Fano axiom

The Fano axiom is an incidence axiom in synthetic geometry for both affine planes and projective planes . It is named after the Italian mathematician Gino Fano . In affine or projective planes over a division ring or body the Fano axiom is true if and only if the characteristic of not second The Fano plane , also named after Fano , the minimal model of a projective plane, does not fulfill the Fano axiom . ${\displaystyle K}$ ${\displaystyle K}$

Affine Fano axiom

Affine Fano axiom: In the parallelogram , the diagonals and intersect at a point . The axiom makes it possible to assign midpoints to a segment. ${\displaystyle P_{1}P_{2}P_{3}P_{4}}$ ${\displaystyle P_{1}P_{3}}$ ${\displaystyle P_{2}P_{4}}$ ${\displaystyle D}$ ${\displaystyle (P_{1},P_{2})}$ ${\displaystyle M}$

An affine plane fulfills the Fano axiom if it applies there (see the figure on the right): ${\displaystyle {\mathcal {A}}}$

• "With every non-degenerate parallelogram, the diagonals intersect." Or equivalent:
• "In no non-degenerate parallelogram are the diagonals parallel."

In more detail and more formally, the axiom reads as follows: If there are points of the affine plane , none of which are on a straight line, then the following applies: Out and follows . ${\displaystyle P_{1},P_{2},P_{3},P_{4}}$ ${\displaystyle A}$ ${\displaystyle P_{1}P_{2}\parallel P_{3}P_{4}}$ ${\displaystyle P_{2}P_{3}\parallel P_{4}P_{1}}$ ${\displaystyle P_{1}P_{3}\not \parallel P_{2}P_{4}}$

For an affine translation plane , the following statements are both equivalent to the Fano axiom:

• No translation has order 2, that is, for every translation it follows that is. ${\displaystyle \tau }$ ${\displaystyle \tau \circ \tau =\operatorname {Id} _{A}}$ ${\displaystyle \tau =\operatorname {Id} _{\mathcal {A}}}$
• The oblique body of the true-track endomorphisms of the translation group has one of two different characteristics . ${\displaystyle S}$

For any affine plane the first of these statements follows from the Fano axiom.

For each affine translation plane the alternative applies:

1. Either the diagonals are parallel in every non-degenerate parallelogram, or
2. the diagonals intersect in every non-degenerate parallelogram.

In the first case every non-identical translation has the order , in the second case all non-identical translations also have the same order, this is either an odd prime number or infinite, then one sets . In all these cases there is also the characteristic of the oblique body described above . ${\displaystyle p=2}$ ${\displaystyle p}$ ${\displaystyle p=0}$ ${\displaystyle p}$ ${\displaystyle S}$

Midpoints of a route

In an affine plane that satisfies the Fano axiom, one can assign midpoints to a segment: ${\displaystyle (P_{1},P_{2})\in {\mathcal {A}}^{2}}$ ${\displaystyle M}$

1. If it is, you bet and name the "midpoint of the route ". ${\displaystyle P_{1}=P_{2}}$ ${\displaystyle M=P_{1}}$ ${\displaystyle M=P_{1}}$ ${\displaystyle (P_{1},P_{2})}$
2. If is, one chooses any point outside the straight line and adds to a non-degenerate parallelogram . The parallel to the diagonal intersection intersects at one point . All points that can be constructed in this way (with changing auxiliary points ) are called "midpoints of the line ". ${\displaystyle P_{1}\neq P_{2}}$ ${\displaystyle P_{3}}$ ${\displaystyle P_{1}P_{2}}$ ${\displaystyle P_{1}P_{2}P_{3}P_{4}}$ ${\displaystyle P_{1}P_{4}}$ ${\displaystyle D}$ ${\displaystyle P_{1}P_{2}}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle P_{3}}$ ${\displaystyle (P_{1},P_{2})}$

Point mirroring

A collineation on an affine Fano plane is called point reflection if there is a point that is a center point for every connecting line . ${\displaystyle \delta }$ ${\displaystyle Z}$ ${\displaystyle (P,\delta (P))}$

• In general, no point reflection on has to exist at any point . ${\displaystyle Z\in {\mathcal {A}}}$ ${\displaystyle Z}$
• In the case of their existence, the point reflection an is clearly determined by. Then the point is the only center point for any point-image point segment. ${\displaystyle \delta }$ ${\displaystyle Z}$ ${\displaystyle Z}$ ${\displaystyle (P,\delta (P))}$ ${\displaystyle Z}$
• Every point reflection is a dilatation and therefore an affinity , because its projective continuation is a plane perspective . The only fixed point of affinity and the center of the projective continuation is the center point of any point-image point segment . ${\displaystyle Z}$ ${\displaystyle (P,\delta (P))}$
• Every point reflection is involutive .
• In an affine translation plane and even more so in a Desarguessian plane, a point reflection exists for every point . It is the centric stretch with the stretching factor . ${\displaystyle Z\in {\mathcal {A}}}$ ${\displaystyle Z}$ ${\displaystyle Z}$ ${\displaystyle -1}$

Fano projective axiom

Two projective forms of the Fano axiom have been formulated which are dual and equivalent to one another. For this, the terms complete quadrangle or complete quadrilateral are required, which are also dual to one another.

Complete square

A complete square. The four “corners” A, B, C, D are marked in red, pairs of opposing sides are each the same color. The intersections of the opposite sides, E, F, G - the "diagonal points" - are gray.

A complete quadrilateral in a projective plane consists of 4 points (the corners of the quadrilateral) in a general position, i.e. no three of them lie on a common straight line. The 6 straight lines connecting the corners are called the "sides" of the square, two sides that do not go through a common corner are called the "opposite sides" of the square.

A complete square is called an "anti-Fano square" if the points of intersection of the opposing sides lie on a straight line, otherwise it is called a "Fano square".

→ A complete square, understood as an ordered set of four points, forms a projective point base .

The projective axiom

Fano's projective axiom is:

"The points of intersection of the opposite sides (diagonal points) in any complete quadrilateral are not collinear ."

The Fano axiom thus requires that every complete quadrilateral in the projective plane is a Fano quadrilateral. Then the projective plane is called a Fano plane. If, on the other hand, every complete square is an anti-Fano square, then the projective plane is sometimes referred to as the anti-Fano plane.

Remarks

Regarding the projective Fano axiom, please note:

• There are projective planes that are neither fano nor anti-fano, see later in this article.
• Each desargue projective level is either a Fano or an anti-Fano level. It is an anti-Fano plane if the characteristic of its coordinate inclined body is 2, and a Fano plane if any other characteristic.
• More generally, even every Moufang plane either Fano or an anti-Fano plane. There the criterion is: If the core of the alternative coordinate body is the plane, then this plane is an anti-Fano plane if the characteristic is this oblique body , and a Fano plane if any other characteristic of . ${\displaystyle S}$ ${\displaystyle S=\operatorname {Kern} (A)=\lbrace x\in A:\forall a,b\in A\quad x(ab)=(xa)b\rbrace }$ ${\displaystyle A}$ ${\displaystyle \operatorname {char} (S)=2}$ ${\displaystyle S}$
• The Fano level is an anti-Fano level in the axiomatic sense !

Relations of the projective to the affine Fano axiom

• By cutting out a projective straight line (“slitting”) or projective extension, a projective desargue Fano plane always becomes an affine desargue plane that fulfills the affine Fano axiom, and vice versa.
• By slitting a Moufang plane in which the projective Fano axiom applies, an affine translation plane always arises in which the affine Fano axiom applies.
• If the projective extension of an affine translation plane that fulfills the Fano axiom is a Moufang plane, then this Moufang plane also fulfills the Fano axiom.
• Slitting a projective Fano plane always creates an affine plane that satisfies the affine Fano axiom.

Complete four-sided

A complete quadrilateral in a projective plane consists of 4 straight lines (the sides of the quadrilateral) in a general position, i.e. no three of them pass through a common point. The 6 intersection points of the sides are called the “corners” of the four-sided, every two corners that are not on one side are called the “opposite corners” of the four-sided.

The dual form of the projective Fano axiom is:

"The straight lines connecting the opposite corners (diagonals) in any complete quadrilateral are not copunctal ."

The following applies: For every Fano level, its dual level is also a Fano level.

This means that for each projective plane the Fano axiom and the dual Fano axiom are equivalent.

Projective planes with Fano and Antifano quadrangles and Desargues' theorem

Finite levels

The following theorem by Gleason says that a finite anti-Fano plane (in American parlance, unfortunately, here by Andrew Gleason , for example , often referred to as a fano plane ...) is always desarguean and therefore one over a finite field : ${\displaystyle \mathbb {P} ^{2}(\mathbb {F} _{q})}$ ${\displaystyle \mathbb {F} _{q},q=2^{r},r\geq 1}$

From the collinearity of the diagonal points of all complete quadrilaterals in a finite projective plane follows the general validity of Desargues theorem in this plane.

Examples of real, finite half-bodies of even order, i.e. quasi-bodies that satisfy both distributive laws but are not alternative fields, were given by Donald Ervin Knuth in his dissertation. For this reference, see the article Half-Body (Geometry) . There, two such half-bodies of order 16 are specifically specified in the Examples section .

The projective levels above all these "Knuthschen" real half-bodies belong to the Lenz-Barlotti class V. According to Gleason's theorem, they can not fulfill the anti-Fano axiom because they are non-desarguic. On the other hand , they contain the Fano plane as a substructure (the prime body with 2 elements is contained in the core of the half body as a partial body) and thus also anti-Fano quadrangles. ${\displaystyle \mathbb {P} ^{2}(\mathbb {F} _{2})}$

Conversely , Günter Pickert suspects : Fano and anti-Fano quadrangles exist in every finite, non- Desarguean plane ! He proves a much weaker sentence by Hanna Neumann :

If p is a prime number, r is a positive integer, which should also apply in the case , then there is a finite projective plane of order in which both a complete quadrilateral with collinear diagonal points and one with non-collinear diagonal points occur. ${\displaystyle r\geq 2}$ ${\displaystyle p=2}$ ${\displaystyle q=p^{2r}}$

The planes constructed by Pickert for proof can be coordinated by real quasi-bodies of order . That means: You and Knuth's planes mentioned above can be slotted in such a way that an affine translation plane of order q arises. In this affine plane either ${\displaystyle q=p^{2r}}$

1. the diagonals of each non-degenerate parallelogram are parallel or
2. the diagonals of each non-degenerate parallelogram intersect.

The first case occurs if and only if is, the second if p is odd. This proof shows at the same time that already for affine translation planes from the validity of the affine “anti-Fano-axiom” or the affine Fano-axiom in general, the validity of the corresponding axiom in the projective closure cannot be inferred. ${\displaystyle p=2}$

Any levels

Pickert replaces the finiteness requirement of Gleason's theorem with a transitivity requirement. See the definitions and language rules that are explained in the article Classification of projective planes . He thus proves: "If there are three non-collinear points in a projective plane , so that the plane - and - is transitive, and if the diagonal points in this plane in every complete quadrilateral are collinear, then the plane is desargue." ${\displaystyle O,U,V}$ ${\displaystyle (OU,V)}$ ${\displaystyle (OV,U)}$

meaning

• The meaning of the Fano axiom for elementary affine geometry is obvious: The Fano axiom must be valid in any planes, and in Desarguese planes it is sufficient for a center point to exist at two different points! Without Fano's axiom there are no bisectors, no perpendicular lines, no point reflections, etc.
• Its usefulness in the investigation of quadratic forms is somewhat less obvious : Here one would like to be able to divide by 2 (for example when adding a square and when "symmetrizing" a form matrix, see projective quadric ).
• If one makes, as is often done in linear algebra, the general prerequisite that the characteristic of the coordinate bodies under consideration is not 2, one also avoids some special cases that only occur in affine geometries with 2 points on each line (see Collineation ) or only in the minimal model of projective geometry (see Fano plane ), but not directly because of the characteristics of the body, but because of the smallness of the models.

literature

• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . Teubner, Stuttgart 1976, ISBN 3-519-02751-8 (simple representation of the axioms, didactic advice for geometry lessons at grammar schools).
• Lothar Wilhelm Julius Heffter : Fundamentals and analytical structure of projective, Euclidean and non-Euclidean geometry . 3rd significantly revised edition. Teubner, Stuttgart 1958 (presentation of the connections between classical (real, Euclidean) geometry and some generalizations in synthetic and absolute geometry).
• Lars Kadison, Matthias T. Kromann: Projective Geometry and Modern Algebra . Birkhäuser, Boston / Basel / Berlin 1996, ISBN 3-7643-3900-4 ( Table of Contents [PDF; accessed on June 6, 2016] Consequences of the Fano axiom for the transitivity properties of projective groups).
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin / Heidelberg / New York 1975, ISBN 3-540-07280-2 , 12.3: Complete rectangles with collinear diagonal points , p. 297–301 (takes into account the then current results, especially on finite levels).
• Hermann Schaal: Linear Algebra and Analytical Geometry . 2nd revised edition. tape II . Vieweg, Braunschweig 1980, ISBN 3-528-13057-1 (meaning of the Fano axiom in linear algebra over bodies and for the classification of conic sections).

References and comments

1. ^ Kadison and Kromann (1996), 5.2: Fano's Axiom P6.
2. ↑ Hauke ​​Klein: Fano axiom (English).
3. ↑ Eric W. Weisstein: Fano's Axiom. From MathWorld - A Wolfram Web Resource .
4. ^ Hermann Schaal: Linear algebra and analytical geometry. Volume II, p. 220.
5. ^ Kadison and Kromann (1996), proposition 5.4.
6. ↑ Andrew M. Gleason: Finite Fano Planes . In: American Journal of Mathematics . tape 78 , no. 4 , October 1956, p. 797-807 .
7. ↑ ^{a b c} Quoted from Günter Pickert: Projective planes. 1975, p. 301.
8. ^ Hanna Neumann: On some finite non-desarguesian planes . In: Archives of Mathematics . tape 6 , no. 1 , September 15, 1954, p. 36-40 , doi : 10.1007 / BF01899210 .
9. ↑ Literally quoted from Günter Pickert: Projective planes. 1975, p. 300, there the theorem is also proven.
10. ↑ The term affine “anti-Fano axiom” is not a common term in the literature . Here the levels with the 1st property are meant.