Pre-Euclidean level

In synthetic geometry, a pre-Euclidean plane is an affine plane over a body whose characteristic is not 2 and on which an orthogonality relation between the straight lines is defined. In absolute geometry , the pre-Euclidean planes form exactly the class of “Euclidean” models for plane geometries. In absolute geometry, the attribute “Euclidean” is to be understood as the opposite of “non-Euclidean” : Among the metric planes , exactly the pre-Euclidean planes, as described in this article, fulfill the Euclidean axiom of parallels.

An orthogonality relation with the required properties can be explained if the coordinate field of the affine plane has more than one class of squares . The possible orthogonality relations can be classified by the square class of their orthogonality constants. In a pre-Euclidean plane, vertical mirror images and bisectors can be defined, but the latter need not exist for all angles. If the orthogonality constant is in the square class of −1, then squares (the geometric figures) exist in the pre-Euclidean plane and a Cartesian coordinate system can be introduced. If there are bisectors for every pair of intersecting lines, then the pre-Euclidean plane is called a freely movable plane .

Every freely movable plane is a pre-Euclidean plane with squares, every Euclidean plane in the sense of synthetic geometry is a freely movable plane.

A pre-Euclidean level is also referred to in the literature as a generalized Euclidean level .

The present article names the axioms by which an orthogonality relation is characterized on an affine plane in synthetic geometry. In detail, however, only the consequences of this orthogonality for a Pappusian plane that satisfies the affine Fano axiom , i.e. a pre-Euclidean plane, are explained in more detail, which already shares many properties of a Euclidean plane equipped with an orthogonality.

Definitions and characteristics

Orthogonality

A two-digit relation ("is perpendicular to") on the set of lines of an affine plane is called an orthogonality relation if it has the following properties: ${\displaystyle \perp }$

1. For every line there is at least one line with . ${\displaystyle g}$ ${\displaystyle h}$ ${\displaystyle g\perp h}$
2. From follows . ${\displaystyle g\perp h}$ ${\displaystyle h\perp g}$
3. Is , then applies if and only if . ${\displaystyle g\perp h}$ ${\displaystyle h'\parallel h}$ ${\displaystyle g\perp h'}$
4. For no straight line is . ${\displaystyle g}$ ${\displaystyle g\perp g}$

The 4th axiom is occasionally left out in the literature. A straight line that is perpendicular to itself is called isotropic ; according to the 3rd axiom, it is then exactly perpendicular to all its parallels.

Equivalently, one can define an orthogonality mapping on the set of directions ( sets of parallels) of the affine plane , which is required to be bijective , involutorial and, if the orthogonality relation should not allow isotropic straight lines, equivalent to the 4th axiom, it must also be free of fixed elements . ${\displaystyle U}$ ${\displaystyle f\colon U\rightarrow U}$

Already based on the first 3 axioms it follows: For every straight line and every point there is exactly one perpendicular to go through . ${\displaystyle g}$ ${\displaystyle P}$ ${\displaystyle g}$ ${\displaystyle P}$

→ This axiomatic definition generalizes the term orthogonality of plane geometry . For the two-dimensional case of a plane, it generalizes the orthogonality concept of linear algebra and analytical geometry .

Pre-Euclidean level

For an affine translation plane with orthogonality relation (all 4 axioms), which satisfies the affine Fano axiom, i.e. in which no translation is involutor, the following sentences are equivalent:

• the height intersection theorem : "In every triangle the heights intersect at a point, the height intersection ." and
• the means solders sentence : "In each triangle to intersect the perpendicular bisectors of the three sides (the perpendicular bisector of Euclidean geometry) at one point." - This is the Euclidean geometry in the center of the circumscribed circle .

If one of these equivalent sentences applies to an affine translation plane without involutorial translations with orthogonality relation, then it follows that the great affine theorem of Pappos applies in it, that is, it is a Pappus plane and isomorphic to an affine plane over a body whose characteristics do not 2 is. Such an affine plane, i.e. a Pappus plane with a fixed element-free orthogonality mapping, in which the Fano axiom and the height intersection theorem apply, is called a pre-Euclidean plane or a generalized Euclidean plane . ${\displaystyle K}$

→ The concept of the pre-Euclidean plane generalizes the concept of the freely movable plane , which is explained in the article Pythagorean solids , and the even more specific concept of the Euclidean plane, which is explained in the article Euclidean solids .

Orthogonality constant

In a pre-Euclidean plane , the orthogonality mapping can be characterized by an orthogonality constant. As described in the article Ternary Bodies , the plane is provided with a coordinate system in such a way that the coordinate axes are perpendicular to each other ( ), so that the coordinate body is identified with the first coordinate axis. Then the orthogonality mapping assigns the family of parallels with the slope factor to the family of parallels with the slope , the parallels to the first to the second coordinate axis and vice versa. The number is called the orthogonality constant. It is unique except for a multiplication by a square number, i. H. it is another rectangular coordinate system to by selection with . The set is called the square class of . Exactly the numbers from the same class of squares lead to equivalent orthogonality relations. Geometrically equivalent in the sense that the orthogonality constant of the one orthogonality is converted into that of the other by choosing a suitable coordinate system. The equivalent orthogonality maps generally each assign different orthogonal directions to a specific direction. ${\displaystyle A}$ ${\displaystyle (O,E_{1},E_{2})}$ ${\displaystyle OE_{1}\perp OE_{2}}$ ${\displaystyle K}$ ${\displaystyle a\in K^{*}}$ ${\displaystyle \left(g_{(a;d)}:a\cdot x_{2}+x_{1}=d;\;d\in K\right)}$ ${\displaystyle {\frac {c}{a}}}$ ${\displaystyle c\in K^{*}}$ ${\displaystyle {\overline {c}}=c\cdot k^{2}}$ ${\displaystyle k\in K^{*}}$ ${\displaystyle Q_{c}=\lbrace c\cdot k^{2}\,|\;k\in K^{*}\rbrace }$ ${\displaystyle c}$

In a pre-Euclidean plane the orthogonality constant is never in the square class of , otherwise an isotropic straight line would exist. Conversely, by choosing a coordinate system in the affine plane over a body ( ) and choosing an orthogonality constant , an unambiguous orthogonality relation is introduced that satisfies the theorem of height intersection. For every body whose characteristic is not 2 and which has at least two different square classes, the affine plane can be made a pre-Euclidean plane. ${\displaystyle 1,\;Q_{1}=\lbrace k^{2}\,|\;k\in K^{*}\rbrace }$ ${\displaystyle A=K^{2}}$ ${\displaystyle K}$ ${\displaystyle \operatorname {char} (K)\neq 2}$ ${\displaystyle c\in K^{*}\setminus Q_{1}}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle K}$ ${\displaystyle K^{2}}$

There are also other agreements about the “orthogonality constant” in the literature: In absolute geometry, the convention is used , where c is the orthogonality constant defined here, and then k is called the orthogonality constant of the geometry. This, too, is only determined up to quadratic equivalence or “choice of a coordinate system with vertical axes”. The constant k defined in this way must then not be quadratic equivalent . ${\displaystyle k=-{\frac {1}{c}}}$ ${\displaystyle -1}$

Circles

In a pre-Euclidean level, an equivalence relation of equality of distance from a certain point can be defined as follows: ${\displaystyle M}$

• Two points are said to be equidistant if they are the same or if their perpendicular goes through . ${\displaystyle M}$ ${\displaystyle M}$

Each class of equidistant points is referred to as a center-centered circle , or shorter as a circle around . The circle that only consists of the center is called the zero circle . For all "circles" that are not zero circles, the Thales theorem and its reverse apply accordingly. In particular, every circle other than the zero circle contains at least 3 non- collinear points and three different points of a circle are never collinear. The center point is clearly determined by an equivalence class of points that are equidistant from one another - already by 3 different points from the class, provided it is not the zero circle. ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \lbrace M\rbrace }$ ${\displaystyle M}$ ${\displaystyle M}$

In this generalization of the concept of a circle , the middle perpendicular is used to define the circle. Since a circle is clearly described as an equivalence class by the phrase “circle through with center ”, the classic “construction steps with compass and ruler” can be formulated and carried out in a pre-Euclidean level . However, the question of whether and when two circles intersect or whether only one circle intersects one of its centers (straight line through its center point) must be checked again in each individual case! → See below the example of the affine plane over the rational numbers with the “usual” orthogonality. ${\displaystyle A}$ ${\displaystyle M}$

Length classes

The circles in a pre-Euclidean plane are invariants under parallel displacements (translations) : Through the translation , a circle with a center is mapped onto a circle with a center . A circle can be used as a representative of a “length”: Two directed lines and belong to the same length class (in short: “are of equal length”) if the translation converts a circle around that contains into a circle around that contains. Since a point reflection at its center can be defined in a pre-Euclidean plane to the directed line (based on Fano's axiom) , it can be shown that this concept of length is independent of the order of the points in , that is, and always belongs to the same length class. ${\displaystyle {\overrightarrow {MN}}}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle (M,A)}$ ${\displaystyle (N,B)}$ ${\displaystyle {\overrightarrow {MN}}}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle N}$ ${\displaystyle B}$ ${\displaystyle (M,A)}$ ${\displaystyle (M,A)}$ ${\displaystyle (M,A)}$ ${\displaystyle (A,M)}$

In general, two classes of length in the pre-Euclidean plane cannot be compared by size, and a directed "segment" is just a pair of points. In order to be able to define the concept of a “line” in the sense of “set of points between and ”, axioms of arrangement are required which cannot be fulfilled in many pre-Euclidean levels. ${\displaystyle (M,A)}$ ${\displaystyle M}$ ${\displaystyle A}$

Squares

A square can be defined in terms of orthogonality alone as a nondegenerate parallelogram with sides and diagonals perpendicular to one another. A square exists in a pre-Euclidean plane if and only if the orthogonality constant c is in the square class of . The existence of squares can be seen as an additional axiom: it is independent of the axioms of a pre-Euclidean level. ${\displaystyle -1}$

In a pre-Euclidean plane with squares , a coordinate system can be selected on any two axes that are perpendicular to one another so that the unit points together with their point mirror images form a square at the origin ${\displaystyle (O,E_{1},E_{2})}$

${\displaystyle \!\,E_{1}(1|0);\;E_{2}(0|1);\;E_{1}'(-1|0);\;E_{2}'(0|-1)}$

form. To do this, the unit points on the mutually perpendicular axes must be selected so that their perpendicular goes through the origin - this is possible precisely when there are squares. Such a coordinate system is called a Cartesian coordinate system of the pre-Euclidean plane with squares. In a Cartesian coordinate system the orthogonality constant takes on the value −1. ${\displaystyle E_{1},E_{2}}$ ${\displaystyle O}$

→ Compare the main article Cartesian coordinate system .

Automorphisms

Every automorphism of a pre-Euclidean plane must be incidence-preserving and therefore, because the plane is affine, a collineation . Because the displacements in each pre-Euclidean plane form a two-dimensional vector space, each such collineation is determined by a semilinear self-mapping of this vector space.

The automorphisms of the pre-Euclidean plane are precisely the collineations under which orthogonality is preserved.

Examples and counterexamples

• The Euclidean plane (and every plane above another Euclidean solid ), with its usual notion of orthogonality, is a pre-Euclidean plane with squares. Since the real numbers and, more generally, every Euclidean field have only two classes of squares and , the only possible orthogonality here is the normal except for coordinate transformation. With this orthogonality, all coordinate planes become freely movable over Euclidean bodies. ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle Q_{1}}$ ${\displaystyle Q_{-1}}$
• The affine plane over the complex numbers cannot be made a pre-Euclidean plane, since the field only has one class of squares, this applies accordingly to every algebraically closed field. ${\displaystyle \mathbb {C} ^{2}}$ ${\displaystyle \mathbb {C} }$
• The affine plane above the rational numbers becomes a pre-Euclidean plane with squares due to the usual orthogonality with the orthogonality constant. However, this level is not freely movable. If one chooses any positive non- square number or its opposite number as the orthogonality constant , then it becomes a pre-Euclidean plane without squares. ${\displaystyle \mathbb {Q} ^{2}}$ ${\displaystyle c=-1}$ ${\displaystyle c}$ ${\displaystyle -c}$
• If the remainder class field is an odd prime number , then there are exactly two square classes in . Therefore, the Pappus plane (except for coordinate transformation) can be made into a pre-Euclidean plane in exactly one way. It contains squares if and only if a quadratic non- remainder is modulo , i.e. if the prime number has the form . In no case are these levels freely movable. ${\displaystyle K=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle K}$ ${\displaystyle K^{2}}$ ${\displaystyle -1}$ ${\displaystyle p}$ ${\displaystyle p=4\cdot k+3,\;k\in \mathbb {N} _{0}}$
• If the finite field with q elements is more general than in the previous example , and the characteristic p of K is not 2, then q is an odd prime power and there exist two square classes , which are a real subgroup of , and their real subsidiary class . Here, too, one can always define an orthogonality by choosing an element of N as the orthogonality constant and here, too, squares exist in the plane if and only if there is no square number. This must be again . ${\displaystyle K=\mathbb {F} _{q}}$ ${\displaystyle q=p^{r},r\geq 1}$ ${\displaystyle Q_{1}=(K^{*})^{2}}$ ${\displaystyle (K^{*},\cdot )}$ ${\displaystyle N=K^{*}\setminus Q_{1}}$ ${\displaystyle -1\in N}$ ${\displaystyle q\equiv 3{\pmod {4}}}$
• The quadratic number field has an infinite number of square classes. The square classes that do not contain a square number (all but ) lead as orthogonality constants to pre-Euclidean levels that cannot be converted into one another by a coordinate transformation. None of these pre-Euclidean planes contain a square, for there is in that body . ${\displaystyle K=\mathbb {Q} (i)}$ ${\displaystyle Q_{1}}$ ${\displaystyle (K^{2},\perp )}$ ${\displaystyle Q_{-1}=Q_{1}}$

Didactic significance of the pre-Euclidean levels

The axiomatic term "pre-Euclidean level" is interesting for the teacher in school lessons :

There can be no thought of treating the axioms of affine planes or the more specific pre-Euclidean planes in class. However, it is worthwhile for the teacher to understand the examples, especially the rational pre-Euclidean level with “ordinary” orthogonality (naive drawing level), that is, orthogonality constant . ${\displaystyle c=-1}$

In constructing teaching parallel lines are initially lotgleiche line called "double Lote". Against the background of absolute geometry , the naive prejudice that non-parallel straight lines intersect, i.e. meet at a point , do not simply “run through each other” and that straight lines do not simply “run through” circles, can be critically questioned. Circles exist as abstract structures in every pre-Euclidean plane, so also in the naive plane of signs. And in the naive plane of drawing they look exactly like real (real) circles. But note: Among other things, because the rational numbers lie close to the real number set! But already the “third simplest straight line”, the first bisector of the standard coordinate system, runs through the circular line without meeting it at a rational point. Such student experiences may be more motivating to get involved in a range expansion of the rational numbers than the classic, mathematically equivalent example that the diagonal in the rational unit square has no rational length: They almost ask to look for other straight lines "through" the circle, who miss the circle! These considerations also come closer to the idea of ​​the linear “continuum” in the sense of the intermediate value theorem than the classical length problem. Against the historical background of the elements of Euclid and the axiom discussion of absolute geometry, it is precisely the naive-evident existence statements of naive geometry, such as “every line has a center” and “every angle can be halved”, which can by no means be fulfilled naturally.

literature

Main source of the article (sword) and on the didactic significance of the pre-Euclidean levels

• Wendelin Degen, Lothar Profke: Fundamentals of affine and Euclidean geometry . Teubner, Stuttgart 1976, ISBN 3-519-02751-8 .
• Lothar Profke: From affine to Euclidean geometry with the help of an orthogonality relation . In: Mathematics Lessons . 22: 4 (axioms of affine and euclidean planes). Friedrich, Hannover 1976, p. 36-86 .
• Günter Pickert : Deductive geometry in high school lessons . In: Mathematical semester reports . tape X . Springer, 1964, p. 202-223 .

To slightly different ways of speaking

• Heinz Lüneburg : The Euclidean plane and its relatives . Birkhäuser, Basel / Boston / Berlin 1999, ISBN 3-7643-5685-5 ( excerpt from google-books [accessed on June 12, 2013]).

To the pre-Euclidean planes in absolute geometry

• Friedrich Bachmann : Structure of geometry from the concept of reflection . 2nd supplemented edition. Springer, Berlin / Heidelberg / New York 1973, ISBN 3-540-06136-3 .

References and comments

1. ↑ Bachmann (1973), §12 Euclidean Geometry
2. ↑ For example in Lüneburg (1999). V: partial relationships and orthogonality , 3: orthogonality relations of Pappos planes , definition 3.2
3. ↑ Bachmann (1973), §13 Algebraic representation of the Euclidean movement groups
4. ↑ After Pickert (1964) and Degen (1976)