface
requires additional treatment.

ρi

Mi χ
Ni L i−1,ii

Ni−1 Mi−1
Li−1

θi,i−1Figure 2: Rotation around a fan.
respectively. χi−1,i is represented as the sequence of a pitch and then yaw (Figure 2).
cos θi−1,i − sin θi−1,i 0100χi−1,i = Yi−1,i Pi =  sin θi−1,i cos θi−1,i 0 0 cos ρi − sin ρi  .
00
1
0 sin ρi cos ρi(1)

In order to maintain consistency as a closed fan, sequentially applying the rotations aroundthe closed loop must transform the orientation of the ﬁrst facet, i.e., 0, 1 to be identical toits original orientation.
R (ρ0 , · · · , ρn−1 ) = χ0,1 χ1,2 · · · χn−1,0 = I.

(2)

Infinitesimal Motion The derivative of the constraint with respect to ρi is calculated asfollows. Deﬁne an orthogonal matrix Ti representing the orientation of the facet i, i + 1 withrespect to the facet 0, 1 as

 x
x
x
N
M
L
i
i
i
[
]
Ti = χ0,1 · · · χi−1,i = Li Mi Ni = Lyi Miy Niy  ,Lzi Miz Niz
where Li is the direction vector of edge i and Ni is the normal of facet i, i + 1 with respect
to the local coordinate system whose x and z axes orient the direction of foldline 0 and the[ ] [ ]T
normal of facet 0, 1. Then, Equation (2) is written as R = Ti Ti = I.

4

T. Tachi: Flexible Origami Based on Reciprocal Figures

By using
0 0 0∂χi−1,i∂Pi= (Yi−1,i Pi ) P−1= χi−1,i 0 0 −1 ,
i
∂ρi
∂ρi
0 1 0

we obtain the derivative of Equation (2) as



y 
z
0
0
0
0
−L
L
i
i
[ ]T
∂R [ ] 0−Lxi  .
= Ti 0 0 −1 Ti =  Lzi
∂ρi
0 1 0
−Lyi Lxi0
The constraints can be written using three independent equations:R(2, 3)f = R(3, 1) = 0.
R(1, 2)(3)(4)

The inﬁnitesimal folding motion (∆ρ1 , · · · , ∆ρn ) satisﬁes
n−1
∑
∂f
i=0

∂ρi

∆ρi =

n−1
∑

Li ∆ρi = 0.

(5)

i=0

Here, Li is the normalized direction vector of edge i from a vertex to the adjacent vertex in
the local coordinate system of facet 0, 1.
Now, we consider the global system given by the intersection of the constraints from eachinterior vertex. We denote the position of vertex v (v = 1, · · · , V ) by xv , and the rotation
along the foldline incident to vertex u and v by ρu,v = ρv,u or ρe (e = 1, · · · , Ein ), where V
and Ein are the numbers of total vertices and fold lines respectively. For later convenience,
we let 1, · · · , Vin be the interior vertices and Vin + 1, · · · , V be the exterior vertices, where Vin
is the number of interior vertices.
We apply a rotation Tu to Equation (5) so that for any edge u, v, L′u,v = Tu Lu,v′
uis the direction vector of the edge in the global coordinate system, i.e., ∂ρ∂fu,v= L′u,v =(xu − xv ) / ∥xu − xv ∥. The inﬁnitesimal folding motion satisﬁes, for each interior vertex
v = 1, · · · , Vin ,∑(xu − xv )∆ρu,v = 0.
(6)
∥xu − xv ∥u adjacent to v
Reciprocal Figure This equation is equivalent to an equilibrium condition when ∆ρu,v
is replaced by the axial force applied to the lines. As in the ﬁeld of graphical statics wherereciprocal ﬁgures have been used to solve an equilibrium of a 2D frameworks [5, 1], we consider
a 3D reciprocal ﬁgure of origami foldlines.
Because we are interested in a 3D reciprocal ﬁgure, we consider reciprocal ﬁgures tobe the dual graph of a foldline network in 3D space whose corresponding edges are parallel(instead of perpendicular), i.e., “reciprocal-parallel” as deﬁned by Sauer [6]. The dual graph isconstructible because the origami surface is a manifold; and the interior vertex, foldlines, andT. Tachi: Flexible Origami Based on Reciprocal Figures

b

b

c

d

U

c

a

d

U
f

V
f
e

5

e

V

a

Mountain
ValleyFigure 3: Left: A foldline network. Right: A reciprocal ﬁgure of the foldline network. The
illustration shows the planar case with zero signed area (Section 2.2).
facets of the polyhedral surface correspond to polygons, segments, and nodes, respectively.
Here, note that we use only foldlines (edges between facets) but not boundary edges (Figure
3).
Theorem 1 (Schief et al. [7]) The existence of a (non-degenerate) reciprocal ﬁgure of the
foldline network of a rigid origami structure homeomorphic to a disk is equivalent to theexistence of a ﬁrst-order folding mode of the rigid origami structure.
Proof: Assume that a valid ﬁrst order folding motion exists. Then, we construct a dualgraph of the foldline network such that the node coordinate x′f is the angular velocity of thecorresponding facet f . For each adjacent pair of facets f and g and the foldline u, v between(xu −xv )them, x′f − x′g = ∥x∆ρu,v ; thus, the graph is a reciprocal ﬁgure of the foldline network.
u −xv ∥
Similarly, assume that a reciprocal ﬁgure of the foldline network exists, in which thecoordinate of the node corresponding to facet f is denoted by x′f . Then, we let the angularspeed of each foldline u, v between facets f and g be ∥x′f − x′g ∥. This set of angular speedsof follines satisﬁes Equation (6), thereby representing a valid inﬁnitesimal folding motion. □2.1. Surface with holes
Non-disk origami, or speciﬁcally non-simply connected origami, e.g., cylindrical origami ororigami with holes, is often applied to obtain collapsible structures. The inﬁnitesimal folding
mode of an origami structure with holes can be represented by the intersection of constraints
around each interior vertex and additional constraints for the loop around each hole. Unlike
the constraint around a vertex, there are six constraints around a loop, which correspond torotation and translation.
Similar to a single vertex, we represent transformation by matrices, but in this case,
we use a 4 × 4 homogeneous matrix so that we can take both translation and rotation into
account.

cos θi−1,i − sin θi−1,i 0 dxi−1,i1000 sin θi−1,i cos θi−1,i 0 dyi−1,i  0 cos ρi − sin ρi 0

,
(7)
χi−1,i = Yi−1,i Pi = 001
0  0 sin ρi cos ρi 0
0
0
0
1
0
0
0
1

6

T. Tachi: Flexible Origami Based on Reciprocal Figures

ρi
Mi L iNi
N
Mi−1 i−1dθi−1,i i−1,iLi−1
Figure 4: Rotation around a hole.
()T
where di−1,i = dxi−1,i , dyi−1,i , 0 represents the origin point of facet i, i + 1 observed from thelocal coordinate of i − 1, i (Figure 4). Then, the orientation and position of facet i, i + 1 withrespect to facet 0, 1 is][
Li Mi Ni Oi
,
Ti =
0 0
0 1
where Oi represents the position of the origin observed from facet 0, 1. The derivative of
Equation (2) is calculated as

0−Lzi Lyi
 Lzi
∂R
0
Lxi
= 
y
 −Li Lxi0∂ρi0

Li × Oi 


0





R(2, 3)
R(1, 4)
We obtain six independent equations: f = R(3, 1) = 0 and, g = R(2, 4) = 0. Here,R(1, 2)R(3, 4)
f = 0 and g = 0 represent the closure conditions of rotation and translation, respectively.
We can similarly use a global coordinate system to replace Li by L′u,v = (xu − xv ) / ∥xu − xv ∥and Oi by O′u = xu .
Since f = 0 reduces to Equation (6), the overall constraints for the inﬁnitesimal folding
motion are as follows.
1. For each interior vertex v and for each hole with incident foldlines u, v (v is on theboundary), Equation (6) is satisﬁed and,2. For each hole,∑incident foldline u,v((xu − xv )∆ρu,v∥xu − xv ∥)× xv = 0.

(8)

Similarly to the case of disk origami, these equations are equivalent to an equilibrium condition when the hole is replaced by a rigid body. Here, Equations (6) and (8) represent theequilibrium of forces and moments, respectively.

T. Tachi: Flexible Origami Based on Reciprocal Figures

7

A valid reciprocal ﬁgure of the foldline network can be constructed similarly to the case ofa disk. However, the reciprocal ﬁgure is only a necessary condition for inﬁnitesimal foldability,
and is not a suﬃcient condition.
2.2. Conditions in Degenerate Planar Surfaces
In the previous sections,we considered a polyhedral surface in general, which is not necessarily
a developable surface, although the term“origami” typically implies developability. Developable origami surfaces can be manufactured just by folding a sheet of paper; in this processof manufacturing, the existence of folding motion matters. We have observed from examples
that when we apply developability and ﬂexibility conditions at the same time, the surface
often converges to a completely ﬂat state. When every vertex lies in a plane, the constraints
given by a reciprocal ﬁgure degenerate, and one of the three constraints for each vertex islost. In this case, the additional ﬂexibility is not desirable since it represents no valid folding
motion, but rather lends a trivial shakiness in the planar state.
In order to ﬁnd a valid folding motion in the ﬂat state, we apply the condition thatthe signed area of the cell in the reciprocal ﬁgure is zero as is introduced by Watanabe andKawaguchi [11], as the second-order folding mode of origami in the ﬂat state. This zero-area
condition is represented as
n−1 ∑
i−1
∑
Lj ∆ρj × Li ∆ρi = 0.
(9)i=0 j=0

Since Li lies in a plane, the orientation of this vector is always kept perpendicular to theplane. Therefore, only one out of three equations is independent, and the vector orientationis always perpendicular to the constraints of Equation (5). Therefore, the intersection of
Equations (5,9) can be represented by their linear combination as
(
)
i−1
n−1
∑
∑
Li ∆ρi +Lj ∆ρj × Li ∆ρi = 0.
(10)
j=0

i=0

Figure 5 shows example patterns with and without the zero-area conditions.

3. Computational Method
We propose a novel interactive method for designing variations of inﬁnitesimally foldable
origami structures. The method is especially useful when achieving a ﬂexible state of anorigami structure in which the number of constraints exceeds the number of variables; thistype of origami generically becomes a static structure. The conﬁguration of an origami structure with an inﬁnitesimal folding mode is represented by its vertex coordinates and reciprocal)T(ﬁgure. We parameterize the conﬁguration by the vertex coordinates X = xT1 , · · · , xTV andthe ratios W = (w1 , · · · , wEin )T of the edges. Each element of W corresponding to edge
e connecting vertices u, v is represented we = ∆ρe /ℓe . The constraints in Equation (6) are
re-written as∑∆fv =wu,v (xu − xv ) = 0(11)u incident to v

8

T. Tachi: Flexible Origami Based on Reciprocal Figures
b

a

b

c'

a
c'

c b b'
a a'
d' d
f
e
e' f'

f'
e'

c' d'
b'

d

e
f

a'

b'
c b
a a'
d' d
e f
e' f'
c'

d'

f'

e
f

e'
a'

d
c

c

b'

Figure 5: Left: A pattern without the zero-area condition. Note the asymmetric quadrangles
and hexagons with positive- and negative- signed area. This pattern is a bistable structure
when the panel is not triangulated. Right: Modiﬁed pattern with the zero-area condition.
This pattern has second-order ﬂexibility in a planar state.
Here, the sign of we indicates wheter the fold angle increases or decreases by the folding. In
order to avoid the trivial solution that the angular velocity of every foldline equals 0, we set
the following conditions:{we < −α folding mountain or unfolding valley,(12)
we > αfolding valley or unfolding mountain
where α can be set to an arbitrary positive value proportional to some scale factor.
Initial Solution We solve Equations (11) and (12) by solving
U=

Vin
∑
v=1

∥∆fv ∥ +2

Ein∑

max (0, α − |we |)2 = 0

e=1

with respect to X and W. The solution is not unique, and we ﬁnd a solution close to theoriginal conﬁguration by the following two steps.
1. Minimize U with respect to W while X is ﬁxed.
2. Minimize U with respect to X and W using the previous result as the initial conﬁguration.
If 3V < E as in a triangular disk mesh, step 1 alone yields U = 0. An interesting case occurs
when 3V ≥ E, e.g., quadrangle based origami pattern or a cylindrical or closed polyhedron,
which suggests that the structure is generically static. The minimization problem in step 1
converges to the least square solution of an over-constrained system, and then the pattern
is modiﬁed to a ﬂexible conﬁguration with singularity in step 2. For solving the above
optimization problems, the conjugate gradient method is used.
Exploring Solution Space Once we obtain a solution, we can get variations of the pat}T{
terns using a perturbation-based method that uses F (X, W) = ∆f1T , · · · , ∆fVTin = 0 as a

T. Tachi: Flexible Origami Based on Reciprocal Figures9

geometric constraint that forms a multi-dimenisonal continuous solution space. A plausible
deformation represented by 3V + E-vector [∆X ∆W]T must satisfy[] [ ∂F ∂F ] [][ ] ∆X
∆X
∂X
∂W
C
= ∂G
= 0,
(13)
∂G
∆W
∆W
∂X
∂W
where G represents other optional constraints such as the penalty constraints derived from
Equation (12), developability, and ﬂat-foldability, i.e., the constraints combined with theconstraints described in [10]. The elements are calculated as follows.
 ∑− kincident to v wv,k if u = v∂∆fv 
= wv,u(14)
if u is incident to v

∂xu

0
else
{
xk − xv if u = v (or xu − xv if v = k)
∂∆fv
=
(15)
∂wu,k
0
else
Equations (8,10) are polynomial functions of wu,v and xu ; the derivative functions in thesecases can be similarly calculated.
We apply the perturbation based method similar to [10] using the Moore-Penrose generalized inverse C+ of the constraint matrix C as follows:
[
]
[
]
(
) ∆X0
∆X
+
= I−C C
,
(16)
∆W
∆W0
[
]
∆X0
where
is an arbitrarily assigned initial deformation, which is projected orthogonally∆W0
to the constrained subspace by multiplying with (I − C+ C); this results in a valid deformationthat is as close a s possible to the given initial deformation.
We implemented the constraints and variables by improving freeform origami [10]. In
this system, the user can freely select and drag points on the rendered origami structure
through a 2D input device to apply a deformation, e.g., translation of a point or a weighed
point-set translation or rotation.
Beforeactually applying
the
[
]
[
] deformation to the current∆X0∆X
state, the deformation vector
is substituted by
to obtain a constrained valid
0∆W
next state. Since the above-mentioned Euler’s integration accumulates the error, the residualis eliminated based on Newton-Raphson’s method for each step. The generalized inverse
solution for each step is calculated using the conjugate gradient method. In this way, we
immediately achieve a valid variation of forms in an interactive manner.

4. Shaky Closed Polyhedron
By using the system, we can design a variation of a shaky, i.e., inﬁnitesimally transformable,
closed polyhedron. Since the number of constraints equals the number of variables in a
generic closed 3D polyhedron, a shaky polyhedron is a non-trivial singular polyhedron. Since
the shaky polyhedron must include valley fold lines, we ﬁnd its initial topology by ﬂipping
the edges of a convex polyhedron. Figures 6 and 7 show examples of shaky polyhedra.

10

T. Tachi: Flexible Origami Based on Reciprocal Figures

Figure 6: Left: A snub dodecahedron with ﬂipped edges. Middle: A shaky polyhedronobtained by two-step optimization to satisfy the inﬁnitesimal foldability condition. Right:
The freeform variation wherein shakiness and planarity of the pentagonal facet are preserved.

Figure 7: A shaky closed polyhedral tower with partial developability (the entire surface isdevelopable except for the top and bottom vertices indicated in the ﬁgure). This polyhedronproduces a twisting motion when it is compressed.

5. Finitely Foldable Quadrivalent Mesh from Crease Pattern
It is known that a quadrivalent mesh with planar facets with developability and ﬂat-foldability
is ﬁnitely transformable if there exists an intermediate 3D state [9]; the limitation of thismethod was that we needed to start from a valid 3D conﬁguration, instead of a 2D crease
pattern. By applying Equation (10) for each interior vertex instead of the existence of 3D
state, we can modify a bi-directionally ﬂat-foldable quadrivalent pattern into a ﬁnite rigid
foldable pattern (Figure 8). An interesting side eﬀect of using this method for designing
discrete version of curved folding represented as planar quad mesh is that the smoothness ofthe folding is naturally obtained. This successfully avoids the jaggy artifacts that existed in
the discrete curved folding representation in [10].

T. Tachi: Flexible Origami Based on Reciprocal Figures

(a)

11

(b)

Figure 8: (a) An invalid pattern that satisﬁes developability and ﬂat-foldability. (b) A valid
ﬁnitely foldable pattern is obtained when Equation (10 is introduced. The method alsoachieves the “Smoothness” of discretized curved folding.

6. Origami and Tensegrity
The equivalence of an inﬁnitesimal folding mode of a polyhedral surface and the equilibriumof forces implies that the same shape can be used as a shaky polyhedron as well as a tensegrity
structure. First-order ﬂexible closed polyhedron is associated with a self-equilibrium form
such as a tensegrity structure. By substituting mountain and valley folds of a polyhedronby cables and struts, i.e., compression and tension members, respectively, we can often (butnot always) obtain a tensegrity structure where axial forces proportional to the folding angle
velocity are applied (Figure 9). In order to ensure that it is a tensegrity structure, we have totake into account stability along with equilibrium when designing tensegrity structure; thisis not the scope of this paper. This graphical connection among diﬀerent structural systems
is an interesting aspect to be investigated in future.

7. Conclusion
The condition for a valid inﬁnitesimal folding mode of an origami structure homeomorphic
to a disk and its equivalence to the edge lengths of its reciprocal ﬁgure were demonstrated.
In the case of an origami structure with holes, additional three conditions are introduced foreach hole. The conditions around each interior vertex and hole are equivalent to the equilibrium conditions of a point and a rigid body, respectively. We also derived the condition
that represents inﬁnitesimal foldability when the structure is unfolded ﬂat. We proposed adesign system in which an inﬁnitesimally transformable polyhedral surface can be designed
interactively by combining with the framework of freeform origami. By using this system,
we were able to design variations of shaky closed polyhedron with or without partial developability and ﬁnitely rigid foldable quad-based origami. The potential applications of thesemodels include the design of deployable structures using ﬁnite foldability and the designs ofenergy absorption devices using inﬁnitesimal transformability.

12

T. Tachi: Flexible Origami Based on Reciprocal Figures

Figure 9: Tensegrity structure vs. shaky polyhedron with its crease pattern.

Acknowledgments
This work was supported by the JST Presto program. I would like to thank anonymous
reviewers for helpful comments and suggestions.

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Received