Research Papers

Designing Freeform Origami Tessellations by Generalizing Resch's Patterns

[+] Author and Article Information
Tomohiro Tachi

Assistant Professor
Department of General Systems Studies,
Graduate School of Arts and Sciences,
The University of Tokyo,
3-8-1 Komaba, Meguro-Ku,
Tokyo 153-8902, Japan
e-mail: tachi@idea.c.u-tokyo.ac.jp

Star tuck is thus a generalization of waterbomb base used for origami tessellation.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 2, 2013; final manuscript received June 4, 2013; published online October 3, 2013. Assoc. Editor: Larry L. Howell.

J. Mech. Des 135(11), 111006 (Oct 03, 2013) (10 pages) Paper No: MD-13-1055; doi: 10.1115/1.4025389 History: Received February 02, 2013; Revised June 04, 2013

In this research, we study a method to produce families of origami tessellations from given polyhedral surfaces. The resulting tessellated surfaces generalize the patterns proposed by Ron Resch and allow the construction of an origami tessellation that approximates a given surface. We will achieve these patterns by first constructing an initial configuration of the tessellated surfaces by separating each facets and inserting folded parts between them based on the local configuration. The initial configuration is then modified by solving the vertex coordinates to satisfy geometric constraints of developability, folding angle limitation, and local nonintersection. We propose a novel robust method for avoiding intersections between facets sharing vertices. Such generated polyhedral surfaces are not only applied to folding paper but also sheets of metal that does not allow 180 deg folding.

Copyright © 2013 by ASME
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Tachi, T., 2010, “Origamizing Polyhedral Surfaces,” IEEE Trans. Vis. Comput. Graph., 16(2), pp. 298–311. [CrossRef] [PubMed]
Fujimoto, S., 1976, “ Souzousei wo kaihatsu suru rittai origami ,” Hyougo-ken Gakkou Kouseikai Tamba Shibu (in Japanese).
Miura, K., 1970, “Proposition of Pseudo-Cylindrical Concave Polyhedral Shells,” Proceedings of IASS Symposium on Folded Plates and Prismatic Structures.
Davis, E., Demaine, E. D., Demaine, M. L., and Ramseyer, J., 2013, “Reconstructing David Huffman's Origami Tessellations,” Proceedings of the ASME 2013 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference.
Resch, R. D., 1968, “Self-Supporting Structural Unit Having a Series of repetitious Geometrical Modules,” U.S. patent no. 3,407,558.
Resch, R. D., and Christiansen, H., 1970, “The Design and Analysis of Kinematic Folded Plate Systems,” Proceedings of IASS Symposium on Folded Plates and Prismatic Structures.
Tachi, T., 2010, “Freeform Variations of Origami,” J. Geom. Graph., 14(2), pp. 203–215.
Bateman, A., 2002, “Computertools and Algorithms for Origami Tessellation Design,” Origami 3: Proceedings of the 3rd International Meeting of Origami Mathematics, Science, and Education, pp. 121–127.
Lang, R. J., and Bateman, A., 2011, “Every Spider Web has a Simple Flat Twist Tessellation,” Origami 5, P. Wang-Iverson, R. J.Lang, and M.Yim, eds., CRC Press, pp. 455–473.
Sheffer, A., and de Sturler, E., 2001, “Parameterization of Faceted Surfaces for Meshing Using Angle-Based Flattening,” Eng. Comput., 17(3), pp. 326–337. [CrossRef]
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Fig. 1

Regular triangular tessellation by Resch

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Fig. 2

Origamizer and Resch's tessellation. Both are comprised of surface polygons and tucks that are hidden. Notice that Resch's pattern can have the tuck folded halfway, whereas origamizer vertex keeps the tuck closed because of the crimp folds.

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Fig. 3

The process for obtaining a freeform origami tessellation

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Fig. 4

Top: Insertion of a star tuck. Bottom: Vertex with odd number of incident edges n can be interpreted as the vertex with 2 n edges by the insertions of digons between the facets.

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Fig. 5

Example tessellations generated from regular planar tilings. (a) Triangular pattern with regular 6-deg vertices. (b)–(d) Triangular, quadrangular, and hexagonal pattern with the insertion of digons.

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Fig. 6

Variations of the tuck structures for a regular triangular mesh

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Fig. 7

Pair of edge-adjacent facets p and q

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Fig. 8

Left: Fold angle ρ(x) of the vertex between two fixed points. Note the singularity at the end points. Right: Modified angular evaluation fp,q(x).

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Fig. 9

Left: Local collision between vertex-adjacent facets. Right: Mesh after the penalty function is applied to avoid the intersection.

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Fig. 10

Detection of invalid vertices

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Fig. 11

View angle αo,i,i + 1 and the vectors

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Fig. 12

Penalty function given from a star-like facet fan. Illustrated on a plane by the Gnomonic projection of the spherical surface.

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Fig. 13

Example designs of bell shape, hyperbolic surface (anticlastic), and spherical surface (synclastic) from star-tuck origami tessellations.

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Fig. 14

Blobby origami tessellation using truncated stars. Note that the mesh is used inside-out to make the tuck visible.

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Fig. 15

Top: Spherical origami tessellation failing to avoid intersection. Bottom: Spherical origami with cuts.

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Fig. 16

Tessellated origami bunny using the initial cut (indicated by thick curve) of the mesh on the back and behind the ears. The tessellation is based on the twist fold.

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Fig. 17

Example folding of a perforated steel sheet

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Fig. 18

Continuous unfolding motion from a 3D form to a planar sheet



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