Journal for Geometry and Graphics
Volume 14 (2010), No. 2, 203–215.

Freeform Variations of Origami
Tomohiro Tachi
Graduate School of Arts and Sciences, The University of Tokyo
3-8-1 Komaba, Meguro-Ku, Tokyo 153-8902
email: tachi@idea.c.u-tokyo.ac.jp

Abstract. This paper presents a novel method to construct a 3D freeform surface that can be manufactured by folding a sheet of paper. More specifically, we
provide a design system using which the user can intuitively vary a known origami
pattern in 3D while preserving the developability and other optional conditions
inherent in the original pattern. The system successfully provides 3D origami designs that have not been realized thus far.
Key Words: Origami, Developable Surface, Discrete Differential Geometry, Computer Aided Design.
MSC 2010: 53A05, 52B10, 52B70

1. Introduction
Origami is the art of folding, or more formally, isometrically transforming, a sheet of paper
into various forms without stretching, cutting, or gluing another piece of paper to it. Using
origami, a complex 3D shape can be produced with an isometric transformation; hence, it canbe applied for forming the shapes of a variety of products, architectural elements, medical
devices, and so on, with a watertight sheet of hard material so that there is no need toassemble multiple parts. In particular, the simultaneous kinetic motion of connected paper ispotentially useful for assisting the manufacturing process and for realizing self-(re)configurable
structures.
Several designs are proposed to produce 3D origami forms applicable in engineering contexts: e.g., polyhedral surfaces using periodic symmetry, such as Miura-ori [11], Resch’spattern [13], and “waterbomb tessellation” [4, 17]; and straight or curved developable surface reflected with respect to planes, such as Huffman’s concentric circular tower (see [3]),Mosely’s “bud” [12], and the extension of Miura-ori by Buri and Weinand [2]. However,the number of degrees of freedom of the shapes that can be produced using these designs isstrictly limited because of the symmetry used in the design approach; this makes it impossible
to freely control the resulting 3D shape.
When applying origami to actual designs for various applications including space structures, architectural structures and facades, medical devices, and other products in civil use,
ISSN 1433-8157/$ 2.50 c 2010 Heldermann Verlag

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Figure 1: Example of freeform variations of origami
it is important to control the customized 3D shape in the resulting folded state so that theshape is consistent with the required functionality, existing structures, and environments.
Therefore, computationally constructing an arbitrarily formed 3D surface using origami is asignificant challenge not only to origami artists but also the engineers, designers, and personal
users who apply origami to practical purposes.
The most generalized approach to realize an arbitrary 3D origami form is to use the“Origamizer” algorithm [16], which provides a crease pattern that folds into a given polyhedron. However, this method produces highly complex patterns comprising an interlocking
structure by crimp folds; in this method, the crimp folds are necessary to exactly align thesurface onto an arbitrarily tessellated polyhedral surface. This does not allow a continuous
folding motion from a planar state to a 3D state; this becomes a significant disadvantage inapplications to industrial manufacturing processes.
Our objective is to realize an easily foldable freeform surface manufactured from a sheet
material. Although the generalization method proposed by the author [15] produces a variety
of non-symmetric forms with valid folding motion, with this method, the topology of the
foldline network was not controllable and only generalized Miura-ori design within a certain
range of deformation can be produced. We extend this perturbation-based method so that it
is possible to design various shapes by modifying and reconfiguring multiple origami patterns
while preserving the conditions inherent in the original patterns (Fig. 1).

2. Geometric model of origami
An origami surface is a surface with creases characterized by the existence of a one-to-one
isometric mapping between the flat and the 3D states. We model such a surface as a 3D
piecewise linear surface under various geometric constraints so that an isometric mapping
to a valid planar figure exists. The configuration is represented by the vertex coordinates
x (where x is a 3nvert vector and nvert the number of vertices) of a triangular mesh whoseedges represent creases assigned with a target folding angle of −π (valley fold), 0 (3D crease),or π (mountain fold). (Note that in origami, folding angles −π and π are different. In

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205

the implementation software, the previous rotation angle value of each edge is stored foreach frame and is used to determine the plausible angle from the candidates calculated fromthe vertex coordinates.) These variables are constrained by origami conditions including
developability, which is a necessary condition to consist origami, and flat-foldability, which
although not always necessary, is a basic condition shared by many origami designs. In
addition, there are several design-specific conditions related to the coincident positions ofelements of surfaces in the 3D and flat states. Therefore, we generalize origami patterns byflexibly changing the mesh configuration while ensuring that the sets of conditions inherent
in the original patterns, i.e., the preserved features, are satisfied.
2.1. Developability
An origami is a developable surface with G1 discontinuity whose most simple case is a polyhedral surface. As the curvature of a polyhedral surface cannot be defined, the developabilityof origami is given by zero Gauss area Gk around every interior vertex k:Xθk,i = 0,(1)
Gk = 2π −iwhere θk,i is the i-th incident sector angle of vertex k. This discretized representation allows
for more flexible variations of form than possible with a smooth developable surface, which isfundamentally a plane, cylinder, cone, or tangent surface.
2.2. Paper boundary
Traditionally, an origami is folded from a square or a fixed convex polygon. As the shapeof the paper is not preserved by (1), we require additional conditions to be defined for the
perimeter. This is represented as follows: For each vertex k on the perimeter,XX0θk,i −
θk,i
= 0,
(2)
i

i

and for every straight element j of the perimeter comprising edges i of length ℓj,i ,XXℓj,i −ℓ0j,i = 0,
i

(3)

i

0
where θk,iis the original incident sector angle of vertex k, and ℓ0j,i is the original length of i.
In an engineering sense, the boundary property is not always important. Hence, on the basisof the design context, we can ignore this condition or instead set up another condition suchas the convexity of the boundary of the paper:Xθk,i ≤ π .
(4)i2.3. Flat-foldability
Several origami models such as Miura-ori, Yoshimura pattern, and waterbomb tessellation
are foldable to another flat state with every foldline folded completely, i.e., by ±π. Thiscondition is called flat-foldability, and it can be applied to design objects and spaces that canbe compactly packed. Flat-foldability of origami can be represented as a combination of two

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T. Tachi: Freeform Variations of Origami

σ=+1
σ=−1
θk,0
θk,1

θk,5
θk,2

θk,4,4

θk,3

Figure 2: Flat-foldable vertex.
conditions — that the coordinates are valid and that a valid overlapping ordering exists. The
former condition implies that there exists a valid isometric mapping to a plane and that thiscondition can be represented locally as follows: For each interior vertex k,XFk =σk,i θk,i = 0,(5)iwhere is σk,i = ±1 is the sign assigned to each facet representing the orientation of the facet
in the flat-folded state (Fig. 2).
The order of overlap of surfaces must also be considered because multiple facets are
mapped onto the same position in the plane when the surface is flat-folded. There havebeen several studies on this topic. Bern and Hayes [1] proved that determining a valid
ordering from a crease pattern is an NP-complete problem with worst-case complexity. A
practical exact algorithm is investigated by Meguro [10] that implements a divide-andconquer approach to solve the overlapping problem in a practical amount of time for mostwell-designed origami. Although these exact approaches help us determine whether the given
set of foldlines is valid, they provide no information for establishing valid foldlines. Therefore,
our method uses angular inequalities that approximately remove local intersections: For four
adjacent crease lines ℓi−1 , ℓi , ℓi+1 , and ℓi+2 , where the two creases in the middle, ℓi andℓi+1 , have the same mountain-valley (MV) assignment (let us say that it is M without loss ofgenerality). The sector angles between them, θi−1 , θi , and θi+1 (Fig. 3), satisfy the following:θi ≥ min(θi−1 , θi+1 ),(6)

θi ≥ θi−1(7)

θi ≥ θi+1(8)

additionally, if ℓi−1 ’s assigment is V,
and if ℓi+2 ’s assigment is V,
It should be noted that (6) is the necessary condition for flat-folding derived by Kawasaki[6] and that (7,8) is an empirical condition applicable to finding a locally closed vertex [14].
2.4. Folding angle conditions
To avoid a local collision between adjacent facets, the folding angle ρ (where, |ρ| = π −dihedral angle) between each pair of adjacent facets must be maintained in the range [−π, π].
More specifically, the range of possible rotation angles is defined by the target rotation angleof the edge.

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T. Tachi: Freeform Variations of Origami

θi+1ℓi−1 θi−1 θi
ℓi
ℓi+1
good

ℓi+2
bad

good

Valleybadbad

Mountain

Figure 3: Angular inequalities for flat-foldability. Left: Kawasaki’s necessary condition.
Right: an empirical condition.
1. If target is π, 0 ≤ ρ ≤ π.
2. If target is −π, −π ≤ ρ ≤ 0.
3. If target is 0, −απ ≤ ρ ≤ απ (0 ≤ α < 1.0).
The first two conditions ensure that the fold is not made in the opposite direction as in the flatfolded state. This is important as creases cannot correct their orientation in a smooth folding
process. The third condition is applied for the extra crease necessary to triangulate each facet
surrounded by more than three creases. The folding angle of such a crease represents the twistof the facet element. The inequality is set up to make the surface practically feasible. Notethat α (0 ≤ α < 1.0) determines the flexibility of the paper in use (if 0, each facet is rigid
and cannot twist).

3. Valid deformation
The given constraints are calculated numerically by iteratively solving linear equations. The
process is an extension of the Newton-Raphson method, where solving f (x) = 0 and g(x) ≤ 0
is understood as minimizing the respective penalty functions f (x)2 and max (g(x), 0)2 .
By combining the required conditions into one vector equation, we can obtain the form ofF(x) = 0 for each step of the calculation.
As the number of constraints nc is lower than the number of variables 3nvert in origami
models, the vector equation forms an under-constrained system; therefore, the solution is not
fixed to one configuration. In order to solve this system, we use a method that modifies aknown pattern that satisfies F(x) = 0 , such as Miura-ori and Resch’s pattern, by numericallyintegrating an infinitesimal deformation of a triangular mesh within the solution space of theequation ∂Fdx = 0 ,(9)∂xwhere the nc × 3nvert rectangular matrix [∂F/∂x] is the Jacobian matrix of the constraints.
The infinitesimal deformation dx that satisfies is calculated using its pseudo-inverse (i.e.,

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T. Tachi: Freeform Variations of Origami

Penrose-Moore generalized inverse) [∂F/∂x]+ .
dx =



∂F
I−
∂x

+ 

∂F
∂x

!

dx0 ,

(10)

where dx0 is an arbitrarily given deformation vector. Note that the typical conditions arelinearly expressed in terms of the sector angles θ and folding angles ρ in the pattern. We canuse the following form:       
∂F
∂F ∂θ
∂F ∂ρ
=
+
,
(11)
∂x
∂θ
∂x
∂ρ ∂x
where the elements of [∂θ/∂x] and [∂ρ/∂x] are represented as follows. As in Fig. 4, the angle
and the normal of the triangle ijk (the coordinates are denoted as xi , xj , and xk , respectively)
are denoted as θj , ki and nijk , respectively, and the folding angle of ji and the length of theperpendicular from k to ji are denoted as ρij and hk,ij , respectively. We get,∂θj,kinijk × (xi − xj ),=∂xi
kxi − xj k2∂θj,kinijk × (xi − xj ) nijk × (xk − xj )
= −
+
,
∂xj
kxi − xj k2
kxk − xj k2∂θj,kinijk × (xk − xj )
= −
;
∂xk
kxk − xj k2

(12)
(13)
(14)

and
∂ρij
∂xk
∂ρij
∂xl
∂ρij
∂xi
∂ρij
∂xj

nijk
,
hk,ij
njil
=
,
hl,ij
− cot θj,kinijk− cot θj,il
njil
=
+
,
cot θj,ki + cot θi,jk hk,ij cot θj,il + cot θi,lj hl,ij− cot θi,jk
nijk− cot θi,lj
njil
=
+
.
cot θj,ki + cot θi,jk hk,ij cot θj,il + cot θi,lj hl,ij
=

(15)
(16)
(17)
(18)

The vertex coordinates are deformed along dx, which is the valid deformation closest to
estimated deformation dx0 . Since dx0 can be specified arbitrarily, we can use a user input of
direct dragging of the surface in 3D to construct the vector. The solution is calculated using
Euler integration. A step of the Newton-Raphson method is applied to each integration stepto numerically eliminate residual error caused by nonlinear and inequality conditions.

4. Mesh modification
When we deform the surface by a large amount, the pattern transforms such that vertices
can no longer traverse along the surface because of the intersection of the surrounding edges.
This limits the amount of deformation that the surface can tolerate, and it is necessary toreconfigure the mesh structure to achieve further deformation. Such limited deformation of
the mesh structure is detected by the presence of degenerated edges. For each edge that is
too short, an operation similar to the edge collapse operation proposed by Hoppe et al. [5]T. Tachi: Freeform Variations of Origami

njil
xi

209

nijk

xl
xk

ρij

θj,ki
hk,ij
xj

Figure 4: Notation used for defining elements of [∂θ/∂x] and [∂ρ/∂x].
is performed. When the flat-foldability of the original pattern is to be preserved, the process
must take into account the target folding angles assigned to the creases. The following are
the rules that must be followed during the collapsing process to keep the configuration of theflat-folded state unchanged.
• When two edges are merged into one, the target angle for the merged edge is the sumof the original target angles.
• The sum of the target angles around the merged vertex must be −2π (convex) or 2π(concave) (Maekawa’s theorem [9]).
• A minimum of four creases should surround each interior vertex.
Therefore, the possible merging processes are limited to the following two cases (double
signs correspond respectively):
1. Vertices with sum of angles ±2π and ±2π are merged by collapsing a ±π edge, forming
a ±2π vertex.
2. Vertices with sum of angles 2π and −2π are merged by collapsing a ±π edge, forminga ∓2π vertex.
In most mesh optimization techniques, edge flipping is used along with edge collapse to
avoid the formation of a jagged surface. However, our method applies this procedure only
to creases that must remain unfolded throughout the folding motion (when target angle is 0
and α = 0) given that nonsmooth small creases play an important role in origami. In otherwords, the variety of forms possible with origami originates in the G1 discontinuity of thesurface. The above described mesh modification process significantly changes the topology of
the crease pattern structure of origami; this results in the creation of a new origami patternthat combines different types of vertices (Fig. 7, Right).

5. Implementation
We propose a system to explore the variations of origami forms through an interactive userinterface that enables the user to freely drag the vertices of a surface in a folded 3D state
while the system calculates the geometric constraints in the background. While editing, theuser can see the crease pattern and the flat-folded pattern (if the model is flat-foldable). In
addition, the user can switch between the design and edit modes: in the design mode, featuresin the design context are preserved; whereas in the simulation mode, preserves the length ofthe elements of the paper and simulates how the designed paper behaves.
The system is implemented in C++ using wxWidgets and OpenGL for the graphical interface and BLAS (Intel MKL) for numerical calculations. The system completes the iterated

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T. Tachi: Freeform Variations of Origami

Figure 5: Variations of waterbomb tessellation. Left: regular pattern.
Right: deformed pattern forming a double-curved surface. For each figure,left: 3D form, upper-right: crease pattern, bottom-right: flat-folded pattern.

Figure 6: Folded model of a variation of the waterbomb tessellation.
Left: flat-folded state. Center: 3D state. Right: the developed state.
step of equation (10) via the conjugate gradient method (number of iterations is limited to
512 for each frame) at 1-5 fps for a 256 vertex model with 452 constraints on a laptop PC
with a quad-core processor (Intel Core i7 1.6 GHz).

6. Model examples and results
An appropriate set of origami conditions enables the generalization of various known origami
tessellation patterns. Here, we show actual examples of generalized origami patterns.
6.1. Waterbomb tessellation
The waterbomb tessellation is a triangle-based flat-foldable pattern folded into a cylindricalsurface. This pattern is well-known to various origami artists, one of the earliest designusing which is by Fujimoto [4], and it is also applied for deployable structures such as astent graft [16]. The curvature of the surface can be altered by varying the pattern using
our method (Fig. 5). As the surface is composed of both convex and concave vertices, theT. Tachi: Freeform Variations of Origami

211

Figure 7: Variations of Miura-ori. Left: regular Miura-ori. Right: deformed pattern.
Notice the topological change in the crease pattern network.

Figure 8: Variation of Yoshimura pattern

resulting generalized pattern is flexible enough to enable the creation of origami forms thatapproximate a 3D freeform surface.
The resulting models were constructed by folding a sheet of paper whose surface wasscored with a CNC cutting plotter (Fig. 6). Although a certain level of folding skill is stillrequired to construct this pattern, it is observed that the pattern can collapse into the folded
state once we have the right precreases on the surface. This hints at the possibility of thedevelopment of an automated manufacturing method in the future.

6.2. Miura-ori
Miura-ori [11] is a quadrilateral-based flat-foldable pattern folded into a 3D corrugated surface
that approximately follows the shape of a plane. Figure 7 shows an example of this design. In
this example, we have applied boundary condition (2) and used α = 0.1 for each triangulatingcrease to approximate the surface’s elastic feature.

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T. Tachi: Freeform Variations of Origami

Figure 9: A variation of Resch’s triangular pattern. Left: Original pattern.
Right: A variation forming a double-curved polyhedral triangular mesh.
6.3. Yoshimura pattern
The Yoshimura pattern is a triangular (diamond) flat-foldable pattern that folds into a cylindrical surface (see [11]). As all vertices are convex and cannot be altered by our mesh modification method, the approximate mean curvature of the surface is always positive. Hence,
although we can produce many variations of the Yoshimura pattern, its very topology significantly limits the scope for variation. Figure 8 shows a rendering of a freeform shell using thedeformation of the Yoshimura pattern.
6.4. Resch’s triangular pattern
The triangular tessellation proposed by Resch [13] is a developable and non flat-foldable
surface in which a group of three vertices forms a vertex in the folded form to hide a tuck-likeflat-folded part behind a composed plane. This type of coincidence of points is calculated
by adding extra constraints or removing the variables and combining the equations incident
at the vertices. The variational patterns form polyhedral surfaces with tucks folded behind.
This can be interpreted as a special case of origamizer method [16], where in our method,

Figure 10: Design of crumpled paper

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213

Figure 11: Left: Isometric triangular grid.
Right: Irregular developable surface with fixed boundary.
the pattern works without crimp folding. While the variation, we do not allow the surface tochange the topology of the crease pattern in order to retain the characteristics of the patternthat forms a polyhedral surface.
6.5. Plane (crumpled paper)
Transforming a planar paper, i.e., a regular triangular mesh without a crease, results in theformation of an irregularly corrugated surface or, put simply, crumpled paper. Hence, ourmethod can also be used for designing and rendering crumpled paper (Fig. 10), and we foundthat it yields better results than directly simulating the non-stretching behavior of a sheet ofpaper; the latter tends to produce visible regular triangular artifacts (Fig. 11).
6.6. Curved folding
We further tested applying the method to designing curved folds. As demonstrated by Liu
et al. [8] and Kilian et al. [7], a smooth developable patch, i.e., a torsion-free ruled surface,
between creases can be approximated by a planar quadrilateral strip. Planar quadrilaterals
can be formed by a triangular mesh with creases whose target folding angle is 0 and α = 0. We
were able to obtain nonsymmetric curved folding shapes by carefully transforming a regular
rectangular grid, but it was found that the surface can easily become jagged (Fig. 12). The
problem arises because of the excess degrees of freedom in the configuration. To solve thisproblem, studies must be conducted on the addition of more constraints to the system topreserve the smoothness of the continuous foldlines and the surface.

Figure 12: Designing curved folding. Top: good curved folding. Bottom: a jagged surface

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T. Tachi: Freeform Variations of Origami

7. Conclusion and future work
This paper described a novel method for obtaining a generalized form of origami model by
numerically integrating infinitesimal deformations under origami constraints and collapsingedges to alter the topology of the crease pattern. The method enables the realization of new
and complex origami patterns that are not achievable with existing design methods. We havedeveloped a system that implements the proposed method (the software is available from theauthor’s web site1 ). Certain features of the original origami patterns, such as foldability froma plane, flat-foldability, and the smooth transformation from one state to another, can also bepreserved in the designed models if the user so desires. The system can be used for different
applications by choosing different sets of active conditions.
A future work of this study is in developing a method that fully support curved folding. In
addition, whereas in this study the topology of the surface is limited only to a that of a disk,
future studies for extending the method to nondisk surfaces, especially cylindrical surfaces,
can widen the field of possible applications of origami designs; examples of such applications
include collapsible containers and portable architectural spaces.

Acknowledgment
This study is supported by the grant in aid for JSPS fellows and by KAKENHI (22800009)
Grant-in-Aid for Research Activity Start-up, by Japan Society for the Promotion of Science.

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Received August 7, 2010; final form November 24, 2010