Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Designing One-DOF Mechanisms for Architecture
by Rationalizing Curved Folding
Tomohiro TACHI1*, Gregory EPPS 2
1*

Graduate School of Arts and Sciences, The University of Tokyo
3-8-1 Komaba, Meguro-Ku, Tokyo, 153-8902, JAPANtachi@idea.c.u-tokyo.ac.jp
2

RoboFold Ltd
London, SW18 2LR, UK
gregory.epps@robofold.com
Abstract
We propose a modeling method based on rationalizing curved folding in order to find theform variations of 1DOF origami mechanism. We interact with a physical paper model ofcurved folding and then discretize a curved folding by identifying and fixing the rulings. The
discretized form is a rigid origami structure with at most one degree of freedom. The form
adjustment follows the discretization so that it is sure to realize a mechanism. The workshop
performed by the authors based on the proposed design method is reported. The objective of
the workshop was to utilize the 1DOF characteristic of discretized curved folds as a constraint
in the design of dynamic architectural components. The results showed the feasibility of themethod and suggested a novel methodology for designing.
Keywords: kinetic architecture, curved folding, rigid origami, form finding, physics-based
design1 Introduction
1.1 Kinetic Surface using Rigid Origami
Kinetic design in architecture is very important, e.g., dynamic solar shading, doors andopenings, acoustic controllers, or portable furniture; virtually any architecture has kinetic
components to be functional. However, the design of such components is morphologicallypoorly investigated field. This can be attributed in part due the challenge imposed by an extra
dimension (time) apparent in the component design. Our objective is to add more design
freedom in such components so that we can explore variations of multi-functional, contextmatching, high-performance, efficient, and super cool-looking designs.
In order to design such a kinetic structure in a relatively big scale, it is useful to use geometricmethod that requires no or few material deformation in order to achieve the overall
transformation. Two major geometric approaches for achieving mechanical structures are 1)
to use bars and rotational hinges to construct scissors mechanism, e.g., the works of Santiago
Calatrava and Chuck Hoberman; and 2) to use plates and hinges to form polyhedral surface
linkage with a synchronized motion, i.e. rigid origami. In this paper, we focus on the designsof the latter, since they have an advantage over the bars and hinges mechanisms in the senseProceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japanthat they can form a kinetic watertight surface; a watertight surface is suitable for partitioningarchitectural spaces. However, unlike the bar linkage models, the form of rigid origami
structures is heavily constrained, and it is has been very hard to find really useful design
solutions.
There have been studies of rigid origami for forming kinetic surface such as Miura-ori [1],Ron Resch’s Pattern [2], and Chuck Hoberman’s designs [3]. Especially, mechanisms basedon a quadrilateral mesh such as Miura-ori produce 1DOF synchronized motions. This implies
that the whole structure transforms by asserting force to a part of the structure. This is very
useful for designing kinetic architectural components because of the following reasons.
1. The structure can be formed by thick panels and hinges and thus is scalable toarchitectural structure.
2. The overall structure is controlled by one manipulator or a human user.
3. It is possible to design the structure that counterbalances itself in the transformation.
4. The structural mechanism is redundant and we can cut out holes to surface while
preserving the mechanism.
Recent studies generalize theses structures to achieve further design freedom. Tachi [4][5]proposes computational design methods to obtain variations of miura-ori-like quadrilateralmesh structures while preserving their flat-foldability and rigid-foldability. However, since
this method uses continuous transformation from one pattern to another, the achievable forms
are restricted by the initial patterns, the finding of which is still an important yet unsolved
problem.
1.2 Contributions
In order to find the form variations of 1DOF rigid origami, we propose a modeling methodthat combines physical curved folding and computational design method. We have performed
a workshop [6] based on this approach. The results demonstrated that the method is feasible
and useful. This also suggests a new paradigm of design methodology. The following is thesummary of contribution.
1. The method includes physical interaction with paper in the brainstorming stage. The
design process starts from physical interaction with paper by cutting, bending, folding
and curved folding. This allows designers to explore wide variety of possibilities andto be inspired from the reaction of the physical paper.
2. Rigid origami interpretation of curved folding enables the flexible designs ofmechanisms. We make 1DOF mechanisms by identifying the rulings using a physicalapproximation method, and linking adjacent vertices in order to rationalize curved
folded paper as a planar quadrangle mesh.
3. The workshop results demonstrate the feasibility and usefulness of our proposed
approach as a design method. They also suggest a novel methodology to design that
results from embedding understanding of the connection between physical material,
geometry, and architectural applications within a predefined process.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

2 Design Method
2.1 Method Overview
The proposing design process is as follows:
1) We first start from interacting with a physical model of curved folding made frompaper or unwoven fabric, which can change its shape by flexibly adapting its rulings
position.
2) Once we get some meaningful form, we discretize a curved folding by identifying andfixing the rulings. The discretized form is a rigid origami structure with at most one
degree of freedom.
3) We use Freeform Origami and Rigid Origami Simulator [7] to simulate folding of theresulting discretized pattern and solve the developability and planarity constraints. We
add other geometric constraints consistent with the design problem, and adjust thepatterns to fit the design intention.
4) We implement the surface designs as physical mechanisms using thin sheet or thickpanels cut by vinyl cutter, laser cutter, or other 3-axis machines.
2.2 Interaction with Curved folding
Curved folding is a piece of paper folded along curved creases; it is formed by combining
folding and bending of the paper. In a geometric sense, a curved folded shape is a developablesurface comprising the patches of smooth (G2 continuous) developable surface: plane,
generalized cylinder, tangent surface, and generalized cone. Even though each of thesesmooth patches only makes relatively simple form, the existence of creases make the surface
freely configurable, enabling globally non-developable complex surface, e.g., approximatedhyperbolic surface and elliptic surfaces [8], the sculptures by David Huffman [9], the earlycomputer graphics works of Ron Resch [10], and product designs by Gregory Epps (Fig. 1-2).

Fig. 1: A car model by Gregory Epps animated by Rigid Origami Simulator [7], using therulings reconstruction by Killian et al. [15]

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Fig. 2: A car model by Gregory Epps.
Each smooth developable surface patch is a ruled surface, where the rulings have non-twisting
relationship with neighborhood (this is because of the developability). A state of curved
folding is parameterized by the position of creases and rulings on the surface, the rotational
angle at the crease, and the maximum curvature of the surface at every point (perpendicular to
the ruling passing through the point).
The differential geometric analyses of curved folding by Huffman [11] and by Fuchs andTabachnikov [12] give the basic understanding of how a deformation of a curved fold such as
folding and twisting affects the change in the rulings and the folding angles. The following
results from [12] are useful to understand the relationship between the three-dimensional
configuration and the crease pattern (ruling pattern) of a curved fold.
1. The curvature of a fold line in space increases as it folds. The relationship is
proportional: k3 D k2 D / cos D , where k3D and k2D are the curvature (the inverse of theradius of the curvature) of the foldline in the space and the crease pattern respectively;
 is the half of the folding angle of the crease at the point. From this fact, we canunderstand that a straight line in the crease pattern cannot be bent and is kept straightin the space until it is folded 180 degrees (Fig. 3).
2. The curved fold does not twist when the left and right rulings have the sameorientation in the crease pattern. The curved fold is twisted when the rulings havekinks at the rulings (Fig. 4).

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

3. The folding angle, i.e., complementary angle of dihedral angle, along a curved fold isconstant when the rulings reflect at the curved fold in the crease pattern, producing
uniform curved folding (Fig. 5).

Fig. 3: Pattern-3D Relation 1: Curvature and fold angle.

Fig. 4: Pattern-3D Relation 2: Rulings and twisting. Left: Non-twisting curved folding bystraight rulings alignment in the crease pattern. Middle & Right: Twisting curved folding bykinked rulings alignment.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Fig. 5: Pattern-3D Relation 3: Uniform and non-uniform folding. Left: Uniform angle folding
using mirror reflecting rulings. Right: Non-uniform folding (generic). The folding angle is notconstant along the crease.
Result 2 implies that a non-twisting curved folding is composed of a developable surface andits mirror reflection with respect to the plane that includes the curved fold. It corresponds to
the known method for creating a curved folding by cutting and mirror reflecting a single
developable surface (Fig. 6). However, this is only a limited case, and we can find more form
variations with twisted curved creases by physical interaction with paper.
Normally, a piece of paper deforms plastically and thus irreversibly by folding, and it does
elastically thus reversibly by bending. Here, folding implies a local rotation around fixedcrease, while bending includes, in addition to the change in the absolute amount of curvature(distributed rotation along the rulings), the position and orientation of rulings (Fig. 7).
Therefore, a physical interaction with a curved folding implies more degrees of freedom thana regular straight line origami, where the configuration is basically represented only by thefolding angles.
Using the difference in the behavior, we can design patterns in the following two steps. First,
we fold the paper to set the crease shape to design overall approximate form. Then, by
reversibly and continuously bending the folded shape, we can find twisted or non-uniformly
transforming variations. An advantage of this physical method is that we can interactively
find twisted variations of the same curved folding pattern with different ruling alignment. The
form is unstable in this stage, but the form can be fixed in the rationalization step described in
the next section since the twisting motion is recorded as the ruling position on the surface.
Since creasing paper or card is irreversible, it is naturally required that the designer makes lotsof test models. When a more fine adjustment of crease shape is required, we can use unwoven
fabric such as felt instead of paper, so that we can achieve reversible folding. This adds
further degrees of freedom of designs by allowing a designer to interactively shift the crease
position in the pattern, rather than defining the crease pattern in the developed state.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Fig. 6: Curved folding design using mirror reflection of a developable surface (a moreelaborate design example is Huffman’s cone design: see [9]).

Fig. 7: The elastic transformation with the change in the rulings position (Bending only).
2.3 Rigid Curved Folding
By discretely sampling the rulings, we can represent developable surfaces and hence curved
folded surfaces as quadrangle meshes. The non-twisting property of developable surface
forces the adjacent rulings to form a planar quadrangle. Thus a curved folded surface isapproximated by a planar-quadrilateral (PQ) mesh (as this representation is used in Figs. 2-6).
PQ mesh representation of developable surface is used to simulate the bending and creasingof paper by Kergosien et al. [13], and is also studied by Liu et al. [14]. A computational
method to reconstruct a PQ mesh from a laser-scanned curved folding is proposed by Kilian
et al. [15]. We use much simpler method for obtaining the PQ mesh from the physical model:aligning a straightedge on the surface to find the rulings, and then drawing lines along them.
We unfold the surface and draw the developable PQ mesh as a straight graph in the plane, i.e.,crease pattern. The smooth curvature of the fold line is also rationalized by a series of straightlines (polyline) that link the ends of adjacent rulings, thus completing the quad facet, andtherefore rationalizing both the bending and folding,
We interpret such an obtained PQ mesh as a rigid origami structure i.e., mechanism by rigid
facets connected by rotational hinges. Such a structure has less flexibility than the originalphysical curved folding since the change in the position and orientation of a ruling (nowdiscretized as a crease) is not allowed in the rigid origami structures. Once the surface
configuration is recorded as a crease and ruling pattern in 2D, the form that can be produced
from the pattern with the rigid origami motion is no longer flexible. They try to reproduce theexact 3D form that we designed through physical interaction. Therefore, rigid origami
interpretation of curved folding encodes 3D and 4D forms into 2D pattern. Interpreting the 2D
as 3D or 4D is a simple matter of defining an angle constraint between any two faces.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, JapanA PQ mesh rigid origami structure is either a 1DOF mechanism or an over-constrained static
structure. In generic cases, the structure is not rigid-foldable, i.e., there exists no mechanism.
This is because each interior vertex composed of four edges is already a 1DOF mechanism, inwhich one of the folding angles determines the entire configuration; if we align these vertices,
the motion of every edge is multiply defined by all incident edges (Fig. 8). Rigid foldabilityof the PQ mesh is ensured in these known special cases, but the general condition is still an
open problem:
1. When the model is representing one curved crease (2n array of facets) or the structure
essentially repeats single curved crease.
2. When the similar vertex structure is repeated to avoid contradiction.
3. When the crease pattern is flat-foldable and there is at least one 3D state (as shown in[4]). Flat-foldable pattern is a discrete version of reflecting rulings that produces
uniform folding.

Fig. 8: Contradiction of fold angles in quadrangle mesh rigid origami.
2.4 Adjustment
We adjust the patterns so that they achieve certain properties that suit the design problems. In
order to keep the developability while we adjust and add extra geometric constraints, we
utilize Freeform Origami [8]. Freeform Origami is the freely available software which enables
interactive pattern variations under several geometric constraints related to origami. Using thesoftware, rigid foldable structures and bi-stable structures can be designed from a PQ mesh.
These patterns can also fit pre-existing geometric conditions given by the design contexts.
Bi-Stable Structure
Even though the reconstructed pattern does not always produce the rigid folding motion, in
most cases, we can at least make a bi-stable structure that transforms from a planar state to athree-dimensional state with a small amount of elastic bending of the sheets. This is because
1) most patterns that represent curved folding in 3D are continuously rigid foldable if we add
a triangulation crease to emulate small amount of bending of the panel; and 2) since there isan original physical model, we can expect that there is at least one 3D configuration that is
valid even without triangulation. We can obtain a bi-stable folding structure by the following
procedure in the software:
1) Fold the mesh in the simulation mode, in which the system simulates rigid origami
motion numerically solving unstable truss model, while allowing triangulation. We

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japanobtain close enough shape to the 3D configuration used for obtaining the PQ mesh.
2) Switch to design mode and add developability constraints to ensure it is made from asheet, and planarity constraints to forbid triangulation. This adjusts the pattern to be avalid 3D configuration without triangulation.
3) Set additional geometric conditions for the target 3D shape, such as the fixed shape of
the pattern in the developed state and the geometric attachment to existing buildings.
In this way, we can build a valid 3D model that can also exist in a developed state. Since thetransformation from the developed state to the 3D state is not generally ensured, we obtain asnap-through mechanism that is both stable in the developed and 3D states.
Rigid Foldable Structure
In the case where the pattern is close to locally flat-foldable (uniform curved folding), we canconfigure the system to force the pattern to be flat-foldable in the step 2) of the procedure. If
the pattern converges to one three-dimensional configuration, the rigid foldability is ensured.
Such mechanism has exactly one degree of freedom.
If the pattern is not close enough to flat-foldable, adding triangulation creases is a general way
to turn bi-stable structure into a rigid foldable structure. If every facet is triangulated, ageneric pattern has N-3 degrees of freedom where N is the number of edges on the perimeter.
We can experimentally increase and decrease the number of creases until we get 1DOF rigidfoldable structure.
2.5 Implementation
Rigid origami can be formed by thick panels and hinges geometrically. This allows us to scale
up the structure to architectural size. A general and effective method for thickening is to use
two layers of constant thickness panel sandwiching ideal zero-thickness hinge material [5].
The essence of the idea is to make a thick groove along the edge at the valley side of the panel
to avoid intersection when folded along the hinge in the center (Fig. 9). Each panel can bemodeled by cutting out a two-dimensional polygon, thus can be manufactured using 3-axis
CNC machines (where the third axis is for the depth of cut) such as cutting plotter, laser cutter,and milling machines. We can place piano hinges at the shared edges of the panels to create
the connection. Also, if we place fabric between the layers and glue them together, the fabricacts as rotational hinges between the panels.

Fig. 9: Implementing rigid origami with thick panels.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

3 Workshop
The authors have performed a workshop [7] based on our proposed design method. The
objective of the workshop was to utilize the 1DOF characteristic of discretized curved folds asa constraint in the design of dynamic architectural components. Curved folded components
demonstrate a continuous transformation of states from flat to folded, and so are ideal for
openings in architectural envelopes – in this case doors and solar shading were identified as
suitable examples. Participants in the workshop developed a tacit understanding of curved
folds through extensive paper folding exercises (Stage 1), and then applied this knowledge tothe workshop briefs; ‘Doors and Openings’ and ‘Dynamic Solar Shading’ (Stage 2), before
learning how to manually identify rulings on the surface in order to translate their designs intoa PQ mesh representation for manipulation and folding simulation in Freeform Origami [6]and Grasshopper [16] (Stage3). The final designs were then printed and re-built using foamcore facets and paper or tape hinges, and assembled into the desired configuration (Stage 4).
The fact that the workshop operates as one continuous design process, and each step further
informs the designs of the next step, allowed for continuity of thinking by participants andallowed for amplified results at the multiple levels that this design methodology operates on.
By employing what could be called a physically responsive process based design
methodology, whereby formal aspects are allowed to develop stepwise from imposed
limitations such as material and geometry, explicit outcomes are unknown, but the parameters
within which they occur are implicitly understood. This differs from a typical design processthat involves thinking of a design mentally and visually, and working back from this tounderstand manufacturability, functional suitability etc. This would again suggest that thedesigns are limited – but to the contrary, the results demonstrate unexpectedly elegant
solutions to the design briefs.
Here, we elucidate on the process in Stages 1 and 2, since they are most novel in theirapproach.
3.1 Abstracting Design Intent
The primary purpose of Stage 1 is as a method of abstraction to encourage participants to be
informed by the medium, rather than their expectations of form generation – this is due to theunpredictability of geometric possibility in curved folding, when compared to the current
modeling paradigm, which could be considered as a clay-modeling paradigm.
Abstracting Briefs
The class therefore starts with a number of abstract briefs that refer to effects and proceduresin paper folding, rather than architectural program: change size, strengthen, articulate, encase,protect, join things and store energy Participants were required to demonstrate successful
meeting of this brief through multiple models (Fig. 10). During modeling, further constraints
were added in order to continuously force abstraction, such as using algorithmic approaches[17][18] and breaking up chunks of time into short ten minute exercises to enlarge, divide,join and multiply as mechanisms to change the geometry using physical methods, rather thanrelying on mental imagery to design.

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Common Library
The outcome of the introductory design stage, as well as familiarizing the participants withthe methods of design that are effective for the curved folding medium, was to create a sharedlibrary of geometries that represent the investigative areas that one could encounter when
designing in this medium. Each group placed their work on a common wall, building acommon toolkit of shape typologies encapsulating the fast paced learning for shared use in
the main design session (Fig. 11). The physical and visual array of form, mechanisms,
perforation techniques, articulating actions, and approaches provided a simple storage andretrieval system for the design ideas during the two day workshop, as it remained on display
for reference. Participants were also required to share their experience and beliefs about theirfavorite work with the class.
3.2 Applied Briefs
As workshop leader of the design phase, Gregory Epps aided in making a down-selection
from the available media, identifying appropriate geometries for the participants to further
pursue. With the brief of “Dynamic Solar Shading” (Fig. 12-13) and “Doors and Openings”(Fig. 14-15) now active, groups of four to seven focused on refining articulation, function andform by increasing scale and adding broader levels of detail as they progressed, such a fixing
locations, methods of actuation, hinge and material types and aesthetics. Because the wholegroup had distributed the task of researching and disseminating these core aspects, theirability to express themselves was evident within the timeframe of the two day workshop (Fig.
10-15).
Further Improvements
In further iterations of this teaching technique, it is suggested the knowledge gained in Stage
1 is explicitly spread through to the Stage 2 by exploding the initial teams from theintroductory session, and using one member form each team to build the teams that will apply
the knowledge.

4 Conclusion
We presented a novel method for designing rigid foldable structure through physicalinteraction with paper to create a curved folding and then rationalizing the folded surface. The
combination of physical interaction and digital/geometric manipulation allows the designers
to understand and to discover the connection between the design material, geometry, and thepotential application in architecture. Our approach can be compared to completely geometricand generative approach in which it is too restrictive to explore new ideas for designs, or
purely physical approach in which we cannot really understand the 4D behavior of thestructure when it is really build with panels and hinges. This approach led the success in thedesign workshop.
The workshop posed multiple challenges for participants on a geometric level – they had tolearn how to conceive physically, rather than mentally, their thoughts on design of a 4D
object, using a surface representation that differs hugely for the standard modeling paradigms,

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japanand transpose these forms to a secondary linkage based system, that could be argued is very
distant from the initial medium of curved folding, and had to meet strict geometricrequirements for this latter stage to function effectively. Contrary to these limitations, thephysically responsive process based design methodology increases variation in aesthetics andfunction of the output.

Fig. 10: Stage 1: Design Abstraction; articulation experimentation of crumpled paper

Fig 11: Stage 1. Common Library (Shape Typology Toolkit); review of all participants

Fig 12: Stage 2. Group Brief: Dynamic Solar Shading (members: Emilie Hergott, Nicolas,
Leduc, Sophie Pennetier, and Simon Schleicher); upward motion actuated by downward force
on exterior edge

Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

Fig 13: Stage 2. Group Brief: Dynamic Solar Shading; hexagonal array of articulated
shading elements

Fig. 14:Stage 2. Group Brief: Doors and Openings (members: Jacek Jaskólski, Thorsten
Lang, Ana Maria Marcu, Ioanna Symeonidou, and Liam Woolley); animation in Rigid
Origami Simulator. Notice the triangulation added to enable a mechanism.

Fig. 15: Stage 2. Group Brief: Doors and Openings; two layer rigid plate model sandwichinga paper hinge, pattern generated using Grasshopper [16] definition by Tomohiro Tachi;
assembly according to [5]Proceedings of the International Symposium on Algorithmic Design forArchitecture and Urban Design,
ALGODE TOKYO 2011
March 14-16, 2011, Tokyo, Japan

References[1] K. Miura, Proposition of pseudo-cylindrical concave polyhedral shells, Proc. IASS
Symposium on Folded Plates and Prismatic Structures, 1970.
[2] R. D. Resch and H. Christiansen. The design and analysis of kinematic folded platesystems, Proc. IASS Symposium on Folded Plates and Prismatic Structures, 1970.
[3] C. Hoberman. “Reversibly Expandable Three-Dimensional Structure.” United States
Patent No. 4,780,344, 1988.
[4] T. Tachi, Generalization of Rigid-Foldable Quadrilateral-Mesh Origami, Journal of IASS,
50(3), pp. 173–179, 2009.
[5] T. Tachi, Rigid-Foldable Thick Origami, Origami5: Proc. the 5th International
Conference on Origami in Science, Mathematics and Education (OSME 2010),
Singapore, July 13–17, 2010, to appear, A K Peters.
[6] T. Tachi, Freeform Origami, www.tsg.ne.jp/TT/software/[7] G. Epps and T. Tachi, One DOF Linkage Manipulation using Rigid Origami, workshop
of Advances in Architectural Geometry 2010, September 18-19, 2010.
[8] E. D. Demaine and M. L. Demaine, History of Curved Origami Sculpture,
http://erikdemaine.org/curved/history/[9] E. D. Demaine, M. L. Demaine, and D. Koschitz, Reconstructing David Huffman's
Legacy in Curved-Crease Folding, Origami5: Proc. OSME 2010, Singapore, July 13–17,
2010, to appear, A K Peters.
[10] R. D. Resch, Portfolio of Shaded Computer Images, Proc. IEEE, 62(4), pp.496-502, 1974.
[11] D. A. Huffman. Curvature and creases: A Primer on Paper. IEEE Transactions on
Computers, C-25(10), pp. 1010–1019, 1976.
[12] D. Fuchs and S. Tabachnikov. More on Paperfolding. The American Mathematical
Monthly, 106(1), pp. 27–35, 1999.
[13] Y. Kergosien, H. Gotoda, and T. Kunii, Bending and Creasing Virtual Paper, IEEE
Computer Graphics and Applications, 14(1), pp. 40–48, 1994.
[14] Y. Liu, H. Pottmann, J. Wallner, Y.-L. Yang, and W. Wang. Geometric modeling withconical meshes and developable surfaces, ACM Transactions on Graphics, 25(3), 681–
689, 2006.
[15] M. Kilian, S. Flöry, N. J. Mitra, and H. Pottmann, Curved folding, ACM Transactions on
Graphics, 27(3), 1-9, 2008.
[16] McNeel.
Grasshopper
Generative
ModelingforRhino,
http://grasshopper.rhino3d.com/.
[17] Vyzoviti, Sophia. Folding Architecture. BIS Publishers, 2003.
[18] Vyzoviti, Sophia. Fold It! Workshop, Industrial Design Engineering, Royal College ofArt, UK, 2005