<P> The art 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 folds to solve mathematical equations . </P> <P> In 1893, Indian mathematician T. Sundara Rao published "Geometric Exercises in Paper Folding" which used paper folding to demonstrate proofs of geometrical constructions . This work was inspired by the use of origami in the kindergarten system . This book had an approximate trisection of angles and implied construction of a cube root was impossible . In 1936 Margharita P. Beloch showed that use of the' Beloch fold', later used in the sixth of the Huzita--Hatori axioms, allowed the general cubic equation to be solved using origami . In 1949, RC Yeates' book "Geometric Methods" described three allowed constructions corresponding to the first, second, and fifth of the Huzita--Hatori axioms . The axioms were discovered by Jacques Justin in 1989 . but were overlooked until the first six were rediscovered by Humiaki Huzita in 1991 . The first International Meeting of Origami Science and Technology (now known as the International Conference on Origami in Science, Math, and Education) was held in 1989 in Ferrara, Italy . </P> <P> The construction of origami models is sometimes shown as crease patterns . The major question about such crease patterns is whether a given crease pattern can be folded to a flat model, and if so, how to fold them; this is an NP - complete problem . Related problems when the creases are orthogonal are called map folding problems . There are three mathematical rules for producing flat - foldable origami crease patterns: </P> <Ol> <Li> Maekawa's theorem: at any vertex the number of valley and mountain folds always differ by two . <Dl> <Dd> It follows from this that every vertex has an even number of creases, and therefore also the regions between the creases can be colored with two colors . </Dd> </Dl> </Li> <Li> Kawasaki's theorem: at any vertex, the sum of all the odd angles adds up to 180 degrees, as do the even . </Li> <Li> A sheet can never penetrate a fold . </Li> </Ol>

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