Rigid origami

Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike with paper origami, the sheets cannot bend during the folding process; they must remain flat at all times. However, there is no requirement that the structure start as a single flat sheet – for instance shopping bags with flat bottoms are studied as part of rigid origami.

Rigid origami is a part of the study of the mathematics of paper folding, and rigid origami structures can be considered as a type of mechanical linkage. Rigid origami has great practical utility.

Mathematics

The number of standard origami bases that can be folded using rigid origami is restricted by its rules.[1] Rigid origami does not have to follow the Huzita–Hatori axioms, the fold lines can be calculated rather than having to be constructed from existing lines and points. When folding rigid origami flat, Kawasaki's theorem and Maekawa's theorem restrict the folding patterns that are possible, just as they do in conventional origami, but they no longer form an exact characterization: some patterns that can be folded flat in conventional origami cannot be folded flat rigidly.[2]

The Bellows theorem says that a flexible polyhedron has constant volume when flexed rigidly.[3]

The napkin folding problem asks whether it is possible to fold a square so the perimeter of the resulting flat figure is increased. That this can be solved within rigid origami was proved by A.S. Tarasov in 2004.[4]

Complexity theory

Determining whether all creases of a crease pattern can be folded simultaneously as a piece of rigid origami, or whether a subset of the creases can be folded, are both NP-hard. This is true even for determining the existence of a folding motion that keeps the paper arbitrarily close to its flat state, so (unlike for other results in the hardness of folding origami crease patterns) this result does not rely on the impossibility of self-intersections of the folded paper.[5]

Applications

The Miura fold is a rigid fold that has been used to pack large solar panel arrays for space satellites, which have to be folded before deployment.

Robert J. Lang has applied origami to the problem of folding a space telescope.[6]

Folding paper shopping bags is a problem where the rigidity requirement means the classic solution does not work.[7]

Recreational uses

Martin Gardner has popularised flexagons which are a form of rigid origami and the flexatube.[8]

Kaleidocycles are toys, usually made of paper, which give an effect similar to a kaleidoscope when convoluted.

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References
  1. Demaine, E. D (2001). Folding and Unfolding. Doctoral Thesis (PDF). University of Waterloo, Canada.
  2. Abel, Zachary; Cantarella, Jason; Demaine, Erik D.; Eppstein, David; Hull, Thomas C.; Ku, Jason S.; Lang, Robert J.; Tachi, Tomohiro (2016). "Rigid origami vertices: conditions and forcing sets". Journal of Computational Geometry. 7 (1): 171–184. doi:10.20382/jocg.v7i1a9. MR 3491092.
  3. R. Connelly; I. Sabitov; A. Walz (1997). "The bellows conjecture". Beiträge zur Algebra und Geometrie. 38 (1): 1–10.
  4. Tarasov, A. S. (2004). "Solution of Arnold's "folded ruble" problem". Chebyshevskii Sbornik (in Russian). 5 (1): 174–187. Archived from the original on 2007-08-25.
  5. Akitaya, Hugo; Demaine, Erik; Horiyama, Takashi; Hull, Thomas; Ku, Jason; Tachi, Tomohiro (2020), "Rigid foldability is NP-hard", Journal of Computational Geometry, 11 (1)
  6. "The Eyeglass Space Telescope" (PDF).
  7. Devin. J. Balkcom, Erik D. Demaine, Martin L. Demaine (November 2004). "Folding Paper Shopping Bags". Abstracts from the 14th Annual Fall Workshop on Computational Geometry. Cambridge, Massachusetts: 14–15.CS1 maint: multiple names: authors list (link)
  8. Weisstein, Eric W. "Flexatube". Wolfram MathWorld.
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