# Division of pages

A straight line d and the two half-planes determined by it. M and N are on the same side of d while M and P are on different sides.

In the elementary geometry of the plane of the drawing, each straight line divides the plane into two (open) half-planes , the sides of the straight line; this observation is initially taken from the point of view. This division of the sides can be described mathematically as an equivalence relation on the set of all points of the plane that do not lie on the dividing straight line.

In analytical geometry this can be specified and generalized: In a - dimensional affine space above an ordered body , every hyperplane , i.e. every - dimensional subspace, provides a side division of the total space into two half spaces. ${\ displaystyle n}$ ${\ displaystyle n-1}$

In synthetic geometry , all page divisions that are determined by straight lines of an affine plane can be introduced by axiomatic description of a page division function with which the plane becomes a weakly arranged plane . Such a page division function then makes it possible to introduce a weak interrelationship at this level , which through an additional axiom becomes an interrelationship in the sense of Hilbert's system of axioms of Euclidean geometry . Levels with a "strong" interrelationship that satisfies Hilbert's axioms of arrangement are called arranged levels . Weak side division functions exist for desargue affine planes if and only if the oblique coordinate of the plane allows a nontrivial quadratic character and can be clearly described by such a character. Each arrangement of a Desargue plane corresponds uniquely to an arrangement of its coordinate oblique body.

This article mainly describes the division of pages in an affine plane in terms of synthetic geometry. The concept of page division from analytical geometry, which is a special case of the synthetic concept for levels, is put in front as the main idea.

## Definitions

### Analytical geometry

A three-dimensional space is used as a basis for a clear presentation. The page division can be carried out in the same way in every finite-dimensional space with a hyperplane instead of the plane. Let be an ordered body and the three-dimensional affine space with the coordinates vector space . Each plane can be described by an inhomogeneous coordinate equation. The sign of the affine function provides the division of the sides. If this function returns values ​​greater than 0 for the position vectors of two points, then they lie on the same side of the (hyper) plane described by, also if both values ​​are less than 0. If one of the values ​​is less and one is greater than 0, then the points are on different sides. ${\ displaystyle K}$${\ displaystyle A}$ ${\ displaystyle K ^ {3}}$ ${\ displaystyle a_ {1} \ cdot x_ {1} + a_ {2} \ cdot x_ {2} + a_ {3} \ cdot x_ {3} -d = 0}$ ${\ displaystyle f (x_ {1}, x_ {2}, x_ {2}) = a_ {1} \ cdot x_ {1} + a_ {2} \ cdot x_ {2} + a_ {3} \ cdot x_ {3} -d}$${\ displaystyle f = 0}$

Although the point coordinates and the (hyper) plane equation depend on the selected coordinate system, the page division does not change when the coordinate system is changed. It depends solely on the order of the ordered body and even determines it unambiguously.

### Synthetic geometry

Regarding the 3rd axiom: Two different straight lines (red), which intersect the connecting line PQ (blue) at the same point T, provide the same value of the lateral division function for P and Q. This enables the lateral division function to be made a (weak) intermediate function .

Let be an affine incidence plane , the set of its lines and the set of triples ${\ displaystyle A}$${\ displaystyle G}$${\ displaystyle \ mathbf {R}}$

${\ displaystyle \ mathbf {R} = \ lbrace (g, P, Q) \ in G \ times A \ times A | P \ not \ in g \ land Q \ not \ in g \ rbrace;}$

then a mapping into the cyclic group is called a (weak) division function if the following axioms are fulfilled: ${\ displaystyle S: \ mathbf {R} \ rightarrow C_ {2}}$ ${\ displaystyle (C_ {2}, \ cdot) = \ left (\ lbrace -1,1 \ rbrace, \ cdot \ right)}$

1. ${\ displaystyle S (g, P, Q) = S (g, Q, P).}$
2. ${\ displaystyle S (g, P, Q) \ cdot S (g, Q, R) \ cdot S (g, R, P) = + 1.}$
3. Are and are copunctal , then is${\ displaystyle P \ neq Q; P, Q \ not \ in g; P, Q \ not \ in h}$${\ displaystyle PQ, g, h}$ ${\ displaystyle S (g, P, Q) = S (h, P, Q).}$
4. Are and , so is${\ displaystyle P \ neq Q, PQ \ parallel g}$${\ displaystyle PQ \ neq g}$${\ displaystyle S (g, P, Q) = + 1.}$
5. For every straight line there are points with and${\ displaystyle g}$${\ displaystyle P, Q \ not \ in g}$${\ displaystyle S (g, P, Q) = - 1.}$

An affine incidence plane with a division function is called weakly arranged .

From the first two axioms it follows that the property ("lie on the same side") for a fixed straight line is an equivalence relation for points that do not lie on the straight line. The 4th axiom says that the parallels of a straight line lie entirely on one side of the straight line, the 5th axiom requires that there are two different sides to every straight line. The 3rd axiom, which is also referred to as the straight line relation , requires that the side division that an intersecting straight line introduces on a certain straight line only depends on the point of intersection . ${\ displaystyle S (g, P, Q) = + 1}$${\ displaystyle g}$${\ displaystyle g}$${\ displaystyle PQ}$${\ displaystyle T \ in g \ cap PQ}$

### Relationship between the definitions

Define the division function for an affine plane over an ordered body

${\ displaystyle S (g, P, Q) = \ operatorname {sgn} (a_ {1} \ cdot p_ {1} + a_ {2} \ cdot p_ {2} -d) \ cdot \ operatorname {sgn} ( a_ {1} \ cdot q_ {1} + a_ {2} \ cdot q_ {2} -d),}$

where is the sign function , an equation of the straight line and the coordinates of the points, then this affine plane becomes a weakly arranged (even an arranged ) plane in the sense of synthetic geometry and all derived terms are applicable. In particular, the side divisions that create two different straight lines are compatible with one another in the sense of the 3rd axiom - for intersecting straight lines - or the 4th axiom - for parallel straight lines. ${\ displaystyle \ operatorname {sgn}}$${\ displaystyle a_ {1} \ cdot x_ {1} + a_ {2} \ cdot x_ {2} -d = 0}$${\ displaystyle g}$${\ displaystyle P (p_ {1} | p_ {2})}$ ${\ displaystyle Q (q_ {1} | q_ {2})}$

## Properties and derived terms

On the concept of the half-plane and half-line. The half-plane Y described in the text is not marked here: It consists of all points of the straight line t (green) and the points "to the right" of t .

The terms given below are defined in Degen's textbook. Deviating from this, there are different conventions in the literature about whether a half-plane without attributes should be "open" or "closed", i.e. whether it contains its "edge line", as well as about whether a "half line" contains its starting point.

• For every straight line there exist two disjoint, nonempty sets, the sides of , where two points belong to the same side if and only if is.${\ displaystyle g}$${\ displaystyle g}$${\ displaystyle P, Q \ in A \ setminus g}$${\ displaystyle S (g, P, Q) = + 1}$
• The side of that contains a certain point is noted as (compare the figure on the right: the set of points is colored reddish), the other side as .${\ displaystyle g}$${\ displaystyle P \ in A \ setminus g}$${\ displaystyle gP ^ {+}}$${\ displaystyle E = gP ^ {+}}$${\ displaystyle gP ^ {-}}$
• It applies ${\ displaystyle A = gP ^ {+} \; \; \! \! {\ dot {\ cup}} \; \; \! \! g \; \; \! \! {\ dot {\ cup} } \; \; \! \! gP ^ {-}.}$
• The union is referred to as a (closed) half-plane , the side as its interior , the straight line as its edge .${\ displaystyle X = gP ^ {+} \ cup g}$ ${\ displaystyle X ^ {\ circ} = gP ^ {+}}$ ${\ displaystyle \ partial X = g}$
• The half-planes and are called opposite to each other .${\ displaystyle gP ^ {+} \ cup g}$${\ displaystyle gP ^ {-} \ cup g}$
• The intersection of two opposite half planes is their common edge.
• A straight line that is parallel to the edge of a half plane and has at least one point in common with the half plane lies entirely in the half plane.
• A straight line is not parallel to the edge of a half-plane if and only if it contains points of the half-plane but is not entirely in the half-plane.
• The intersection of a half-plane with a straight line that is not parallel to the edge is referred to as a half- line, as its starting point , as the carrier line of the half-line. (Compare the figure above right: the straight line t “cutting out” the half line h is marked in green.) Two half lines that arise as the intersection of two opposite half planes are called opposite .${\ displaystyle Y}$${\ displaystyle g}$${\ displaystyle \ partial Y = t}$${\ displaystyle T \ in \ partial Y \ cap g}$${\ displaystyle g}$${\ displaystyle g}$
• Due to the 3rd axiom of the lateral division function, there are exactly two half-lines for a straight line and a starting point and these, as quantities, do not depend on which straight line ( ) cuts through the defining opposite half-planes.${\ displaystyle g}$${\ displaystyle T \ in g}$${\ displaystyle t}$${\ displaystyle t \ neq g}$${\ displaystyle T}$

The terms defined here are based on topological terms (“interior”, “edge”). If one takes the set of all sides (the "interior" for all half-planes) as a sub- basis, then a topology is actually defined on the plane. If the plane contains an infinite number of points, which is always the case for an affine plane over an ordered body, then the defined terms also apply in the topological sense. In a finite level, however, a trivial one arises, namely the discrete topology .

## Intermediate relation

For each page division function on an affine plane, there is a clearly determined mapping on the set of collinear point triples of the plane ${\ displaystyle S}$${\ displaystyle {\ overline {S}}}$

${\ displaystyle {\ overline {\ mathbf {R}}} = \ lbrace (T, P, Q) \ in A \ times A \ times A | T, P, Q \ quad {\ text {are collinear}}, T \ neq P, T \ neq Q \ rbrace.}$

It is defined by for any straight line through which neither nor goes ${\ displaystyle g}$${\ displaystyle T}$${\ displaystyle P}$${\ displaystyle Q}$

${\ displaystyle {\ overline {S}} (T, P, Q) = S (g, P, Q)}$

is set. That this definition is independent of the choice of follows from the 3rd axiom for the page division function. The surjective function is called the weak intermediate function induced by the side division function . The weak intermediate function is invariant under parallel projections . ${\ displaystyle g}$${\ displaystyle {\ overline {S}}: \, {\ overline {\ mathbf {R}}} \ rightarrow C_ {2}}$

To the axiom (Z) of a weak intermediate function: In a non-degenerate triangle (red points) 3 points (blue), which lie between different pairs of points of the starting triangle, form a non-degenerate triangle again.

One then says " lies between and " when and is and calls this three-digit relation a weak interrelationship . This relation fulfills the following axioms (→ compare axioms of arrangement (group II) in Hilbert's system of axioms): ${\ displaystyle T}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle (T, P, Q) \ in {\ overline {\ mathbf {R}}}}$${\ displaystyle {\ overline {S}} (T, P, Q) = - 1}$

(A1) If lies between and , then the three points are collinear and also lies between and .${\ displaystyle T}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle T}$${\ displaystyle Q}$${\ displaystyle P}$
(A2) If and are different points, there is a point lying between them.${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle T}$
(A4, Pasch's axiom ) If there are three non-collinear points, and if there is a straight line that does not go through any of these points and contains a point between and , then also contains a point between and or one between and .${\ displaystyle P_ {1}, P_ {2}, P_ {3}}$${\ displaystyle g}$${\ displaystyle P_ {1}}$${\ displaystyle P_ {2}}$${\ displaystyle g}$${\ displaystyle P_ {2}}$${\ displaystyle P_ {3}}$${\ displaystyle P_ {3}}$${\ displaystyle P_ {1}}$
(Z) If there are three non-collinear points and intermediate points of the three connecting lines of this triangle, then they are also non-collinear.${\ displaystyle P_ {1}, P_ {2}, P_ {3}}$${\ displaystyle Q_ {1}, Q_ {2}, Q_ {3}}$${\ displaystyle Q_ {1}, Q_ {2}, Q_ {3}}$

The axiom (Z) can also be formulated as a supplement to (A4), the axiom of Pasch: "... then also contains a point between and or one between and , never both !" - The "or" in the axiom (A4 ) thus becomes the exclusive or through (Z) . ${\ displaystyle g}$${\ displaystyle P_ {2}}$${\ displaystyle P_ {3}}$${\ displaystyle P_ {3}}$${\ displaystyle P_ {1}}$

The following applies: If a three-digit relation in an affine incidence plane satisfies the axioms (A1), (A2), (A4) and (Z), then there is a uniquely determined division function , from which this interrelation is defined via the intermediate function induced by as described above leaves. The page division function can be calculated directly from the intermediate function: ${\ displaystyle S}$${\ displaystyle S}$

${\ displaystyle S (g, P, Q) = {\ begin {cases} +1, \; {\ text {if}} \; P = Q {\ text {or}} PQ \ parallel g \\ {\ overline {S}} (T, P, Q) \; {\ text {with}} T \ in g \ cap PQ, \; {\ text {if}} P \ neq Q \, {\ text {and} } PQ \ not \ parallel g. \ End {cases}}}$

## Weak arrangement on desargue levels

A Desargue plane is isomorphic to a coordinate plane over an inclined body . For three collinear points with there is always exactly one element , the stretching factor with , and vice versa for each there is a collinear point triple with (→ see Affine translation plane ). Because both the stretching factor and the intermediate function are invariant under parallel projections, a well-defined assignment can be made ${\ displaystyle K}$${\ displaystyle T, P, Q}$${\ displaystyle T \ neq P}$${\ displaystyle \ alpha \ in K}$${\ displaystyle \ alpha = \ operatorname {SF} (T, P, Q)}$${\ displaystyle \ alpha ({\ overrightarrow {TP}}) = {\ overrightarrow {TQ}}}$${\ displaystyle \ alpha \ in K}$${\ displaystyle \ alpha = \ operatorname {SF} (T, P, Q)}$

${\ displaystyle K ^ {*} \ rightarrow C_ {2}, \ quad \ alpha \ mapsto {\ overline {S}} (T, P, Q); \ quad \ alpha = \ operatorname {SF} (T, P , Q)}$

which is a nontrivial quadratic character of . ${\ displaystyle K}$

Conversely, any nontrivial quadratic character of the oblique body can be used to define a weak intermediate function: ${\ displaystyle \ chi}$

${\ displaystyle {\ overline {S}}: {\ overline {\ mathbf {R}}} \ rightarrow C_ {2}, \ quad (T, P, Q) \ mapsto \ chi \ left (\ operatorname {SF} (T, P, Q) \ right).}$

Every weak side division on a desargue plane is induced by a nontrivial quadratic character of the oblique coordinate and vice versa. On a desargue level there is a weak side division function if its coordinate oblique allows a nontrivial quadratic character and for every such character there is a weak arrangement of the plane.

## Arranged level

Hilbert's third axiom of arrangement reads

(A3) If lies between and , then does not lie between and .${\ displaystyle T}$${\ displaystyle P}$${\ displaystyle Q}$${\ displaystyle Q}$${\ displaystyle P}$${\ displaystyle T}$

Axiom (Z) follows from Hilbert's axioms (A1) to (A4). An affine plane with an intermediate relation that fulfills the axioms (A1) to (A4) and thus automatically also (Z) is called an arranged plane .

### properties

• In an arranged affine incidence plane, exactly one of three different collinear points lies between the other two.
• Each arranged affine incidence plane is also weakly arranged.
• The arrangement ( intermediate relationship ) can be described with a page division function that is clearly determined by it or with the intermediate function that is determined by it. In the case of a Desargue plane, like the weak arrangement, it is clearly determined by a nontrivial quadratic character.
• In an arranged plane and in a finite, weakly arranged plane, Fano's affine axiom applies , so in these cases there is always a center point for any two points. This center point is then always (in the sense of the respective interrelationship) between the two points.
• If the affine translation plane, which arises from slotting from a Moufang plane , can be made into an arranged plane, then both planes are desarguesch.

### Desargue arranged plane

The arrangement of a sloping body must fulfill the same axioms as the order of an ordered body, from which it follows that it is uniquely determined by a positive area that has the following properties (→ compare ordered bodies ): ${\ displaystyle K}$${\ displaystyle K ^ {+} = \! \, \ lbrace \ alpha \ in K | \; 0 <\ alpha \ rbrace}$

1. ${\ displaystyle a, b \ in K ^ {+} \ Rightarrow a + b \ in K ^ {+},}$
2. ${\ displaystyle a, b \ in K ^ {+} \ Rightarrow a \ cdot b \ in K ^ {+},}$
3. for each one of the relationships applies .${\ displaystyle a \ in K}$${\ displaystyle a \ in K ^ {+}, \, - a \ in K ^ {+}, a = 0}$

A quadratic character is determined by each such positive area , which is positive, negative and fulfills. Conversely, every square character defines an area , which, however, generally only has the 2nd and 3rd properties of a positive area. ${\ displaystyle \ chi}$${\ displaystyle K ^ {+}}$${\ displaystyle -K ^ {+}}$${\ displaystyle \ chi (-1) = - 1}$${\ displaystyle \ chi (-1) = - 1}$${\ displaystyle L = \ chi ^ {- 1} (1)}$

The following sentence clarifies the connection between the arrangement of a Desargue plane and the arrangement of its coordinate inclined body:

The oblique coordinate of an arranged Desargue plane allows an arrangement. Conversely, the affine plane can be arranged over an arranged oblique body. Every arrangement of the oblique body induces a “strong” intermediate function on the plane and vice versa. The connection is made by equivalence
${\ displaystyle {\ overline {S}} (T, P, Q) = + 1 \ Leftrightarrow \ operatorname {SF} (T, P, Q)> 0}$
mediated for everyone .${\ displaystyle (T, P, Q) \ in {\ overline {R}}}$

## Examples

• The affine plane over an ordered body becomes an arranged plane by the page partitioning function described above. The square character assigns every number in the body except 0 to its sign as a number in .${\ displaystyle C_ {2} = \ lbrace -1, + 1 \ rbrace}$
• A formally real body allows at least one order that makes it an ordered body. As in the previous example, each of these body orders determines a nontrivial quadratic character and thus an arrangement of the affine plane above the body.
• A Euclidean field only allows a nontrivial quadratic character, which assigns 1 to every square number and −1 to every non- square number (see also square class ). Therefore, on the affine level above such a body, exactly one weak side division is possible, with which this level becomes a "strongly" arranged level.
• The field of complex numbers and, more generally, every algebraically closed field only allows the trivial character as a quadratic character. Therefore, no weak arrangement is possible on an affine plane over such a body.
• An infinite number of nontrivial quadratic characters can be defined on the field of rational numbers : If the set of prime numbers is arbitrarily divided into two disjoint subsets and , then the choice of a sign for a character is uniquely determined by and . This describes all the square characters of . Each of these characters except the trivial ( ) determines a weak arrangement of the affine plane . Exactly for the character described by the weak arrangement is an arrangement, the "ordinary" arrangement of the rational plane.${\ displaystyle P _ {+}}$${\ displaystyle P _ {-}}$${\ displaystyle \ chi (P _ {+}) = + 1, \ chi (P _ {-}) = - 1}$${\ displaystyle \ chi (-1)}$${\ displaystyle \ chi}$${\ displaystyle \ mathbb {Q} ^ {*}}$${\ displaystyle P _ {-} = \ emptyset, \ chi (-1) = 1}$${\ displaystyle \ mathbb {Q}}$${\ displaystyle P _ {-} = \ emptyset, \ chi (-1) = - 1}$
• Every remainder class field for an odd prime number has an element that creates the cyclic multiplicative group of the field. A nontrivial quadratic character is clearly defined by as a group homomorphism. This is the only nontrivial quadratic character of the body, so the Desargue plane above such a body can be weakly placed in exactly one way.${\ displaystyle K = \ mathbb {Z} / p \ mathbb {Z}}$${\ displaystyle p}$${\ displaystyle \ beta}$ ${\ displaystyle K ^ {*}}$${\ displaystyle \ chi (\ beta) = - 1}$${\ displaystyle K ^ {*}}$
• The affine plane above the remainder class field becomes a weakly arranged plane due to its single nontrivial quadratic character. 2 is a generating element of . For a point triple with , the intermediate function does not change when the points in the triple are cyclically interchanged . Therefore, none of the three different collinear points lies between the other two.${\ displaystyle K = \ mathbb {Z} / 13 \ mathbb {Z}}$${\ displaystyle K ^ {*}}$${\ displaystyle (T, P, Q)}$${\ displaystyle \ operatorname {SF} (T, P, Q) = 2 ^ {2} = 4}$
• If an arbitrary field and the rational function field is above this field, then a nontrivial quadratic character can be introduced through the degree of the rational function : if and only if is odd. This character allows the affine level to be arranged over weak. - So this is also possible over certain infinite bodies with characteristic 2!${\ displaystyle F}$${\ displaystyle K = F (X)}$${\ displaystyle \ textstyle {\ frac {f} {g}} \ neq 0}$${\ displaystyle K}$${\ displaystyle \ textstyle \ chi \ left ({\ frac {f} {g}} \ right) = - 1}$${\ displaystyle \ textstyle \ deg \ left ({\ frac {f} {g}} \ right) = \ deg (f) - \ deg (g)}$${\ displaystyle K}$
• A finite field with characteristic 2 does not allow a nontrivial quadratic character, therefore a side division never exists on an affine plane over such a field.

## literature

Original literature
• David Hilbert : Fundamentals of Geometry . 14th edition. Teubner, Stuttgart / Leipzig 1899, ISBN 3-519-00237-X ( online copy of the 1903 edition [accessed on July 25, 2013]).
• Erich Glock: The orientation functions of an affine space . In: Mathematical Journal . tape 78 , 1962, pp. 319-360 .
• Erich Glock: Organizational functions that lead to page divisions of a special kind . In: Mathematics and Statistics, Archives of Mathematics . tape 12 , no. 1 , p. 71-77 , doi : 10.1007 / BF01650526 .
• Emanuel Sperner : The ordering functions of a geometry . In: Math. Ann. tape 121 , 1949, pp. 107-130 .
• Emanuel Sperner: Relationships between geometrical and algebraic arrangement . In: Meeting report Heidelberger Akad. Wiss. Math. Natural sciences Kl. 1949, p. 413-448 .
Textbooks

## Individual evidence

1. Hilbert (1899), 1 §3: Axioms of the arrangement
2. Sword (1976)
3. ↑ On this, for example, Lüneburg (1999)
4. Pickert (1975) p. 240
5. Degen (1976) p. 105