Construções com régua e compasso

Abstract

The problems of constructions have always occupied a prominent position in geometry. Only with the use of ruler and compass we can carry out a huge diversity of constructions and in all these constructions, the ruler is used only to draw straight lines. Although the greeks use other instruments, the classical restriction to the use of only the ruler and the compass was a matter of great importance to them. Among the problems of constructions with ruler and compass, that of constructing a regular polygon of n sides is probably of greater interest. The constructions of the equilateral triangle, the square, the regular pentagon and the regular hexagon have been known since Antiquity and occupy (or have occupied) position in the study of geometry in schools. However, for some regular polygons this construction (only with the use of ruler and compass) is not possible. As example we can mention the regular heptagon. There are other construction problems that deserve prominent position and for which a construction with ruler and compass is not possible. As examples we can mention the three classic problems of the Greeks: the duplication of the cube (or the construction of the edge of a cube whose volume is the double of that of a cube of a given angle), the trisection of any angle (or the construction of dividing an arbitrary angle in three equal parts) and the quadrature of the circle (or the construction of a square with area equal to the area of a given circle). The importance of studying these problems lies in the fact that they cannot be solved with ruler and compass only, although these instruments are used to solve many other construction problems. The attempt to find a solution to these problems influenced greek geometry, leading to important discoveries. As examples of these discoveries we can mention: the conic sections, some cubic and quartic curves and several transcendent curves. Subsequently a result of great importance was the development of the theory of equations related to domains of rationality, algebraic numbers and group theory. We note that attempting to solve problems like these without solution has resulted in one of the most significant developments in mathematics. The purpose of this dissertation is to show some classic constructions, as the case of the regular polygon of seventeen sides, and to deal with the impossibility of construction by means of ruler and compass, such as: duplication of a cube, trisection of an angle, quadrature of a circle (in this case we will only indicate the proof) and constructions of regular polygons.