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.

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Fig. 1

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

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Fig. 2

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

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Fig. 3

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

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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.

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Fig. 5

A fan with four vertices is embedded (right image)

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Fig. 6

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

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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.

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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.

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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

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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.




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