Rigid origami

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.{{cite journal

The Bellows theorem says that a flexible polyhedron has constant volume when flexed rigidly.{{cite journal

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.{{cite journal |url-status = dead

Blooming 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 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.{{cite journal | doi-access = free

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.{{cite journal

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 rigid origami to the problem of folding a space telescope.

Although paper shopping bags 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 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.

References

References

1. Demaine, E. D.. (2001). "Folding and Unfolding".
2. "The Eyeglass Space Telescope".
3. Devin. J. Balkcom, [[Erik Demaine. (November 2004). "Folding Paper Shopping Bags". Cambridge, Massachusetts.
4. Weisstein, Eric W.. "Flexatube". Wolfram MathWorld.