Desargues theorem

The Desargues theorem , named after the French mathematician Gérard Desargues , is, together with the Pappos theorem, one of the closure theorems that are fundamental to affine and projective geometry as axioms . It is formulated in an affine or a projective variant depending on the underlying geometry. In both forms, the desarguessche can be deduced from the Pappos sentence. Since there are both affine and projective levels in which Desargues' theorem, but not Pappos' theorem, is general, it represents a real weakening of Pappos' theorem.

Example of Desargues' theorem (projective form)

Projective shape: If the connecting lines between the corresponding corner points of two triangles located in one plane intersect at one point (the “ center ”), the intersection points of the correspondingly extended sides lie on a straight line (the “axis”). The reverse is also true.

The figure opposite shows two yellow triangles and . The straight lines through corresponding corner points , and intersect at a point . With this the assumption of Desargues's theorem is given. The result is that the straight lines of corresponding sides of the triangle have intersections , and (intersection of and ), which lie on a straight line , which is also referred to as the axis. ${\displaystyle \triangle ABC}$ ${\displaystyle \triangle A'B'C'}$ ${\displaystyle {\overline {AA'}}}$ ${\displaystyle {\overline {BB'}}}$ ${\displaystyle {\overline {CC'}}}$ ${\displaystyle P}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle {\overline {AB}}}$ ${\displaystyle {\overline {A'B'}}}$ ${\displaystyle a}$

If the center of a configuration is on the axis , one also speaks of the Desargues small theorem . ${\displaystyle P}$ ${\displaystyle a}$

Affine shape: If the connecting lines between corresponding corner points of two triangles located in one plane intersect at one point and two pairs of corresponding sides of the triangles are parallel , the third pair of corresponding sides is also parallel.

The affine shape of the small set of Desargues results if, instead of the common intersection of the parallelism of the support lines , , is assumed. ${\displaystyle P}$ ${\displaystyle {\overline {AA'}}}$ ${\displaystyle {\overline {BB'}}}$ ${\displaystyle {\overline {CC'}}}$

Significance for synthetic geometry

→ Main article : Classification of projective planes

In the classification of projective planes according to Hanfried Lenz and Adriano Barlotti in synthetic geometry , projective planes are formally classified according to group theory . However, each class can also be characterized equally by a specialization of Desargues' theorem and a negation of another specialization. The following terms denote groups of Lenz-Barlotti classes that can be identified by fulfilling Desargues' theorem or one of its specializations:

• An affine or projective level is referred to as a Desargue's level if the statement of the (respective "large") theorem of Desargues is generally valid. There are also affine and projective planes that are not desarguic, such as the Moulton planes . They have been studied extensively, see the books by Pickert and Hughes-Piper.
• An affine or projective plane in the sense of incidence geometry is an affine or projective plane in the sense of linear algebra if and only if it is desargue. It can therefore be described precisely under this condition with the aid of a two- or three-dimensional left vector space over a sloping body . In linear algebra, one usually limits oneself to the investigation of the more specific Pappos planes , which can be described by two- or three-dimensional vector spaces over a body .
• In general, ternary bodies (a generalization of the oblique bodies) are used to describe, in particular, the non-Desargue planes by means of a coordinate range. In general, no specialization of Desargues' theorem has to be fulfilled here.
• Affine planes are referred to as affine translation planes, in which the small affine theorem of Desargues applies. They can be studied using the group of their parallel displacements , a generalization of the vector space concept used for the affine planes of linear algebra. Their projective extensions are called projective translation planes .
• A projective plane in which Desargues' little projective theorem is universal is called the Moufang plane in honor of Ruth Moufang . By cutting out a projective straight line (“slitting”), a Moufang always becomes an affine translation plane. In this way, structures similar to those in affine translation planes can also be established in these projective planes. However, projective expansion of a translation level does not necessarily result in a Moufang level! In particular, there are an infinite number of non-isomorphic, non-desarguean, finite translation planes (see for example quasibodies #quasibodies of finite Moulton planes ), but every finite Moufang plane is a plane above a (commutative) field and therefore even more desarguean. Hence the projective closure of a non-Desarguean finite translation plane can never be a Moufang plane.

In at least three-dimensional affine and projective spaces, Desargues' theorem always holds and is relatively easy to prove. This is one of the reasons why planes are usually studied particularly intensively in synthetic geometry . See also Veblen-Young's axiom .

Finite levels

The order of a finite affine plane is the number of points on one (and therefore each) of its line. Which orders can occur at finite affine planes is a largely unsolved problem. In finite Desargues planes (in which Desargues 'theorem applies) the order is necessarily a prime power , because coordinates from a finite (and therefore commutative) field can be introduced into them, and Pappos' theorem automatically applies in them. For every prime power q there is a Desarguean level of order q. All finite affine planes known so far have prime power orders. The smallest order for which a non-Desarguean plane exists is 9, see the examples in Ternary bodies . Whether there are affine levels 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 about? The Bruck-Ryser-Chowla theorem says the following: n leave the remainder 1 or 2 when dividing by 4, do not be the sum of two squares and do not be a prime power (as in the examples above). Then there is no affine plane of order n.

The non-existence of an affine plane of order 10 has been proven with extensive computer use. For all orders n not mentioned here, starting with 12, 15, 18, 20, 24, ..., the question of existence is unsolved.

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry. From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X ( table of contents ).
• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . Teubner, Stuttgart 1977, ISBN 3-519-02751-8 .
• Daniel Hughes, Fred Piper: Projective planes (=  Graduate texts in mathematics . Volume 6 ). Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-90044-6 .
• Helmut Karzel, Kay Sörensen, Dirk Windelberg: Introduction to Geometry . Vandenhoeck and Ruprecht, Göttingen 1973, ISBN 3-525-03406-7 .
• Rolf Lingenberg : Fundamentals of Geometry I . 3rd, revised edition. Bibliographical Institute, Mannheim / Vienna / Zurich 1978, ISBN 3-411-01549-7 .
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 .

Individual evidence

1. ^ Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 .
2. ^ Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry. From the basics to the applications (=  Vieweg Studium: advanced course in mathematics ). 2nd, revised and expanded edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , 2.7 Spatial geometries are desarguic .