Folding

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Kawasaki's theorem

A theorem giving a criterion for an origami construction to be flat. Kawasaki's theorem states that a given crease pattern can be folded to a flat origami iff all the sequences of angles , ..., surrounding each (interior) vertex fulfil the following conditionNote that the number of angles is always even; each of them corresponds to a layer of the folded sheet.The rule evidently applies to the case of a rectangular sheet of paper folded twice, where the crease pattern is formed by the bisectors. But there are many more interesting examples where the above property can be checked (see, for example, the crane origami in the above figure).

Stamp folding

The number of ways of folding a strip of stamps has several possible variants. Considering only positions of the hinges for unlabeled stamps without regard to orientation of the stamps, the number of foldings is denoted . If the stamps are labeled and orientation is taken into account, the number of foldings is denoted . Finally, the number of symmetric foldings is denoted . The following table summarizes these values for the first .SloaneA001010A001011A0001361111221232264451656145068381447181204628203531392956114845361048352714060

Map folding

A general formula giving the number of distinct ways of folding an rectangular map is not known. A distinct folding is defined as a permutation of numbered cells reading from the top down. Lunnon (1971) gives values up to .OEIS, , ...1A0001361, 2, 6, 16, 50, 144, 462, 1392, ...2A0014152, 8, 60, 320, 1980, 10512, ...The number of ways to fold an sheet of maps is given for , 2, ..., etc. by 1, 8, 1368, 300608, 186086600, ... (Lunnon 1971; OEIS A001418).The limiting ratio of the number of strips to the number of strips is given by

Flexatube

A flexagon-like structure created by connecting the ends of a strip of four squares after folding along diagonals. Using a number of folding movements, it is possible to flip the flexatube inside out so that the faces originally facing inward face outward. Gardner (1961) illustrated one possible solution, and Steinhaus (1999) gives a second.

Unfolding

An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).Examples..

Folding

There are many mathematical and recreational problems related to folding. Origami,the Japanese art of paper folding, is one well-known example.It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular..

Origami

Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in the Wolfram Language by L. Zamiatina.To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming..

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