The trisection of the angle is a traditional problem of Mathématiques. It is a geometrical problem , belonging to the Three major problems of Antiquity, with the Quadrature of the circle and the Duplication of the cube. This problem consists in dividing an angle into three equal parts, using a rule and of a compass.
If it is easy to share an angle into two by building its Bissectrice, if it is easy to share the right angle into three using equilateral triangles, much of mathematicians sought a long time, in vain, a geometrical method to carry out the trisection of an unspecified angle. As of the third century BC, Archimedes proposed a method by adjustment. At the second century BC, Nicomède conceived the conchoïde of right-hand side to approach the solution. But in 1837, Pierre-Laurent Wantzel showed a theorem which allowed exhiber the form of the equations of the problems impossible to solve with the rule and the compass. The equation of the trisection of the angle being of this form, construction is thus impossible to realize according to these rules.
The trisection of the angle is on the other hand realizable by folding a sheet of paper, by a construction due to Hisashi Abe (1980), which the figure illustrates opposite:
- One plots the straight line D passing by the corner has sheet so that it forms, with the lower edge H 0 of the sheet, the angle to be crossed into three.
- Two horizontal bands of the same width (arbitrary) are traced in bottom of the sheet (this can be done easily by folding.) One calls H 1 and H 2 the new lines which delimit them.
- It is now necessary to fold the sheet along a fold p at the same time so that the corner has is moved on the line H 1 (in a A' point), as the point B (intersection of the left edge with the line H 2) is moved on the line D in a B' point.
- the line T passing by has and A' is then the trisecting one of the angle given: the angle formed by H 0 and T is worth 1/3 of the angle formed by H 0 and D .
There exists a method equivalent to this folding using a square.
The relation between the geometrical construction and the algebraic theory is developed in the constructible article Nombre. The algebraic demonstrations are in the article quadratic Tour of extension.
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