Desargues' theorem

Together with the set of Pappus 's theorem Desargues, named after the French mathematician Gérard Desargues, is one of the lock sets, which are fundamental for the affine and projective geometry as axioms. It is formulated depending on the underlying geometry of an affine or a projective variety. In both forms, the desarguesche be inferred from the papposschen sentence. Since there are both affine and projective planes in which the set of Desargues, but not the set of Pappus is generally valid, it represents a real weakening of the set of Pappus

Projective shape: If the lines connecting corresponding vertices of two situated in a plane triangles in a point cut ( the "Center " ), the intersections of the corresponding extended sides lie on a line (the " axis "). The reverse is also true.

The figure shows two yellow triangles and. The straight lines through corresponding vertices and intersect at one point. Thus, the hypothesis of the theorem of Desargues is given. As a result, it follows that the line of corresponding sides of the triangle points of intersection, and (intersection of and ) are lying on a straight line which is also referred to as an axis.

If, in one configuration, the center on the axis, so one speaks also of small set of Desargues.

Affine shape: When cutting the connecting lines between corresponding vertices of two triangles lying in a plane at one point and two pairs of corresponding sides of the triangles are parallel, so also the third pair of corresponding sides are parallel.

The affine shape of the small set of Desargues obtained if, instead of the common intersection of the parallel line of the carrier, is provided.

Importance for the Synthetic geometry

In the classification of projective planes by Hanfried Lenz and Adriano Barlotti in synthetic geometry of projective planes are formally classified by group theory. Each class can be also characterized by a specialization of the equivalent set of Desargues and a negation of another specialization. The terms listed below denote groups of Lenz - Barlotti classes that can be characterized by the fulfillment of the set of Desargues or one of its specializations:

• A affine or projective plane is called desarguesian plane if the statement of the ( respective "big" ) set of Desargues is generally valid. There are affine and projective planes that are not desarguessch, about Moulton levels. They have been studied extensively, see the books by Pickert and Hughes - Piper.
• A affine or projective plane in terms of the incidence geometry is exactly then an affine or projective plane in the sense of linear algebra if it is desarguessch. You can thus describe exactly under these conditions using a two - or three-dimensional left vector space over a division ring. - Most are limited in linear algebra to the study of more specific papposschen levels, which can be described by two or three-dimensional vector spaces over a field.
• General serve Ternärkörper ( a generalization of the skew field ) to a description of the particular nichtdesargueschen levels by a range of coordinates. Here must be satisfied in general, no specialization of the set of Desargues.
• As an affine translation planes affine planes are referred to, in which applies the small affine set of Desargues. They may be using the group of their parallel shifts, a generalization of the vector space concept used for the affine planes of linear algebra studied. Your projective extensions are called projective translation planes.
• A projective plane in which the small set of projective Desargues is universal, ie Moufangebene in honor of Ruth Moufang. By cutting a projective line ( " slots " ) arises from a Moufang - always an affine translation plane. This can also be in these projective planes similar structures in affine translation planes establish. By projective extension of a translational level, however, arises not necessarily a Moufangebene! In particular, there exist infinitely many non-isomorphic, nichtdesarguessche, finite translation planes (see for example Quasi Quasi body # body finite Moulton planes), but every finite Moufangebene is a plane over a ( commutative ) body and therefore a fortiori desarguessch. Therefore, the projective completion of a nichtdesarguesschen finite translation plane can never be a Moufangebene.

In at least three-dimensional affine and projective spaces, the set of Desargues always applies and is relatively easy to prove. This is one of the reasons from which investigated intensively usually levels in synthetic geometry. → See also Axiom of Veblen - Young.

Finite levels

The order of a finite affine plane is the number of points on a (and hence any ) of their line. What orders can occur in finite affine planes is a largely unsolved problem. In finite Desargues planes ( where the set of Desargues applies ) the order is necessary is a prime power, because the coordinates of a finite (and hence commutative ) can introduce in them body, and in them will automatically apply the theorem of Pappus. For every prime power q, there exists a desarguesian plane of order q. All known finite affine planes have prime power order. The smallest order for which there exists a nichtdesarguessche level is 9, see the examples in Ternärkörper. Whether there are affine planes of non- prime -power order, is an unsolved problem.

The order n is not possible for n = 6, 14, 21, 22, 30, 33, 42, 46, ...

How do these numbers come from? The set of Bruck- Ryser- Chowla says the following: let n when divided by 4 remainder 1 or 2, is not the sum of two squares and is not a prime power ( as in the above examples ). Then there is no affine plane of order n

The non-existence of an affine plane of order 10 was demonstrated with extensive computer use. For all not listed above, orders n, starting with 12, 15, 18, 20, 24, ...., the existence question is unresolved.