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.
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February 2017
Research-Article
On the Incompressibility of Cylindrical Origami Patterns
Friedrich Bös,
Friedrich Bös
Institute for Numerics and Applied Mathematics,
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
e-mail: f.boes@math.uni-goettingen.de
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
e-mail: f.boes@math.uni-goettingen.de
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Max Wardetzky,
Max Wardetzky
Institute for Numerics and Applied Mathematics,
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
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Etienne Vouga,
Etienne Vouga
Department of Computer Science,
The University of Texas at Austin,
Austin, TX 78712
The University of Texas at Austin,
Austin, TX 78712
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Omer Gottesman
Omer Gottesman
School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
Harvard University,
Cambridge, MA 02138
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Friedrich Bös
Institute for Numerics and Applied Mathematics,
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
e-mail: f.boes@math.uni-goettingen.de
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
Georg-August-Universität Göttingen,
37081 Göttingen, Germany
Etienne Vouga
Department of Computer Science,
The University of Texas at Austin,
Austin, TX 78712
The University of Texas at Austin,
Austin, TX 78712
Omer Gottesman
School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
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. Feb 2017, 139(2): 021404 (9 pages)
Published Online: December 22, 2016
Article history
Received:
April 5, 2016
Revised:
September 19, 2016
Citation
Bös, F., Wardetzky, M., Vouga, E., and Gottesman, O. (December 22, 2016). "On the Incompressibility of Cylindrical Origami Patterns." ASME. J. Mech. Des. February 2017; 139(2): 021404. https://doi.org/10.1115/1.4034970
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