Journal for Geometry and Graphics
Volume 16 (2012), No. 2, 223–234.

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 these graphicalconditions from a matrix representation of rigid origami, and extend the conditions to cases when the surface includes holes. We propose an algorithm to obtainforms that satisfy the conditions and an interactive system to freely design infinitesimally foldable forms. We show design examples of shaky polyhedron andorigami, and finitely foldable quadrivalent mesh origami.
Key Words: Origami, rigid origami, infinitesimal flexibility, shaky polyhedron,reciprocal figure
MSC 2010: 52C25, 52B10, 52B70, 53A17

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 applications ofdeployable structures and flexible meta-material at different scales, e.g., from medical devices
inside the human body to architectural and space structures. From a theoretical viewpoint,
rigid origami structures are especially interesting when they acquire flexibility from redundantmechanical constraints based on their non-trivial singular geometric configuration, such as
flexible closed polyhedra and rigid-foldable quadrivalent mesh origami structures. In designs
using rigid origami mechanisms, such redundancies can be used for removing a part of thestructure 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 that
ISSN 1433-8157/$ 2.50 c 2012 Heldermann Verlag

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T. Tachi: Flexible Origami Based on Reciprocal Figures

Figure 1: Contraction of a closed loop. This also implies that a non-simply
connected surface requires additional treatment.
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 formfindingof finitely foldable rigid origami structures with redundant constraints. The drawback of thismethod 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 system thatsolves the condition of the existence of an infinitesimal folding mode, which can be graphically
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 anda reciprocal figure. We first obtain an initial configuration based on optimization and use thesame 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 andan interior vertex condition, which further decomposes into multiple interior vertex conditions
after several iterations (Fig. 1). This implies that this condition is sufficiently applicable to asimply connected 2-manifold.

T. Tachi: Flexible Origami Based on Reciprocal Figures

ρi

225

Mi χ
Ni L i−1,ii

Ni−1 Mi−1
Li−1

θi,i−1Figure 2: Rotation around a fan.
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 ρi andthe angle between adjacent foldlines i and i + 1 be denoted by θi,i+1 . Here, note that there aren incident foldlines incrementally numbered counterclockwise, and their index is numbered inmodulo 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 coordinateof facet i − 1, i have its x and z axes along the foldline and the facet’s normal, respectively.
χi−1,i is represented as the sequence of a pitch and then yaw (Fig. 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  .
(1)
0
0
1
0 sin ρi cos ρi
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 to its
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
Li Mix Nix
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, Eq. (2) is written as R = Ti Ti = I. 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 Eq. (2) as
0 0 0  
0−Lzi Lyi∂R
T
0−Lxi  .
= Ti 0 0 −1 Ti =  Lzi
∂ρi
0 1 0
−Lyi Lxi
0

(3)

226

T. Tachi: Flexible Origami Based on Reciprocal Figures

b

b

c

d

U

c

a

d

U
f

V
f

e

V

a

Mountain
ValleyeFigure 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).
The constraints can be written using three independent equations:R(2, 3)f = R(3, 1) = 0.
R(1, 2)(4)

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

∂ρi

∆ρi =

n−1
X

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 Eq. (5) so that for any edge u, v, L′u,v = Tu Lu,v is the direction
vector of the edge in the global coordinate system, i.e.,∂fu′= L′u,v = (xu − xv ) / kxu − xv k.
∂ρu,vThe infinitesimal folding motion satisfies, for each interior vertex v = 1, . . . , Vin,X(xu − xv )∆ρu,v = 0.
kxu − xv ku adjacent to v

(6)

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 to be thedual graph of a foldline network in 3D space whose corresponding edges are parallel (insteadof 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, andfacets 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 (Fig. 3).

T. Tachi: Flexible Origami Based on Reciprocal Figures

227

ρi
Mi Li
Ni
N
Mi−1 i−1dθi−1,i i−1,iLi−1
Figure 4: Rotation around a hole.
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(x − xv )them, x′f − x′g = u∆ρu,v ; thus, the graph is a reciprocal figure of the foldline network.
kxu − xv k

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 kx′f − x′g k. This set of angular speeds offollines satisfies Eq. (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

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 (Fig. 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

228

T. Tachi: Flexible Origami Based on Reciprocal Figureswhere Oi represents the position of the origin observed from facet 0, 1. The derivative ofEq. (2) is calculated as
0−Lzi Lyi Lzi∂R0
Lxi Li × Oi =  −Lyi Lxi

0
∂ρi
0
0
We obtain six independent equations:R(2, 3)
R(1, 4)f = R(3, 1) = 0 and g = R(2, 4) = 0.
R(1, 2)R(3, 4)Here, 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 ) / kxu − xv k and Oi by O′u = xu .
Since f = 0 reduces to Eq. (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), Eq. (6) is satisfied and,2. For each hole,X(xu − xv )∆ρu,v × xv = 0.
(8)
kxu − xv k
incident foldline u,v
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, Eqs. (6) and (8) represent the equilibriumof forces and moments, respectively.
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 thisprocess of manufacturing, the existence of folding motion matters. We have observed fromexamples that when we apply developability and flexibility conditions at the same time, thesurface often converges to a completely flat state. When every vertex lies in a plane, theconstraints given by a reciprocal figure degenerate, and one of the three constraints for eachvertex is lost. In this case, the additional flexibility is not desirable since it represents novalid 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 X
i−1
X
Lj ∆ρj × Li ∆ρi = 0.
(9)i=0 j=0229

T. Tachi: Flexible Origami Based on Reciprocal Figures
a

b

b

c'

a
c'

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

f'
e'

d

c' d'
b'

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 bistablestructure when the panel is not triangulated. Right: Modified pattern with the zero-area
condition. This pattern has second-order flexibility in a planar state.

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 Eq. (5). Therefore, the intersection of Eqs. (5)and (9) can be represented by their linear combination asn−1 XLi ∆ρi +

i=0

i−1
X
j=0


Lj ∆ρj × Li ∆ρi = 0.

(10)

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; this typeof origami generically becomes a static structure. The configuration of an origami structurewith 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
andT
the ratios W = (w1 , · · · , wEin ) of the edges. Each element of W corresponding to edge econnecting vertices u, v is represented we = ∆ρe /ℓe . The constraints in Eq. (6) are re-written
as
X
∆fv =
wu,v (xu − xv ) = 0(11)u incident to v

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.

230

T. Tachi: Flexible Origami Based on Reciprocal Figures

Initial solution We solve Eqs. (11) and (12) by solving
U=

Vin
X
v=1

2

k∆fv k +EinX

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 caseoccurs 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 thepattern 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Tterns using a perturbation-based method that uses F (X, W) = ∆f1T , . . . , ∆fVTin= 0 as ageometric constraint that forms a multi-dimensional continuous solution space. A plausible
deformation represented by 3V + E-vector [∆X ∆W]T must satisfy ∂F ∂F   ∆X
∆X
∂X
∂W


C
= ∂G ∂G
= 0,
(13)
∆W
∆W
∂X

∂W

where G represents other optional constraints such as the penalty constraints derived from
Eq. (12), developability, and flat-foldability, i.e., the constraints combined with the constraints
described in [10]. The elements are calculated as follows.
 P
− k incident to v wv,k if u = v∂∆fv 
= wv,u(14)
if u is incident to v

∂xu

0
else
(
∂∆fv
xk − xv if u = v (or xu − xv if v = k)
=
(15)
∂wu,k
0
else
Equations (8) and (10) are polynomial functions of wu,v and xu ; the derivative functions inthese cases 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
+
,
(16)
= I−C C
∆W0
∆W

∆X0
is an arbitrarily assigned initial deformation, which is projected orthogonally to
where
∆W0


the constrained subspace by multiplying with (I − C+ C); this results in a valid deformationthat is as close as possible to the given initial deformation.

T. Tachi: Flexible Origami Based on Reciprocal Figures

231

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.

We implemented the constraints and variables by improving freeform origami [10]. In thissystem, the user can freely select and drag points on the rendered origami structure througha 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 state, thedeformation vector∆X0∆X
is substituted by
to obtain a constrained valid next state.
0∆W

Since the above-mentioned Euler’s integration accumulates the error, the residual is eliminatedbased on Newton-Raphson’s method for each step. The generalized inverse solution for eachstep is calculated using the conjugate gradient method. In this way, we immediately achievea 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.

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 Eq. (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 (Fig. 8). An interesting side effect of using this method for designing discrete versionof curved folding represented as planar quad mesh is that the smoothness of the folding isnaturally obtained. This successfully avoids the jaggy artifacts that existed in the discrete
curved folding representation in [10].

232

T. Tachi: Flexible Origami Based on Reciprocal Figures

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

(a)

(b)

Figure 8: (a) An invalid pattern that satisfies developability and flat-foldability. (b) A
valid finitely foldable pattern is obtained when Eq. (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 equilibrium offorces 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 (Fig. 9). In order to ensure that it is a tensegrity structure, we have totake into account stability along with equilibrium when designing tensegrity structure; this is
not the scope of this paper. This graphical connection among different structural systems is
an interesting aspect to be investigated in future.

T. Tachi: Flexible Origami Based on Reciprocal Figures

233

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

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 showed 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, wewere able to design variations of shaky closed polyhedron with or without partial developability and finitely rigid foldable quadrivalent origami or discrete curved folding. The potential
applications of these models include the design of deployable structures using finite foldability
and the designs of energy absorption devices using infinitesimal transformability.

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

References[1] P. Ash, E. Bolker, H. Crapo, W. Whiteley: Convex polyhedra, dirichlet tessellations, and spider webs. In Shaping space, Birkhäuser 1988, pp. 231–250.
[2] S.-M. Belcastro, T. Hull: A mathematical model for non-flat origami. In Origami 3,
Proceedings of the 3rd International Meeting of Origami Mathematics, Science, and
Education, 2002, pp. 39–51.
[3] R. Connelly: A flexible sphere. The Mathematical Intelligencer 1 (3), 130–131 (1978).

234

T. Tachi: Flexible Origami Based on Reciprocal Figures[4] T. Kawasaki: On the relation between mountain-creases and valley-creases of a flat
origami. In H. Huzita (ed.): Proceedings of the 1st International Meeting of Origami
Science and Technology, 1989, pp. 229–237.
[5] J.C. Maxwell: On reciprocal figures and diagrams of forces. Philosophical Magazine
27, 250–261 (1864).
[6] R. Sauer: Differenzengeometrie. Springer, 1970.
[7] W.K. Schief, A.I. Bobenko, T. Hoffmann: On the integrability of infinitesimaland finite deformations of polyhedral surfaces. In Discrete Differential Geometry (Oberwolfach Proceedings), 2007, pp. 67–93.
[8] H. Stachel: A kinematic approach to Kokotsakis meshes. Comput.-Aided Geom. Design27, 428–237 (2010).
[9] T. Tachi: Generalization of rigid-foldable quadrilateral-mesh origami. Journal of theInternational Association for Shell and Spatial Structures 50 (3), 173–179 (2009).
[10] T. Tachi: Freeform variations of origami. J. Geometry Graphics 14 (2), 203–215 (2010).
[11] N. Watanabe, K. Kawaguchi: The method for judging rigid foldability. In R. Lang(ed.): Origami 4: The Fourth International Conference on Origami in Science, Mathematics, and Education, A.K. Peters 2009, pp. 165–174.
Received August 2, 2012; final form December 13, 2012