Affine level

In synthetic geometry, an affine plane is an incidence structure comprising points and straight lines , which is essentially characterized by two requirements, namely that every two points have a (unique) connecting straight line and that there are unique parallel straight lines. In linear algebra and analytic geometry , a two- dimensional affine space is called an affine plane . The term synthetic geometry described in this article generalizes this more well-known term from linear algebra.

An affine plane that contains only a finite number of points is called a finite affine plane and is also examined as such in finite geometry . The term order of the plane is particularly important for these planes : It is defined as the number of points on one and therefore every straight line of the plane.

Every affine plane can be expanded to a projective plane by introducing improper points and an improper straight line consisting of these . Conversely, an affine plane arises from a projective plane by removing a straight line with its points. → See also projective coordinate system .

Each affine plane can be coordinated by assigning a coordinate range and algebraized by additional links that result from the geometric properties of the plane in this coordinate range. An affine plane in the sense of linear algebra, i.e. an affine space whose vector space of the parallel displacements is a two-dimensional vector space over a body , results exactly when the coordinate area becomes isomorphic to this body due to the geometric structure. This description of the affine plane with the help of a coordinate range, in which the algebraic term body is generalized, and an overview of the structures that result when important closure theorems are valid, can be found in the main article ternary body . ${\displaystyle K}$

On the other hand, one can examine the group of parallel shifts in an affine plane, which leads to a different algebraization in which the term parallel shift, which in linear algebra can be described by a vector , leads to the term translation . This approach, which complements the coordinate-related approach, is described in the main article Affine Translation Plane.

Definitions

An incidence structure , which consists of a point space , a straight line space and an incidence relation between them, is an affine plane if and only if the following axioms hold: ${\displaystyle \langle {\mathcal {P}},{\mathcal {G}},\mathbf {I} \rangle }$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \mathbf {I} }$

1. Two different points of lie on exactly one straight out . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$
2. The parallel postulate applies , that is, for every straight line and for every point that is not on, there is exactly one further straight line that contains and does not contain a point of . ${\displaystyle g\in {\mathcal {G}}}$ ${\displaystyle A\in {\mathcal {P}}}$ ${\displaystyle g}$ ${\displaystyle h\in {\mathcal {G}}}$ ${\displaystyle A}$ ${\displaystyle g}$
3. There are three different points out (a "triangle"), which are not all on a straight line out . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$

The three axioms can be formalized as:

1. ${\displaystyle \forall A,B\in {\mathcal {P}}\left(A\neq B\Rightarrow \exists !g\in {\mathcal {G}}:\;A\mathbf {I} g\land B\mathbf {I} g\right)}$,
2. ${\displaystyle \forall g\in {\mathcal {G}},A\in {\mathcal {P}}\left({\neg }(A\mathbf {I} g)\Rightarrow \exists !h\in {\mathcal {G}}:\;A\mathbf {I} h\land {\neg }\exists B\in {\mathcal {P}}\quad \left(B\mathbf {I} g\land B\mathbf {I} h\right)\right)}$
3. ${\displaystyle \exists A,B,C\in {\mathcal {P}}:\neg \exists g\in {\mathcal {G}}:\left(A\mathbf {I} g\land B\mathbf {I} g\land C\mathbf {I} g\right)}$.

parallelism

The relation ( parallelism ) between straight lines is defined by: ${\displaystyle g\parallel h}$ ${\displaystyle g,h\in {\mathcal {G}}}$

${\displaystyle g\parallel h}$exactly if or if and have no common intersection. ${\displaystyle g=h}$ ${\displaystyle g}$ ${\displaystyle h}$

The straight line which is uniquely determined according to the 2nd axiom and which goes through a certain point is called the parallel to through and is notated. ${\displaystyle h}$ ${\displaystyle A\in {\mathcal {P}}}$ ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle h=\left[A;\parallel g\right]}$

This relation is an equivalence relation . The equivalence class of the straight lines parallel to a straight line is referred to as the family of parallels and also as the direction of . ${\displaystyle g}$ ${\displaystyle g}$

Ways of speaking

• The straight line , uniquely determined according to the 1st axiom , on which two different points lie, is called the straight line connecting the points and is noted as , sometimes also as . ${\displaystyle g}$ ${\displaystyle A,B}$ ${\displaystyle AB}$ ${\displaystyle A\vee B}$
• The family of parallels in a straight line is noted as. ${\displaystyle g}$ ${\displaystyle [g]}$
• The straight line clearly determined by a straight line and any point is referred to as the parallel to through and is noted as being. ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle h\in [g]}$ ${\displaystyle g}$ ${\displaystyle A}$ ${\displaystyle [A;\parallel g]}$

The conventional point of view, in which the set of points and the set of lines were initially understood as independent sets, is also taken as a basis more often in the more recent mathematical literature. In this context, the set of points that lie on a straight line is referred to as the set of points of the straight line and is often noted as. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle g}$ ${\displaystyle g^{\circ }=\lbrace A\in {\mathcal {P}}|A\mathbf {I} g\rbrace }$

However, since a straight line is completely determined by the incidence relation , it is also often identified with this point set, so that the relation is superfluous. The axioms are then described as properties of the line set , which is a subset of the power set of the point set , the role of the incidence relation is then taken over by the element relation: ( if and only if is). ${\displaystyle \mathbf {I} }$ ${\displaystyle \mathbf {I} }$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle A\mathbf {I} g}$ ${\displaystyle A\in g^{\circ }=g}$

Order of the affine plane

The order of an affine plane is defined as the thickness of the set of points on a straight line . The term is independent of the straight line , because all straight lines of an affine plane (as sets of points) are of equal power, since two different straight lines can always be mapped onto one another by a bijective parallel projection. The following applies: ${\displaystyle g}$ ${\displaystyle g}$

1. An affine plane is finite if and only if its order is finite, that is, it contains finitely many points.
2. If in this case the order of the plane is, then it contains points, straight lines, parallel sets and every parallel set contains straight lines. ${\displaystyle q}$ ${\displaystyle q^{2}}$ ${\displaystyle q\cdot (q+1)}$ ${\displaystyle q+1}$ ${\displaystyle q}$
3. If the affine plane contains an infinite number of points, then as a set of points it is of equal power to the set of points of each of its straight lines and to each of its sets of parallels. The number of its straight lines and its families of parallels also has the thickness of the plane. → See Cantor's first diagonal argument .

projective planes

Each affine level can be assigned a projective level that is unambiguous except for isomorphism by projective closure , that is, by adding an “improper straight line” including its points (as remote elements of the affine level) . Any projective plane can be created in this way. The concept of order is transferred to the projective closure: The projective plane has the order of any affine plane, as the projective closure of which it can be constructed. These affine planes do not have to be isomorphic, but they always have the same order. If this order is equal to the finite number , then the projective plane has points and just as many straight lines, there are exactly points on every straight line and exactly straight lines go through every point . ${\displaystyle q\geq 2}$ ${\displaystyle q^{2}+q+1}$ ${\displaystyle q+1}$ ${\displaystyle q+1}$

Finite levels and open questions

• All currently known finite affine levels have a prime power as an order and affine levels with this order exist for every prime power (status: 2013). Which numbers occur as orders of affine planes is an unsolved problem. From the theorem of Bruck and Ryser, a non-existence statement results for levels with certain orders: e.g. the numbers 6, 14, 21, 22, 30, 33, 38, 42, ... are not orders of affine levels. Order 10 could be excluded through massive use of computers. 12 is the smallest number for which the existential question is unsolved.
• Is every affine level of prime order desargue? That is an unsolved problem.
• Is the order of each affine plane a prime power? This question has not yet been clarified either.

→ As a rule, the investigation of finite levels concentrates on their projective conclusion, the finite projective levels . The article Ternary Bodies gives an overview of the relationships between affine levels and their projective conclusion . Examples of and structural statements about non-desargue projective planes can be found in the article Classification of projective planes .

Examples

smallest model of an affine plane

• The two-dimensional vector space over the real numbers, where it applies, includes all one-dimensional affine subspaces and the incidence relation is given by the containment relation . ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {P}}=\mathbb {R} ^{2}}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \in }$
• Likewise, the two-dimensional vector space over any body (or also: oblique body ) . Every affine plane, in which Desargues' theorem applies, is isomorphic to an affine plane over a sloping body . If Pappos's theorem (also "Pappus-Pascal's theorem") applies in this level , the oblique body is a body (with commutative multiplication). ${\displaystyle K^{2}}$ ${\displaystyle K}$ ${\displaystyle K^{2}}$ ${\displaystyle K}$

The non- Desargues levels, in which Desargues' theorem does not hold, have proven of particular interest . In them, coordinates from ternary bodies have been introduced, especially from quasi-bodies (also called Veblen-Wedderburn systems , with non-associative multiplication) or almost bodies (in which only one of the two distributive laws applies).

• In this case the smallest affine plane is obtained. It consists of four points. ${\displaystyle K=\mathbb {F} _{2}}$

• There are affine planes with a finite number, say n, points on a (and then every) straight line. They are called of the n -th order or of the order n . For every prime power q there are affine levels of order q . Whether there are affine levels whose order is not a prime power is an unsolved problem. A partial result is given by the Bruck and Ryser theorem .

This says the following: Does n when divided by 4 is the radical 1 or 2, and n -order affine plane, then n sum of two squares of natural numbers. Examples: 6 is not the order of an affine plane. 10 is not excluded after the sentence.

However, with a lot of computer use, the non-existence of an affine plane of order 10 has been shown. The question of existence is unsolved. B. for the orders 12, 15, 18, 20, 26, 34, 45, ..., and the existence is excluded for n = 14, 21, 22, 30, 33, 38, 42, 46, ...

• The figures below show the minimal model of an affine plane (left) and its projective extension, the minimal model of a projective plane.

smallest model of an affine plane ( ) ${\displaystyle AG(2,2)}$	${\displaystyle AG(2,2)}$is extended to , the projective Fano plane , by adding a straight line {5,6,7} ${\displaystyle PG(2,2)}$

Generalizations

• The affine plane is the two-dimensional special case of an affine geometry .
• Finite affine planes belong to the networks . An affine plane of order n is a -net. ${\displaystyle (n+1,n)}$
• Even more generally, the finite affine planes, like all networks, belong to the block plans and thus to the finite incidence structures . An affine plane of order n is a block diagram. A finite affine plane has the type as the incidence structure . ${\displaystyle 2-(n^{2},n,1)}$ ${\displaystyle (2,1)}$

literature

• Günter Pickert : Projective levels. 2nd Edition. Springer, Berlin a. a. 1975, ISBN 3-540-07280-2 .
• Daniel R. Hughes, Fred C. Piper: Projective Planes. Springer, Berlin a. a. 1973, ISBN 3-540-90044-6 .

Web links

• Joachim Mohr: The axiomatic structure of the affine levels. In: kilchb.de (private website)