Research Papers: Design Automation

On the Incompressibility of Cylindrical Origami Patterns

[+] Author and Article Information
Friedrich Bös

Institute for Numerics and Applied Mathematics,
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
e-mail: f.boes@math.uni-goettingen.de

Max Wardetzky

Institute for Numerics and Applied Mathematics,
Georg-August-Universität Göttingen,
37081 Göttingen, Germany

Etienne Vouga

Department of Computer Science,
The University of Texas at Austin,
Austin, TX 78712

Omer Gottesman

School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 5, 2016; final manuscript received September 19, 2016; published online December 22, 2016. Assoc. Editor: Nam H. Kim.

J. Mech. Des 139(2), 021404 (Dec 22, 2016) (9 pages) Paper No: MD-16-1271; doi: 10.1115/1.4034970 History: Received April 05, 2016; Revised September 19, 2016

The art and science of folding intricate three-dimensional structures out of paper has occupied artists, designers, engineers, and mathematicians for decades, culminating in the design of deployable structures and mechanical metamaterials. Here we investigate the axial compressibility of origami cylinders, i.e., cylindrical structures folded from rectangular sheets of paper. We prove, using geometric arguments, that a general fold pattern only allows for a finite number of isometric cylindrical embeddings. Therefore, compressibility of such structures requires either stretching the material or deforming the folds. Our result considerably restricts the space of constructions that must be searched when designing new types of origami-based rigid-foldable deployable structures and metamaterials.

Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.


Yoshimura, Y. , 1955, “ On the Mechanism of Buckling of a Circular Cylindrical Shell Under Axial Compression,” National Advisory Committee for Aeronautics, Washington, DC, Technical Report No. 1390.
Coppa, A. P. , 1967, “ Inextensional Buckling Configurations of Conical Shells,” AIAA J., 5(4), pp. 750–754. [CrossRef]
Lobkovsky, A. , and Witten, T. A. , 1997, “ Properties of Ridges in Elastic Membranes,” Phys. Rev. E, 55(2), pp. 1577–1589. [CrossRef]
Witten, T. A. , 2007, “ Stress Focusing in Elastic Sheets,” Rev. Mod. Phys., 79(2), pp. 643–675. [CrossRef]
Ciarlet, P. G. , 1993, Mathematical Elasticity: Three-Dimensional Elasticity, Elsevier, Amsterdam, The Netherlands.
Miura, K. , 1980, “ Method of Packaging and Deployment of Large Membranes in Space,” 31st Congress International Astronautical Federation, pp. 1–10.
Silverberg, J. L. , Evans, A. A. , McLeod, L. , Hayward, R. C. , Hull, T. C. , Santangelo, C. D. , and Cohen, I. , 2014, “ Using Origami Design Principles to Fold Reprogrammable Mechanical Metamaterials,” Science, 345(6197), pp. 647–650. [CrossRef] [PubMed]
Silverberg, J. L. , Na, J.-H. , Evans, A. A. , Liu, B. , Hull, T. C. , Santangelo, C. D. , Lang, R. J. , Hayward, R. C. , and Cohen, I. , 2015, “ Origami Structures With a Critical Transition to Bistability Arising From Hidden Degrees of Freedom,” Nat. Mater., 14(4), pp. 389–393. [CrossRef] [PubMed]
Zhang, X. , Cheng, G. , You, Z. , and Zhang, H. , 2007, “ Energy Absorption of Axially Compressed Thin-Walled Square Tubes With Patterns,” Thin-Walled Struct., 45(9), pp. 737–746. [CrossRef]
Song, J. , Chen, Y. , and Lu, G. , 2012, “ Axial Crushing of Thin-Walled Structures With Origami Patterns,” Thin-Walled Struct., 54, pp. 65–71. [CrossRef]
Kuribayashi, K. , Tsuchiya, K. , You, Z. , Tomus, D. , Umemoto, M. , Ito, T. , and Sasaki, M. , 2006, “ Self-Deployable Origami Stent Grafts as a Biomedical Application of Ni-Rich TiNi Shape Memory Alloy Foil,” Mater. Sci. Eng. A, 419(1–2), pp. 131–137. [CrossRef]
Abel, Z. , Connelly, R. , Demaine, E. D. , Demaine, M. L. , Hull, T. C. , Lubiw, A. , and Tachi, T. , 2015, “ Rigid Flattening of Polyhedra With Slits,” Origami6, p. 109.
Balkcom, D. J. , Demaine, E. D. , Demaine, M. L. , and Ochsendorf, J. A. , 2009, “ Folding Paper Shopping Bags,” Origami4, Vol. 4, R. Lang, ed., CRC Press, Boca Raton, FL.
Wu, W. , and You, Z. , 2011, “ A Solution for Folding Rigid Tall Shopping Bags,” Proc. R. Soc. A, 467(2133), pp. 2561–2574. [CrossRef]
Dudte, L. H. , Vouga, E. , Tachi, T. , and Mahadevan, L. , 2016, “ Programming Curvature Using Origami Tessellations,” Nat. Mater., 15(5), pp. 583–588. [CrossRef] [PubMed]
Saito, K. , Tsukahara, A. , and Okabe, Y. , 2016, “ Designing of Self-Deploying Origami Structures Using Geometrically Misaligned Crease Patterns,” Proc. R. Soc. A, 472(2185), pp. 1–16. [CrossRef]
Kilian, M. , Flöry, S. , Chen, Z. , Mitra, N. , Sheffer, A. , and Pottmann, H. , 2008, “ Curved Folding,” ACM Trans. Graph., 27(6), pp. 75:1–75:9.
Filipov, E. T. , Paulino, G. H. , and Tachi, T. , 2016, “ Origami Tubes With Reconfigurable Polygonal Cross-Sections,” Proc. R. Soc. A, 472(2185), p. 20150607. [CrossRef]
Guest, S. D. , and Pellegrino, S. , 1994, “ The Folding of Triangulated Cylinders—Part II: The Folding Process,” ASME J. Appl. Mech., 61(4), pp. 778–783. [CrossRef]
Guest, S. D. , and Pellegrino, S. , 1996, “ The Folding of Triangulated Cylinders—Part III: Experiments,” ASME J. Appl. Mech., 63(1), pp. 77–83. [CrossRef]
Yasuda, H. , Yein, T. , Tachi, T. , Miura, K. , and Taya, M. , 2013, “ Folding Behaviour of Tachi-Miura Polyhedron Bellows,” Proc. R. Soc. A, 469(2159), pp. 1–18. [CrossRef]
Cauchy, A. L. , 1813, “ Recherche sur les polyèdres—premier mémoire,” J. Ec. Polytech., 9, pp. 68–86.
Connelly, R. , 1979. “ The Rigidity of Polyhedral Surfaces,” Math. Mag., 52(5), pp. 275–283. [CrossRef]
Connelly, R. , Sabitov, I. , and Walz, A. , 1997, “ The Bellows Conjecture,” Beitr. Algebra Geom., 38(1), pp. 1–10.
Huffman, D. A. , 1976. “ Curvature and Creases: A Primer on Paper,” IEEE Trans. Comput., C-25(10), pp. 1010–1019. [CrossRef]
Belcastro, S.-M. , and Hull, T. C. , 2002, “ Modelling the Folding of Paper Into Three Dimensions Using Affine Transformations,” Linear Algebra Appl., 348(1–3), pp. 273–282. [CrossRef]
Streinu, I. , and Whiteley, W. , 2004, “ Single-Vertex Origami and Spherical Expansive Motions,” Discrete and Computational Geometry, Springer, Berlin, pp. 161–173.
Wu, W. , and You, Z. , 2010, “ Modelling Rigid Origami With Quaternions and Dual Quaternions,” Proc. R. Soc. A, 466(2119), pp. 2155–2174. [CrossRef]
Guest, S. D. , and Pellegrino, S. , 1994, “ The Folding of Triangulated Cylinders—Part I: Geometric Considerations,” ASME J. Appl. Mech., 61(4), pp. 773–777. [CrossRef]
Miura, K. , 1969, “ Proposition of Pseudo-Cylindrical Concave Polyhedral Shells,” ISAS Rep., 34(9), pp. 141–163.
Tachi, T. , 2009, “ Generalization of Rigid Foldable Quadrilateral Mesh Origami,” Symposium of the International Association For Shell And Spatial Structures, pp. 2287–2294.
Wang, K. , and Chen, Y. , 2011, “ Folding a Patterned Cylinder by Rigid Origami,” Origami5, P. Wang-Iverson, R. J. Lang, and M. Yim, eds., CRC Press, Boca Raton, FL.
Tachi, T. , 2009, “ One-DOF Cylindrical Deployable Structures With Rigid Quadrilateral Panels,” IASS Symposium, pp. 2295–2305.
Miura, K. , and Tachi, T. , 2011, “ Synthesis of Rigid-Foldable Cylindrical Polyhedra,” pp. 1–10.
Massey, W. S. , 1962, “ Surfaces of Gaussian Curvature Zero in Euclidean 3-Space,” Tohoku Math. J., Second Ser., 14(1), pp. 73–79. [CrossRef]


Grahic Jump Location
Fig. 3

Two origami cylinders with regular (left) and irregular (right) fold patterns (white lines) and illustration of strip construction (emphasized lines)

Grahic Jump Location
Fig. 1

Compressing an origami cylinder made from ordinary paper suggests that it can rigidly collapse. But is this indeed so?

Grahic Jump Location
Fig. 2

An origami cylinder to which the bellows theorem does not apply. Its apparent collapsibility is disproven by our result.

Grahic Jump Location
Fig. 4

Top row: a fold pattern is split into strips, with the bottom one in detail. Bottom row: The lowermost strip is cut open along the emphasized fold. For different heights, the embedded strip may be closed (left) or open (right). For open strips, we measure the gap (the length of the dashed line) of either boundary.

Grahic Jump Location
Fig. 6

To obtain Pi+1, draw rays which enclose an angle of φi with the line A¯Pi

Grahic Jump Location
Fig. 5

A fan with four vertices is embedded (right image)

Grahic Jump Location
Fig. 7

For n = 3, there are four different possibilities (P2 and P3 can be placed at two locations each). Since the construction is symmetric about the line P1A¯, there exist only two “properly distinct” possible embeddings.

Grahic Jump Location
Fig. 8

The lines P1P2 and A0¯A¯ enclose the same angle with P1A¯. Hence, they are either parallel or intersect at an angle of ±2φi.

Grahic Jump Location
Fig. 9

For one of the embeddings for n = 3 depicted in Fig. 7, there are four different ways to embed the points A0¯ and A2¯, resulting in four different possible values ofα

Grahic Jump Location
Fig. 10

Plot of normalized gap magnitude as a cylindrical Miura pattern is compressed. We prove this function vanishes at only finitely many values of λ, and so the pattern is not rigid-foldable.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In