Fold Mapping: Parametric Design of Origami Surfaces

Fold Mapping: Parametric Design of Origami Surfaces with Periodic Tessellations M. Gardiner, R. Aigner, H. Ogawa, R. Hanlon

Abstract: We present a design method that prioritises in-context design for origami surfaces with periodic tessellations in a parametric CAD workflow using Grasshopper 3D. The key design criteria are: target geometry surface, userdefined folding patterns as periodic tessellations, and fold resolution. Using an error minimisation solver, we generate developable crease patterns from nondevelopable meshes. We evaluate our method through a study of a target geometry, Fold Mapped with various fold molecules at variable resolutions, and present a visual analysis as proof of form-fit to the target. This method affords rapid development of origami surfaces, bypassing significant trial and error in by-hand design processes.

1

Background and Related Work

Software for origami design is a highly specialised topic and the utility of existing tools tends towards solving specific categories of design problems. Examples of this intent can be as specific as Ronald Resch’s software to solve configurations of the triangular Resch pattern [Resch and Christiansen 70], towards more generalised tools like Origamizer by [Demaine and Tachi 10,Tachi 08] integrated into FreeForm Origami [Tachi 10] to find crease patterns for any given mesh, or Tachi’s earlier software Rigid Origami [Tachi 2009] intended to explore and evaluate the rigidity of a given geometry or crease pattern, or an algorithm by [Dudte et al. 16] intended to solve geometric surfaces with detailed analysis and composition of the Miuraori. Overall, algorithms are specialised and design processes vary from engineers to artists, and from engineering requirements to aesthetic criteria. In this paper, we focus on a design method to approximate geometric surfaces with periodic tessellations–also known as open corrugations–with the intent to explore off-grid and irregular configurations and to compare a range of patterns for a particular design target. In addition to the aesthetic qualities of these patterns, we have a particular interest in open corrugations due to their deployability and application to kinetic origami systems such as Oribotics [Gardiner 09, Gardiner 15a]. In folded-plate structures, design criteria such as rigid-foldability, developability, target shape, and unit pattern have emerged in released software packages such as: Freeform Origami with Origamizer function a [Demaine and Tachi 10] and point editing tools, [Tachi 13], oricreate [Chudoba et al. 15] as a command line interface,

G ARDINER , A IGNER , O GAWA , H ANLON

Figure 1: Natural Folding: Top: Illustration of Force + Matter. Middle: Expression of Folded Form. Bottom: crease patterns. a/A) Yoshimura-ori, b/B) Miura-ori, c/C) Waterbomb-ori (pinecone-ori/ananas-ori) d/D) Kresling-ori, e/E) Resch4-ori.

the Rigid Origami Toolbox for use in Matlab [Gattas 13], and [Dudte et al. 16] show a method for approximating curved surfaces with miura-ori, considering material properties to determine fold scale versus accuracy of the finished form.

2

Natural Folding

The background for this research is the study of Natural Folding: how nature uses the language of folding, and we apply our findings in the domain of art and design. We examined literature to decide which geometries to use in our work. We found that nature applies folding in many ways: to structurally code strings of proteins in DNA Origami [Rothemund 05], leaf unfurling patterns [Kobayashi et al. 98], insect wings [Kresling 12, Kresling 97], as a way to pattern the crushing power of folded laminations in the earth’s crust [Ramsay 67]. The effect of buckling and bifurcation in materials under stresses from forces is intriguing and has been the source of many discoveries of folding patterns. Natural Folding is the folding we discussed as by-chance in [Gardiner 15b]. Figure 1. summarises our research into the primary Natural Folding patterns, and from these patterns, we will experiment

F OLD M APPING : PARAMETRIC D ESIGN T ESSELLATIONS

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with irregular arrangements by parametric means. Figure 1. shows the geometric relationship between forces and fold patterns. The patterns and forces are described as follows, we also cite our key reference for the description of force: a. Yoshimura cylindrical compression [Yoshimura 51] b. Miura transverse planar compression [Miura 69] c. Waterbomb conical compression [Kresling et al. 08]1 d. Kresling rotational-twist-compression of cylinder [Kresling et al. 08]2 e. Resch torsional compression of the plane [Resch 73, pp. 643-644]. Our intention with this work is to examine Natural Folding by parametric design, and to do so we start with this collection of patterns and also adopt the concept of chance and randomness in buckling and bifurcation to inform our computational method.

3

Method

Our software prototype merges several concepts from packages discussed in related work. Our aim was to extend the methods from existing packages by prioritising the following criteria: 1. Simple and Flexible Pattern Definitions: Simple pattern definitions to allow the end-user to virtually prototype one design with many tessellation patterns, add new ones, and to switch patterns at design-time. 2. Mapping fold pattern to a target surface: the intention being that the artist designs a form, as the target geometry, and the software generates the foldable pattern. 3. Build in a Parametric CAD environment: to support end-to-end visualisation and parameterization of the process and in-context design with related geometries. This enables design-time visualisation of criteria such as target geometry, pattern choice, fold mapping resolution, and to visualise differences between input and output geometries. 4. Allow chance in the computation: to examine a natural aesthetic in the geometric outputs. As discussed in Section 2 Natural Folding is the result of buckling and bifurcation, or chance, the aim of this criteria is to compare deterministic developable solutions, such as those from FreeForm Origami, our solutions that allow randomness. 1 Kresling’s discussion in this paper does not introduce the waterbomb as such, but rather discusses the natural formation via compression of a conical shell, after which she renamed the pattern Pine Cone Pattern as the pinecone is the natural analogue for this geometry. 2 Again, this work describes the torsional force required to create this pattern. Extensive work on the same pattern by Guest in [Guest and Pellegrino 94].


Figure 2: Simplified ORI*GH grasshopper patch, without optional solver parameters, and error display options. In this patch, Surface A and B are defined by manually created geometry.

3.1 Grasshopper in Rhino Based on these claims, we chose Rhinoceros 3D3 combined with the visual programming plug-in Grasshopper 3D4 as our design platform, especially as potential use cases are architecture and design. In particular, Grasshopper 3D provides a popular design environment for parametric design in those disciplines, featuring an active community. Our Grasshopper 3D patch is arranged in several clusters, see the patch shown in Figure 2: (a) Pattern Input (see Section 3.2) takes surfaces, pattern resolution, and fold unit type as principal inputs. Constructed Origami mesh and symmetry data are essential outputs. ORI*gh SOLVER (b) accepts a mesh, and optimizes according to options, such as thresholds and weighting factors as described in Section 3.3, outputs are mesh and error values, including but not shown colour codings for visual feedback by highlighting problematic regions. Crease pattern generation (c) accepts mesh input and outputs edge data with optional mountain, valley, and border specifiers for potential handling differences during fabrication, depending on edge type. Edges and Mesh are key outputs for the crease pattern or flattened configuration. 3.2 Fold Mapping Input 3.2.1 Panelling Tools Our research into existing tools and plugins in Grasshopper revealed a plugin called Panelling Tools [McNeel 11], that is designed to map patterns across geometric surfaces, and therefore, a good candidate for the idea of mapping folds. The essential steps are: 1. 2. 3. 4.

Create surface A, by modelled or parametric method, to generate Grid A. Create surface B, commonly an offset, to generate Grid B. Adjust the resolution of the grids Populate the grids with a pattern 3 “Rhinoceros

3D http://www.rhino3d.com” 3D http://www.grasshopper3d.com”

4 “Grasshopper


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Figure 3: An illustration of the Panelling Tools workflow used. For input to the pattern generator, we specify an initial surface A, surface B is usually calculated by extruding by an offset distance. Grids A and B define the fold resolution via input parameters. The fold patterns are specified by strings containing faces description, i.e. a number of vertex positions in the format {xyz} for each face.

Figure 4: Fold Molecule Definition Pleat. From left: isometric side, isometric, and top view, with the overlaid unit grid.

3.2.2 Fold Molecule Definition Definition of patterns or Fold Molecules requires the abstraction of the folding pattern into a simplified grid. Panelling Tools has limitations in pattern definitions: patterns must be described by integer values within only two z-planes. The method was to use Blender, an open-source 3D tool with Blender Python (bpy), to model and then generate the pattern strings by script. The latest version of Panelling Tools now supports this directly in Rhino. Figure 4. shows a simple pleat pattern, with the unit grid highlighted, the important aspect is to model vertices on the unit grid. Table 1 shows pattern definitions used in the results. 3.3 ORI*gh Solver For generating a developable approximation from the initial input surface, we use a numeric solver, minimizing a multi-dimensional error function, ensuring requirements such as developability and planarity of individual origami segments. Our algorithm is also able to optionally maintain overall object symmetry and to prior-


Table 1: Miura 6090, Kresling and Yoshimura Fold Molecule definitions

Miura 6090 Molecule

Kresling Molecule

Yoshimura Molecule

(2,4,2)(0,2,2)(1,4,0)(3,6,0)(2,4,2) (2,4,2)(3,6,0)(5,4,0)(4,2,2)(2,4,2) (2,4,2)(4,2,2)(5,0,0)(3,2,0)(2,4,2) (2,4,2)(3,2,0)(1,0,0)(0,2,2)(2,4,2)

(1,0,0)(2,1,0)(0,0,0)(1,0,0) (1,1,0)(0,0,0)(2,1,0)(1,1,0) (1,1,0)(2,1,0)(0,2,0)(1,1,0) (0,2,0)(2,1,0)(1,2,0)(0,2,0)

(1,0,0)(0,1,0)(1,2,0)(1,0,0) (1,2,0)(2,1,0)(1,0,0)(1,2,0) (2,1,0)(1,2,0)(2,3,0)(2,1,0) (0,3,0)(1,2,0)(0,1,0)(0,3,0)

itize a selected subset of vertex positions. 3.3.1

Optimization Software

Our solver implementation iteratively changes mesh configurations towards an unknown optimum, by modifying single vertex positions, until a candidate solution satisfying a user-specified tolerance is found. We chose local over global optimization since previous work showed that the latter comes with high computational cost [Wang and Tang 04]. For sake of simplicity, we assume the following: • The polyhedron is composed of faces with a maximum vertex count of 4, thus we only support triangle and quadrilateral faces. • Our pattern tiles are of infinitesimally small or zero thickness. • All crease edges are straight lines, thus we do not consider curved folding. • The polyhedron provides an adequate boundary edge loop count. Closed surfaces, such as toroids or spheres cannot be flattened, as the algorithm does not introduce extra cuts. • We assume the designers’ ability to choose suitable topologies for the basic shape they are targeting, as some patterns curve towards specific shapes better than others. As we offer flexibility in the design process, it is possible to input unsolvable geometry to the solver. We start from a non-developable input mesh, generated by the pattern generator from user input as described in 3.2.2. We then optimize the resulting mesh repeatedly until convergence of a global objective function, which is composed as a weighted sum of the following terms: developability (dev), planarity (plan), convexity (conv), collision (coll), and vertex position constraints (pos). Let Terms = {dev, plan, conv, coll, pos}, then the per-vertex objective function, specifying the error at vertex v out of the set of all vertices V of the model, is defined as E(v) =

∑

λτ Eτ (v)

τ∈Terms

where λτ ∈ R+ 0 is a respective weighting factor and Eτ is the according error measure. We found this approach to be highly powerful since arbitrary undesirable features can be avoided by adding conditions while unused terms can be ignored


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by setting the respective weighting factor to 0. E.g., our algorithm does not preserve mountain and valley folds. By adding an additional penalty-term, this could easily be accomplished. Moreover, by exposing the terms’ weighting factors to the designer, we afford control over the generated output, as different applications and designs may allow different tradeoffs. 3.3.2 Error Minimization Since the objective function is highly non-convex, the solver is likely to run into a local minimum instead of finding a global optimum. Introducing a Monte-Carlo based approach for finding a candidate solution, meeting our chance criteria, proved superior in our early tests when compared to deterministic approaches. For a single optimization step, we iterate each vertex of the mesh and propose a displacement with user-controllable direction. If this displacement decreases the objective function, it is applied, otherwise, it is discarded and we move on to the next vertex. After all vertices of the mesh are updated, we evaluate the mesh configuration and repeat until the condition E max ≤ Eb is met, with Eb being the highest tolerable error value as specified by the user, and E max being the maximum per-vertex error measure E max = max E(vi ) vi ∈V

Regarding computational performance, our implementation was designed such that this part of the pipeline is interchangeable for any other energy minimization algorithm, such as gradient descent, Gauss-Newton, or Levenberg-Marquardt, which are expected to find solutions in much shorter time. For this reason, we skip a more in-depth explanation of the prototype and direct the interested reader to related literature [Nocedal and Wright 99]. Due to the randomness introduced by the Monte-Carlo method, meshes are deformed in a non-deterministic, and thus non-symmetrical manner. Therefore we implemented an optional symmetry-preserving variant of our algorithm. For this purpose, the designer specifies a plane of symmetry in Rhinoceros 3D, while at solver initialization, all vertices on its positive side have to be on perfectly mirrored positions of their counterparts on its negative side, so the respective counterparts can be detected by the algorithm.

4

Results

Of the many experiments we conducted, including a masterclass at the Ars Electronica Festival in 2016, the example illustrated in Figure 5. provides an overview of how the process reflects our initial aims. The piece was fabricated (see Figure 7.) by our method Fold Printing (see our companion paper Fold Printing) was shown during the exhibition Artists as Catalysts [Schöpf et al. 16, 68-153]


Figure 5: Proof-of-concept Parametric Designs for origami headwear with various periodic Tessellations. Shown in pairs of mesh and crease pattern. From left to right, top to bottom: design-to-fit surfaces of target geometry and 3D scan data; Waterbomb pattern; Kresling pattern; low-resolution Yoshimura; medium resolution Resch4; high resolution Resch4 pattern.


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Figure 6: Form fit visual evaluation. Left to right: top two rows: target surface (in red), waterbomb mapping (WB-23.20), Kresling mapping (K-10.13). Bottom two rows: Yoshimura mapping (Y-12.9), Resch4-3030 (R4-30.30) mapping and Resch45050 (R4-50.50) mapping. The waterbomb is the lowest resolution closest match to the target form, and the high-resolution Resch4-50.50 pattern is a very close visual fit.

within the theme of fashion and technology, where the body becomes our design context to work with. We captured 3D scans of design targets, and began constructing target geometries and experimenting with folding patterns. One of the distinctive projects was the ORI*hat, and we present a number of solutions to the target geometry. A key aspect of our evaluation was to compare the form fit of the mapped pattern to the target surface. Visual analysis of Figure 6. shows that some pattern geometries are more suited to the target surface. The pattern WB23.20 was selected for fabrication for its aesthetic qualities, its close fit to the original geometry with a low number of creases.

5

Evaluation

Our results aligned tightly with our aims, and can be summarised as follows: 1. Simple and Flexible Pattern Definitions: due to the relative simplicity of the target geometry, ORI*gh successfully generated developable crease patterns


Figure 7: Fold Print Fabricated Geometries. Left: ORI*hat from a waterbomb pattern. Right: ORI*fox, a varible width Yoshimura patterned cylinder created during the masterclass.


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from a range of design candidates from different patterned inputs. 2. Mapping fold pattern to a target surface: the parametrically generated patterns are visually close fits to the target surface. With higher resolution patterns forming the closest fit. 3. Build in a Parametric CAD environment: Rhino 3d and Grasshopper 3d allowed the integration of scan data, design of the scanned geometry, and visualisation in-context. Feedback from workshop participants was that parametrically altering their designs in-context allowed them to make critical design decisions. 4. Allow chance in the computation: the small permutations visible in the patterns, that break the designs away from high regularity patterns, do give a visual impression of a more natural order. However, feedback from workshop participants indicated that the processing time for some larger geometric sets was at times an impediment to making quick decisions. This suggests that a deterministic solver designed to shorten computation time may be of more benefit to design-time decisions.

6

Further work

The method presented contributes to art and design focussed approaches to parametric origami design. We see potential in implementing a faster developability solver, and introducing a number of new constraints such as flat-foldability and rigid-foldability, thus reproducing more criteria from other packages within the parametric environment. We discovered some interesting kinetic properties in irregular, non-rigid foldable patterns. We found that certain geometric arrangements were still kinetic, but had movement constraints due to self-intersection, in one sense this could be seen as a problem, however we see potential in exploring the design possibilities soft-robotic actuators made with irregular geometries, as many of natural articulations have motion limitations.

7

Conclusion

We presented our aims and method with sufficient detail for replication, as the researcher could use the methods described in 3.2, and then use a tool such as Freeform Origami to generate a developable pattern. We also presented a detailed report of successful candidates as virtual and physical prototypes. To satisfy our aims for parameterisation we selected Rhino/Grasshopper for our software prototype, as it affords visualisation of the complete process that is critical for designtime modification and its tools support digital fabrication. Our fold molecule definitions are comprised of arrays of vertices that construct an idealised folded geometry that can be sketched on graph paper and noted by hand, or designed in 3D software and produced by script. Our solver is compiled as a Grasshopper Plugin. In our evaluation, we compared patterns for a given design target and discussed fundamental geometric properties of well-known tessellations such as Yoshimura, Miura-ori, Waterbomb, Kresling and Resch patterns.


Our results support our case for simplified unit pattern definitions and parameterisation of the complete design process. We conducted workshops with design students and found visualisation and selection of patterns for functional and aesthetic reasons had a radical impact on design-to-fit problems. The ability to design target geometry at scale in context with other design criteria, such as in Figure 5. showing scan data and designs of an origami cap designed-to-fit with iterations in different tessellation patterns, illustrates the benefits of Parametric CAD and Computational Origami in a single environment. Finally, we discussed potential applications of the method from bespoke garments to designs for soft robotics actuators in advanced oribotic applications.

8

Acknowledgements

PEEK Project AR272-G21. Funded through the FWF, PEEK Program, Program Management: Dr Eugen Banauch. Special thanks to Ars Electronica Futurelab for their continued support of this research.

References [Chudoba et al. 15] R Chudoba, JD Van der Woerd, and J Hegger. “Oricreate: Modeling Framework for Design and Manufacturing of Folded Plate Structures.” Origami 6 (2015), 523–536. [Demaine and Tachi 10] E Demaine and Tomohiro Tachi. Origamizer: A Practical Algorithm for Folding Any Polyhedron. Manuscript, 2010. [Dudte et al. 16] Levi H Dudte, Etienne Vouga, Tomohiro Tachi, and L Mahadevan. “Programming Curvature Using Origami Tessellations.” Nature materials. [Gardiner 09] Matthew Gardiner. “A Brief History of Oribotics.” In Origami 4, edited by Robert J. Lang, pp. 51–60. AK Peters, 2009. [Gardiner 15a] Matthew Gardiner. “Folding and Unfolding a Million Times over: Longevity, Origami, Robotics and Biomimetics as Material Thinking in Oribotics.” Symmetrion 26:2 (2015), 189–202. [Gardiner 15b] Matthew Gardiner. “On the Aesthetics of Folding and Technology: Scale, Dimensionality, and Materiality.” In Origami 6, edited by Koryo Miura, Toshikazu Kawasaki, Tomohiro Tachi, Ryuhei Uehara, Robert J. Lang, and Patsy Wang-Iverson. Natick, MA: AK Peters, 2015. [Gattas 13] Joe Gattas. “Rigid Origami Toolbox.”, 2013. [Guest and Pellegrino 94] Simon D. Guest and Sergio Pellegrino. “The Folding of Triangulated Cylinders, Part I: Geometric Considerations.” Journal of Applied Mechanics 61 (1994), 773–777. [Kobayashi et al. 98] Hidetoshi Kobayashi, Biruta Kresling, and Julian FV Vincent. “The Geometry of Unfolding Tree Leaves.” Proceedings of the Royal Society of London. Series B: Biological Sciences 265:1391 (1998), 147–154.


OF

O RIGAMI S URFACES

WITH

P ERIODIC

[Kresling et al. 08] Biruta Kresling, R Abel, and J Cooke Robert. “Natural Twist Buckling in Shells: From the Hawkmoth’s Bellows to the Deployable Kresling-Pattern and Cylindrical Miuraori.” In Proceedings of the 6th International Conference on Computation of Shell and Spatial Structures, John F. Abel and J. Robert Cooke, Eds., Ithaca, 2008. [Kresling 97] Biruta Kresling. “Folded and Unfolded Nature.” In Origami Science and Art: Proceedings of the Second International Meeting of Origami Science and Scientific Origami, pp. 93–106. Seian University, Otsu, Shiga, Japan: Seian University of Art and Design, 1997. [Kresling 12] Biruta Kresling. “Origami-Structures in Nature: Lessons in Designing Smart Materials.” In MRS Proceedings, 1420, 1420, pp. mrsf11–1420. Cambridge Univ Press, 2012. [McNeel 11] McNeel. “PanelingTools Plug-in for Rhino 5 [McNeel Wiki].”, 2011. Available online (https://wiki.mcneel.com/labs/panelingtools). [Miura 69] Koryo Miura. “Proposition of Pseudo-Cylindrical Concave Polyhedral Shells.” ISAS report 34:9 (1969), 141–163. [Nocedal and Wright 99] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer-Verlag New York, Inc., 1999. [Ramsay 67] John G Ramsay. Folding and Fracturing of Rocks. McGraw-Hill Companies, 1967. [Resch and Christiansen 70] Ronald D Resch and Hank Christiansen. “The Design and Analysis of Kinematic Folded Plate Systems.” In Proceedings of IASS Symposium on Folded Plates and Prismatic Structures, 1970. [Resch 73] Ronald D Resch. “The Topological Design of Sculptural and Architectural Systems.” In Proceedings of the June 4-8, 1973, National Computer Conference and Exposition, pp. 643–650. ACM, 1973. [Rothemund 05] P. W. K Rothemund. “Design of DNA Origami.” In Proceedings of the 2005 IEEE/ACM International Conference on Computer-Aided Design, ICCAD ’05, pp. 471–478. IEEE Computer Society, 2005. Available online (http://dl.acm. org/citation.cfm?id=1129601.1129669). [Schöpf et al. 16] C. Schöpf, G. Stocker, and H. Leopoldseder. Radical Atoms and the Alchemists of Our Time. Ars Electronica, Hatje Cantz, 2016. Available online (https://books.google.com.au/books?id=1iy7jwEACAAJ). [Tachi 08] T Tachi. “Origamizer.”, 2008. [Tachi 10] T Tachi. “Freeform Origami.”, 2010. [Tachi 13] Tomohiro Tachi. “Designing Freeform Origami Tessellations by Generalizing Resch’s Patterns.” Journal of mechanical design 135:11 (2013), 111006.


[Wang and Tang 04] Charlie C.L. Wang and Kai Tang. “Achieving Developability of a Polygonal Surface by Minimum Deformation: A Study of Global and Local Optimization Approaches.” The Visual Computer 20:8 (2004), 521–539. doi:10.1007/s00371004-0256-0. [Yoshimura 51] Yoshimaru Yoshimura. “On the Mechanism of Buckling of a Circular Cylindrical Shell under Axial Compression.” Technical report, National Advisory Committee for Aeronautics, 1951.

Matthew Gardiner Ars Electronica Futurelab, Ars Electronica Strasse 1, Linz 4040, Austria. University of Newcastle, School of Creative Industries, Newcastle Australia. e-mail: [email protected] Roland Aigner, Hideaki Ogawa, and Rachel Hanlon Ars Electronica Futurelab, Ars Electronica Strasse 1, Linz 4040