Angle trisection

Under the trisection of an angle (also: Trisection the angle ) is understood in the geometry of the problem, if one only with the help of ruler and compass ( the Euclidean tools ) can be divided into three equal angle any angle constructively and accurately. The trisection of an angle is one of the three classical problems of ancient mathematics and is not feasible for any angle.

Classic problem

Here you have an arbitrary given angle equal parts to be distributed only with the help of a compass and a straightedge not scaled in three.

At specific angles, the three-division of the angle is also possible, such as any integer multiple of 90 °. Even the ancient Greeks tried in vain to find a general solution for arbitrary angles. Around the year 1830, the French mathematician Evariste Galois laid the foundation, which was later proven that this is not generally possible. For example, it is not possible to make the angle to be constructible thirds of 60 ° as 20 ° can not be constructed.

The first published proof of the impossibility Pierre Wantzel 1837 (regardless of methods of Galois theory ).

A tripartite division is only possible if you use other tools as ruler and compass - about a Trisektrix - or when installing on the ruler markings. On the other hand, one can specify with compass and ruler arbitrarily good approximate solutions.

Generalization

A generalization of the problem is to characterize exactly which angles can be constructed and which are not. Equivalents are questions for which natural numbers can divide a circle into equal pieces using compass and straightedge or which regular polygons are constructible. The exact characterization of constructible corner was achieved in 1837 by Pierre Wantzel ( after substantial preparatory work of Carl Friedrich Gauss and Évariste Galois ) and says that this is exactly then is the case when a product of a power of two and mutually different Fermat primes. The first crucial step on the mathematics of antiquity addition was due to the young Gauss with his discovery that the regular seventeen- is constructible. The known Fermat primes are 3, 5, 17, 257 and 65537th General Fermat numbers are the numbers, which is a power of itself. It is believed that the Fermat numbers with no prime numbers are, and know it for many.

For the problem of angle trisection you do not need the advanced theories of Gauss and Galois. Here the realization that a route length that satisfies an irreducible equation of the third degree, is not constructible enough; because every constructible route length can be algebraically by the sequential solving quadratic equations win, so it is algebraically by a power of two degrees.

Non - classical methods

Do not limit yourself to the classic design requirements for ruler and compass, but also permits the use of other design tools and mathematical auxiliary objects to or is content with approximate solutions, so there is a variety of possible methods to three share an arbitrary angle. In the following sections, some of them are presented as examples.

The method of Archimedes

Archimedes was a pragmatist, he gave a solution in his Liber Assumptorum to. Be the three distributive angle as shown in the adjacent drawing. Then go as follows:

In support, it is noted that because of the special position of the ruler, the length of the path is equal to the distance between the marks, ie equal to the radius of the circle can also be found as and. In particular, the triangle is isosceles, so the angle also occurs. The angle of the triangle in one hand equal to (sum of angles in a triangle ), on the other hand, the addition of angle, so. Since the triangle is isosceles, also, the angle also appears in, and the angle of the triangle is equal to at. You Now notice that adding the angle at to, arises.

Each angle can be divided into three parts as shown that with this method, is not in contradiction to the indissolubility of the classical problem, since the above construction was not carried out according to the rules classically allowed. A mark on the ruler and a dexterous application of the ruler are not allowed construction methods. A divergent set of instruments so it was used and the possible constructions are from the instrument set -dependent.

Sharing with Tomahawk

The Tomahawk is a figure from the mathematical point of view from two perpendicular lines and a voltage applied to one of the straight semi-circle is ( see drawing). The name Tomahawk is because the figure vaguely of a Tomahawk ( Indian battle-ax ) recalls. In order to share a three angle using the Tomahawks, you have to position it so that its " stem " is located on the top angle, while the semi-circle and an end of the other line each touch the side of the angle. In this position, the " stem " with one of the leg forms an angle that is exactly one-third of the output angle. If one connects the center of the semicircle with the tip angle of the output angle, so this route is with the leg another angle, which is exactly one-third of the output angle. Overall, one gets the output angle now divided into three equal angles.

Trisection of an angle with Origami

While the trisection of an angle with the classic instruments of the geometry is not possible, the task can be solved with the paper folding origami. This solution is similar to the Archimedean method, geometrically recreate with markings on the ruler. That this is not necessary with the Origami apparently, is on an automatic limitation of the " ruler " traceable - the fanfold paper is in any case finite.