Compass-and-straightedge construction

In Euclidean geometry, a mathematical branch of geometry, is meant by a construction with ruler and compass the development of accurate drawings of a figure based on predetermined sizes - usually is the restriction on the use of the " Euclidean tools " Circle and ruler required. The latter has no markings; you can so so only draw straight lines, but no measure distances. In antiquity it was used initially only collapsing circle while later the non- collapsing circle was permitted for constructions (not least because the amounts of all constructible points from non- kollabierendem compass and ruler and kollabierendem compass and ruler are identical).

Solutions that rely on other tools, were by the Greeks of the classical period (and later by most geometry -rubbing until the 20th century ) regarded as unsatisfactory.

Euclidean tools

The restriction to the " Euclidean tools " was derived from the postulates that Euclid had put together the elements at the beginning of his textbook. This results in the only approved applications of these tools:

• Drawing a straight line of unlimited length by two arbitrarily given, distinct points,
• Drawing a circle having a given desired point as center and passing through a given any other point, and
• Transferring or knocking off a segment on a line or a circle.

An example would be constructing a triangle of three standards, such as two sides of an angle.

History

The construction only means of compass and ( unskaliertem ) ruler was many centuries as the crown of mathematical logic. But was long considered largely exhausted. The discovery of a method of construction for the regular seventeen- March 29, 1796 Carl Friedrich Gauss was the first major innovation for two thousand years. With the help of the developed in the 19th century Galois theory about zeros of polynomials statements about constructible polygons and the trisection of any angle could be made.

Many mathematicians have turned to for years - as we now know unsolvable - tries tasks like squaring the circle. Over the last more than 100 years, the Euclidean restriction, however, was more and more seen as an unnecessary limitation of the possibilities. Some critics saw it as even a so-called mental block. Therefore, the range of tools has been expanded. A general division of the angle may take place by means of a mask, the edge of which forms an Archimedean spiral. In the second half of the 19th century a device came with the " Tomahawk " on the general trisection of an angle.

After the set of Mohr- Mascheroni ( after Georg Mohr and Lorenzo Mascheroni ) can design tasks with ruler and compass with a compass to run alone and by the theorem of Poncelet -Steiner ( after Jean -Victor Poncelet, Jakob Steiner) with the ruler and a given circle.

Algebraic operations

With ruler and compass, you can also the following algebraic operations (ie, the result of which in the representation on the number line ) construct:

• The addition of two real numbers,
• The multiplication of two real numbers,
• The inverse of a non-zero real number,
• The square root of a positive real number (in order can be constructed at a given rectangular area of ​​a square the same ).

A geometric structure which has been especially developed to illustrate the possibilities of algebraic constructions with ruler and compass, are the Euclidean planes ( in the sense of synthetic geometry) on Euclidean bodies.

Impossible constructions

Many geometric figures can not be constructed exactly alone with ruler and compass. Among them are the classic problems of ancient mathematics:

• The trisection of an angle,
• The duplication of the cube,
• Squaring the circle

As well as

• The conic sections ( with the exception of the circle) and
• Many regular polygons.

The proof that these problems are not always solved with ruler and compass, but it was not until the 19th century. Nevertheless caused the attempts to do the impossible, a number of services. The Greeks found some solutions of the "classical" problems with other auxiliaries, they discovered many results of the higher geometry.

Approximate construction

For some characters that can not be constructed with compass and straightedge or for which the design is too complex, there are ways to construct at least approximately. This approximation to the true object constructions come close. Known approximation constructions, for example, the approximate structure for the circle constant pi, the approximate structure for the quadrature of the circle, the approximate construction of the regular heptagon and the approximate construction for the regular nonagon.

Application

The basic geometric constructions play an important role, especially in the descriptive geometry and technical drawing. Your mediation begins with school mathematics and finds in the training of professional technical illustrator diverse applications.