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The Huzita–Hatori axioms or Huzita–Justin axioms are a set of rules related to the mathematical principles of paper folding, describing the operations that can be made when folding a piece of paper. The axioms assume that the operations are completed on a plane (i.e. a perfect piece of paper), and that all folds are linear. These are not a minimal set of axioms but rather the complete set of possible single folds. The axioms were first discovered by Jacques Justin in 1989.〔Justin, Jacques, "Resolution par le pliage de l'equation du troisieme degre et applications geometriques", reprinted in ''Proceedings of the First International Meeting of Origami Science and Technology'', H. Huzita ed. (1989), pp. 251–261.〕 Axioms 1 through 6 were rediscovered by Italian-Japanese mathematician Humiaki Huzita and reported at ''the First International Conference on Origami in Education and Therapy'' in 1991. Axioms 1 though 5 were rediscovered by Auckly and Cleveland in 1995. Axiom 7 was rediscovered by Koshiro Hatori in 2001; Robert J. Lang also found axiom 7. ==The seven axioms== The first 6 axioms are known as Huzita's axioms. Axiom 7 was discovered by Koshiro Hatori. Jacques Justin and Robert J. Lang also found axiom 7. The axioms are as follows: # Given two points ''p''1 and ''p''2, there is a unique fold that passes through both of them. # Given two points ''p''1 and ''p''2, there is a unique fold that places ''p''1 onto ''p''2. # Given two lines ''l''1 and ''l''2, there is a fold that places ''l''1 onto ''l''2. # Given a point ''p''1 and a line ''l''1, there is a unique fold perpendicular to ''l''1 that passes through point ''p''1. # Given two points ''p''1 and ''p''2 and a line ''l''1, there is a fold that places ''p''1 onto ''l''1 and passes through ''p''2. # Given two points ''p''1 and ''p''2 and two lines ''l''1 and ''l''2, there is a fold that places ''p''1 onto ''l''1 and ''p''2 onto ''l''2. # Given one point ''p'' and two lines ''l''1 and ''l''2, there is a fold that places ''p'' onto ''l''1 and is perpendicular to ''l''2. Axiom 5 may have 0, 1, or 2 solutions, while Axiom 6 may have 0, 1, 2, or 3 solutions. In this way, the resulting geometries of origami are stronger than the geometries of compass and straightedge, where the maximum number of solutions an axiom has is 2. Thus compass and straightedge geometry solves second-degree equations, while origami geometry, or origametry, can solve third-degree equations, and solve problems such as angle trisection and doubling of the cube. The construction of the fold guaranteed by Axiom 6 requires "sliding" the paper, or neusis, which is not allowed in classical compass and straightedge constructions. Use of neusis together with a compass and straightedge does allow trisection of an arbitrary angle. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Huzita–Hatori axioms」の詳細全文を読む スポンサード リンク
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