Rigid origami is a branch of origami which is concerned with folding structures using flat rigid sheets joined by hinges. It is a part of the study of the mathematics of paper folding, it can be considered as a type of mechanical linkage, and has great practical utility. There is no requirement that the structure start as a flat sheet — for instance shopping bags with flat bottoms and airbags are studied as part of rigid origami.
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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. However, Kawasaki's theorem and Maekawa's theorem are still applicable.
The Bellows theorem says that a flexible polyhedron has constant volume when flexed.
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.
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 airbags and to folding a space telescope.
Folding paper shopping bags is a problem where the rigidity requirement means the classic solution does not work.
Recreational uses
Martin Gardner has popularised flexagons 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.