Pappus's hexagon theorem

The set of Pappus ( pappus ), sometimes also called set of Pappus -Pascal is a central set in the affine and projective geometry. He first appears as Proposition 139 in Book VII of Mathematical collections of ancient Greek mathematician Pappus of Alexandria. Blaise Pascal in the 17th century was a generalization of the theorem named after him Pascal's theorem, in which the six basic points of the set lie on a conic.

The sentence reads in its more general projective shape:

6 are points of a projective plane alternately on two straight lines g, h, are the points

Collinear, that is, they lie on a straight u (see picture).

If the two lines g, h by the six vertices and the line u kopunktal, one also speaks of the small set of Pappus.

Since two straight lines in an affine plane does not necessarily intersect, the set is additionally formulated in a more specific affine form:

Lying six points of an affine plane alternately on two lines g, h, and both the

And so are parallel (see picture).

In the projective completion of the underlying affine plane, the three parallel pairs of straight lines intersect on the straight line at infinity u, and there is the projective form of the set of Pappus.

Meaning: Pappossche levels

The set of Pappus does not apply in any projective plane. It applies only in those levels that can be koordinatisieren with the help of a ( commutative ) body. Conversely, it follows from the validity of the principle of Pappus the Koordinatisierbarkeit the plane with a coordinate body. Such planes, affine or projective are thus characterized by the set of Pappus and hot pappossche levels.

For an overview of affine and projective planes in which the set of Pappus or weaker lock sets to apply generally, and the inferences which thus result in each case for the algebraic structure of the coordinate range, see the article " Ternärkörper " and " classification of projective planes ".

The set of projective Pappus as an axiom and equivalent statements

As explained earlier meaning in the section, the projective set of Pappus is independent of the incidence axioms of a projective plane, so he is or to him ( based on the incidence axioms ) abbreviated equivalent statements as an axiom here as (PA). This axiom is also independent of the Fano axiom here shortly (FA), because there exist

• Projective planes over any commutative field with characteristic different from 2. They meet (FA ) and ( PA)
• Projective planes over any commutative field with characteristic 2 meet (FA) never, but always (PA)
• Projective planes that are not pappossch nor (FA) meet, as there are non-commutative skew field with characteristic p for each prime p, so even those with the characteristic 2,
• Projective planes that are not pappossch, but (FA) meet, as ever there is to be any odd prime characteristic p and characteristic 0 at least one non-commutative division ring.

→ Compare also the set of Gleason and the set of Hanna Neumann in Fano Axiom # AntiFano.

The following synthetic and analytic statements about a projective plane are equivalent:

Fundamental theorem of projective geometry

The following statement was referred to in the geometry of the location of the 19th century as the fundamental theorem of projective geometry:

In this formulation of the sentence is to be observed:

A modern formulation that takes into account these three conditions to be observed, is:

The following variants of the main theorem are mutually equivalent statements about a projective plane and equivalent to the set of Pappus (PA ):

• Axiom P7 '
• Axiom P7: Let a straight line in a projective plane. There are two triples from three different points. Then there exists at most one projective mapping, with respect to the true.
• Axiom P7 '': There are different lines in a projective plane, their intersection. Then each projective mapping, the fixed (that is, with ) a perspective mapping.

For any projective plane, the three mentioned P7- looking statements are equivalent to the dualized statements. As an example, the dual form of P7:

Related to the set of Desargues Theorem of Hessenberg

As a set of Hessenberg is in projective geometry, the statement

Referred to. This theorem was proved by Gerhard Hessenberg, after which it is named, in 1905 ( incomplete ). It is of fundamental importance for the synthetic geometry. A complete proof ( through various lemmas ) can be found in the textbook of Lüneburg.

This means: the axiom of Pappus (PA ) the axiom of Desargues follows. The fact that the reversal in general: is false (more precisely for infinite projective planes ), is proved by the existence of projective planes of non- comma tive skew fields.

With the set of Wedderburn follows that for finite projective or affine are levels of the set of Pappus and Desargues, the set of equivalent.