Rigidorigami 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 standardorigami 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 has constant volume when flexed rigidly. The napkin folding problemasks 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.
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 this result does not rely on the impossibility of self-intersections of the folded paper.
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. Folding paper shopping bags is a problem where the rigidity requirement means the classic solution does not work.
Recreational uses
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.
