Rigid origami is a branch of

origami ) is the Japanese art of paper folding. In modern usage, the word "origami" is often used as an inclusive term for all folding practices, regardless of their culture of origin. The goal is to transform a flat square sheet of paper into a f ...

which is concerned with folding structures using flat rigid sheets joined by hinges. That is, unlike in traditional origami, the panels of the paper cannot be bent during the folding process; they must remain flat at all times, and the paper only folded along its hinges. A rigid origami model would still be foldable if it was made from glass sheets with hinges in place of its crease lines. However, there is no requirement that the structure start as a single flat sheet – for instance

shopping bag Shopping bags are medium-sized bags, typically around 10–20 litres (2.5–5 gallons) in volume (though much larger versions exist, especially for non-grocery shopping), that are used by shoppers to carry home their purchases. Some are intended ...

s with flat bottoms are studied as part of rigid origami. Rigid origami is a part of the study of the

mathematics of paper folding The discipline of origami or paper folding has received a considerable amount of mathematical study. Fields of interest include a given paper model's flat-foldability (whether the model can be flattened without damaging it), and the use of paper f ...

, and rigid origami structures can be considered as a type of

mechanical linkage A mechanical linkage is an assembly of systems connected to manage forces and movement. The movement of a body, or link, is studied using geometry so the link is considered to be rigid. The connections between links are modeled as providing i ...

. 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. 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. The Bellows theorem says that a

flexible polyhedron In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

has constant volume when flexed rigidly. The

napkin folding problem The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter. The problem is known under several names, including the Margulis na ...

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.

Blooming Bloom or blooming may refer to: Science and technology Biology * Bloom, one or more flowers on a flowering plant * Algal bloom, a rapid increase or accumulation in the population of algae in an aquatic system * Jellyfish bloom, a collective n ...

is a rigid origami motion of a net of a polyhedron from its flat unfolded state to the folded polyhedron, or vice versa. Although every

convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

has a net with a blooming, it is not known whether there exists a blooming that does not cut across faces of the polyhedron, or whether all nets of convex polyhedra have bloomings.

Complexity theory

Determining whether all creases of a

crease pattern A crease pattern is an origami diagram that consists of all or most of the creases in the final model, rendered into one image. This is useful for diagramming complex and super-complex models, where the model is often not simple enough to diagram e ...

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.

Applications

The

Miura fold The is a method of folding a flat surface such as a sheet of paper into a smaller area. The fold is named for its inventor, Japanese astrophysicist Kōryō Miura. The crease patterns of the Miura fold form a tessellation of the surface by ...

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 rigid origami to the problem of folding a space telescope. Although paper

s are commonly folded flat and then unfolded open, the standard folding pattern for doing so is not rigid; the sides of the bag bend slightly when it is folded and unfolded. The tension in the paper from this bending causes it to snap into its two flat states, the flat-folded and opened bag.

Recreational uses

Martin Gardner Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writings of Lew ...

has popularised

flexagon In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be ''flexed'' or folded in certain ways to reveal faces besides the two that were originally on the back and front. Flexagons are usually square or ...

s which are a form of rigid origami and the flexatube. Kaleidocycles are toys, usually made of paper, which give an effect similar to a kaleidoscope when convoluted.

References

External links

* {{Mathematics of paper folding Paper folding Linkages (mechanical) Origami