Minkowski level

A Minkowski plane , named after Hermann Minkowski , is in the classic case an incidence structure that essentially describes the geometry of the hyperbolas given by an equation of the form and the straight lines in the real plane of observation. Points with the same x or y coordinates have no connection, they are therefore called (+) - parallel or (-) - parallel. ${\displaystyle y={\tfrac {a}{x-b}}+c,a\neq 0,\ }$ ${\displaystyle y=ax+b,a\neq 0\ }$

classic Minkowski plane: 2D / 3D model

Obviously, the following applies: exactly one hyperbola passes through 3 points that are not parallel in pairs. However: A straight line is clearly defined by 2 points. Two hyperbolas can intersect at two points or touch at one point (common tangent) or avoid one another. As with Möbius and Laguerre planes , simpler geometric relationships are obtained if the geometry of the hyperbola / straight line is homogenized by adding further points: the two points are added to a hyperbola and the point is added to a straight line and the hyperbola / straight line expanded in this way is called Cycles (see picture). The new incidence structure now has similar properties to a Möbius or Laguerre plane (see section Axioms) and (like Möbius and Laguerre planes) has a spatial model: The classic Minkowski plane is isomorphic to the geometry of the plane sections of a single-shell hyperboloids (see picture) in real projective space . A single-shell hyperboloid is a quadric that contains straight lines and non-degenerate projective conic sections . ${\displaystyle y={\tfrac {a}{x-b}}+c}$ ${\displaystyle y=(b,\infty ),(\infty ,c)}$ ${\displaystyle y=ax+b}$ ${\displaystyle (\infty ,\infty )}$

In addition to these geometric models of the classic real Minkowski plane, there is also the representation above the ring of abnormally complex numbers (analogous to the description of the classic Möbius plane above the complex numbers ). An abnormally complex number (like a complex number ) has the form with . ${\displaystyle a+jb,a,b\in \mathbb {R} ,j\notin \mathbb {R} }$ ${\displaystyle j^{2}={\color {red}{+}}1}$

A Minkowski level is one of the 3 Benz levels : Möbius level, Laguerre level and Minkowski level. The classic Möbius plane is the geometry of the circles and the classic Laguerre plane is the geometry of the parabolas.

The name Minkowski plane comes from the Minkowski metric used to describe pseudo-Euclidean "circles" (hyperbolas).

The axioms of a Minkowski plane

Let there be an incidence structure with the set of points , the set of cycles and two equivalence relations ((+) - parallel) and ((-) - parallel) on the set of points . For one point we define: and . An equivalence class or is called (+) - generating or (-) - generating . (In the spatial model of the classical Minkowski plane, a generating line is a straight line on the hyperboloid.) Two points are called parallel ( ) if or applies. ${\displaystyle ({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {Z}}}$ ${\displaystyle \parallel _{+}}$ ${\displaystyle \parallel _{-}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\overline {P}}_{+}:=\{Q\in {\mathcal {P}}\ |\ Q\parallel _{+}P\}}$ ${\displaystyle {\overline {P}}_{-}:=\{Q\in {\mathcal {P}}\ |\ Q\parallel _{-}P\}}$ ${\displaystyle {\overline {P}}_{+}}$ ${\displaystyle {\overline {P}}_{-}}$
${\displaystyle A,B}$ ${\displaystyle A\parallel B}$ ${\displaystyle A\parallel _{+}B}$ ${\displaystyle A\parallel _{-}B}$

An incidence structure is called a Minkowski plane if the following axioms hold: ${\displaystyle {\mathfrak {M}}:=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$

Minkowski plane: axioms C1, C2

Minkowski plane: axioms C3, C4

(C1): For every two non-parallel points there is exactly one point with . ${\displaystyle A,B}$ ${\displaystyle C}$ ${\displaystyle A\parallel _{+}C\parallel _{-}B}$

(C2): For every point and every cycle there are exactly two points with . ${\displaystyle P}$ ${\displaystyle z}$ ${\displaystyle A,B\in z}$ ${\displaystyle A\parallel _{+}P\parallel _{-}B}$

(C3): For every 3 pairs of non-parallel points there is exactly one cycle that contains. ${\displaystyle A,B,C}$ ${\displaystyle z}$ ${\displaystyle A,B,C}$

(C4): ( Axiom of contact ) For every cycle , every point and every point and there is exactly one cycle such that , i. H. touched in point . ${\displaystyle z}$ ${\displaystyle P\in z}$ ${\displaystyle Q,P\not \parallel Q}$ ${\displaystyle Q\notin z}$ ${\displaystyle z'}$ ${\displaystyle z\cap z'=\{P\}}$ ${\displaystyle z}$ ${\displaystyle z'}$ ${\displaystyle P}$

(C5): Each cycle contains at least 3 points. There is at least one cycle and one point does not give up . ${\displaystyle z}$ ${\displaystyle P}$ ${\displaystyle z}$

The following statements, which are equivalent to (C1) and (C2), are advantageous for investigations of a Minkowski plane.

(C1 '): For any two points applies: . ${\displaystyle A,B}$ ${\displaystyle |{\overline {A}}_{+}\cap {\overline {B}}_{-}|=1}$

(C2 '): For every point and every Cycles applies: . ${\displaystyle P}$ ${\displaystyle z}$ ${\displaystyle |{\overline {P}}_{+}\cap z|=1=|{\overline {P}}_{-}\cap z|}$

Analogous to Möbius and Laguerre planes, the following local structures are also here affine planes.

For a Minkowski plane and we define ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$ ${\displaystyle P\in {\mathcal {P}}}$

${\displaystyle {\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus {\overline {P}},\{z\setminus \{{\overline {P}}\}\ |\ P\in z\in {\mathcal {Z}}\}\cup \{E\setminus {\overline {P}}\ |\ E\in {\mathcal {E}}\setminus \{{\overline {P}}_{+},{\overline {P}}_{-}\}\},\in )}$

and call this incidence structure derivation in the point ${\displaystyle P}$ .

In the classical real Minkowski plane is the real affine plane (see 1st picture). ${\displaystyle {\mathfrak {A}}_{(\infty ,\infty )}}$ ${\displaystyle \mathbb {R} ^{2}}$

A direct consequence of axioms (C1) - (C4) and (C1 '), (C2') is:

Theorem: For a Minkowski plane every derivative is an affine plane. ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel ,\in )}$

This gives the alternative definition

Theorem: Let there be an incidence structure with two equivalence relations and on the set of points . ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$ ${\displaystyle \parallel _{+}}$ ${\displaystyle \parallel _{-}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle {\mathfrak {M}}}$is a Minkowski plane if and only if for every point the derivative is an affine plane. ${\displaystyle P}$ ${\displaystyle {\mathfrak {A}}_{P}}$

The minimal model

The minimal model of a Minkowski plane can be defined using the set of 3 elements: ${\displaystyle {\overline {K}}:=\{0,1,\infty \}}$

${\displaystyle {\mathcal {P}}:={\overline {K}}^{2},\qquad {\mathcal {Z}}:=\{\{(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\}\ |\ \{a_{1},a_{2},a_{3}\}=\{b_{1},b_{2},b_{3}\}={\overline {K}}\}}$,

${\displaystyle (x_{1},y_{1})\parallel _{+}(x_{2},y_{2})}$exactly when and exactly when is. ${\displaystyle x_{1}=x_{2}\ }$ ${\displaystyle \ (x_{1},y_{1})\parallel _{-}(x_{2},y_{2})\ }$ ${\displaystyle \ y_{1}=y_{2}}$

So: the number of points is and that of the cycles . ${\displaystyle |{\mathcal {P}}|=9}$ ${\displaystyle |{\mathcal {Z}}|=6}$

Minkowski plane: minimal model (only the cycles through are drawn) ${\displaystyle (\infty ,\infty )}$

For finite Minkowski planes we get from (C1 '), (C2'):

• Let it be a finite Minkowski plane, i.e. H. . For each pair of Zykeln and each pair is true of generators: . ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$ ${\displaystyle |{\mathcal {P}}|<\infty }$ ${\displaystyle z_{1},z_{2}}$ ${\displaystyle e_{1},e_{2}}$ ${\displaystyle |z_{1}|=|z_{2}|=|e_{1}|=|e_{2}|}$

This gives rise to the following definition:
For a finite Minkowski plane and a cycle of we call the natural number the order of . ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle z}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle n=|z|-1}$ ${\displaystyle {\mathfrak {M}}}$

Simple combinatorial considerations result in:

• For a finite Minkowski plane the following applies: ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$

a) Every derivative (affine plane) has the order . ${\displaystyle n}$

b) c) . ${\displaystyle |{\mathcal {P}}|=(n+1)^{2}\quad ,}$ ${\displaystyle \ |{\mathcal {Z}}|=(n+1)n(n-1)}$

The classic real Minkowski plane

The formal definition of the classical real Minkowski plane specifies the homogenization of the geometry of the hyperbolas described in the introduction:

${\displaystyle {\mathcal {P}}:=\mathbb {R} ^{2}\cup (\{\infty \}\times \mathbb {R} )\cup (\mathbb {R} \times \{\infty \})\ \cup \{(\infty ,\infty )\}\ ,\ \infty \notin \mathbb {R} }$, the amount of points ,

${\displaystyle {\mathcal {Z}}:=\{\{(x,y)\in \mathbb {R} ^{2}\ |\ y=ax+b\}\cup \{(\infty ,\infty )\}\ |\ a,b\in \mathbb {R} ,a\neq 0\}}$

${\displaystyle \cup \{\{(x,y)\in \mathbb {R} ^{2}\ |y={\frac {a}{x-b}}+c,x\neq b\}\cup \{(b,\infty ),(\infty ,c)\}\ |\ a,b,c\in \mathbb {R} ,a\neq 0\},}$the set of cycles ..

The incidence structure is called the classical real Minkowski plane . ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

The set of points consists of , two copies of and the point . Every straight line is supplemented by the point , every hyperbola by the two points (see 1st picture). ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle (\infty ,\infty )}$
${\displaystyle y=ax+b,a\neq 0,}$ ${\displaystyle (\infty ,\infty )}$ ${\displaystyle y={\frac {a}{x-b}}+c,a\neq 0}$ ${\displaystyle (b,\infty ),(\infty ,c)}$

Two points cannot be connected by a cycle if and only if or is. We define: Two points are (+) - parallel ( ) if is, and (-) - parallel ( ) if holds. Both relations are equivalence relations on the set of points. Two points are called parallel ( ) if or is true. ${\displaystyle (x_{1},y_{1})\neq (x_{2},y_{2})}$ ${\displaystyle x_{1}=x_{2}}$ ${\displaystyle y_{1}=y_{2}}$
${\displaystyle P_{1},P_{2}}$ ${\displaystyle P_{1}\parallel _{+}P_{2}}$ ${\displaystyle x_{1}=x_{2}}$ ${\displaystyle P_{1}\parallel _{-}P_{2}}$ ${\displaystyle y_{1}=y_{2}}$

${\displaystyle P_{1},P_{2}}$ ${\displaystyle P_{1}\parallel P_{2}}$ ${\displaystyle P_{1}\parallel _{+}P_{2}}$ ${\displaystyle P_{1}\parallel _{-}P_{2}}$

• The incidence structure defined here fulfills the axioms of a Minkowski plane. ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

Like the classic Möbius or Laguerre plane, there is also a spatial model for the classic real Minkowski plane. However, an affine quadric is not sufficient for the description:

• The classic Minkowski plane is isomorphic to the geometry of the plane sections of a single-shell hyperboloid in 3-dimensional real projective space.

Miquelsche Minkowski planes

The most important non-classical Minkowski planes are obtained by simply replacing the real numbers in the classical model with an arbitrary field . The incidence structure obtained in this way is a Minkowski plane for each body. Analogous to Möbius and Laguerre planes, they are characterized by the corresponding version of Miquel's theorem: ${\displaystyle K}$ ${\displaystyle {\mathfrak {M}}(K)=({\mathcal {P}},{\mathcal {Z}};\parallel _{+},\parallel _{-},\in )}$

Miquel's theorem (circles instead of hyperbola)

Theorem (MIQUEL): For a Minkowski plane the following applies: ${\displaystyle {\mathfrak {M}}(K)}$

If for any 8 points there are pairs of non-parallel points that can be assigned to the corners of a cube in such a way that 4 points with 5 faces each lie on a cycle, this is also the case for the 4 points on the 6th face (see Sect. Image: For a better overview, circles were drawn instead of hyperbolas). ${\displaystyle P_{1},...,P_{8}}$

The meaning of Miquel's sentence is shown in the following sentence by Chen:

Theorem (CHEN): Only one Minkowski plane satisfies Miquel's theorem. ${\displaystyle {\mathfrak {M}}(K)}$

Because of this theorem, a Miquelian is called a Minkowski plane . ${\displaystyle {\mathfrak {M}}(K)}$

Comment: The minimal model of a Minkowski plane is Miquelian.

It is isomorphic to the Minkowski plane with (body ). ${\displaystyle {\mathfrak {M}}(K)}$ ${\displaystyle K=GF(2)}$ ${\displaystyle \{0,1\}}$

That is an amazing result

Theorem (Heise): Every Minkowski plane of even order is Miquelian.

Comment: A suitable stereographic projection shows: is isomorphic to the geometry of the plane sections on a single-shell hyperboloid ( quadric of index 2 ) in 3-dimensional projective space . ${\displaystyle {\mathfrak {M}}(K)}$ ${\displaystyle K}$

Not Miquelian Minkowski planes

There are numerous non-Miquelian Minkowski planes (see Weblink circle geometries). But: There are no ovoid Minkowski planes (in contrast to Möbius and Laguerre planes), because a quadric set of index 2 in 3-dimensional projective space is already a quadric (see quadric set ). Many non-Miquelian examples are obtained by generalizing the relationship between the hyperbolas / straight lines and the broken linear mappings (projective group ). But even the mere replacement of the hyperbolas in the classical model by the similar curves does not yield Miquelian Minkowski planes. ${\displaystyle PGL(2,K)}$ ${\displaystyle y={\tfrac {a}{(x-b)^{3}}}+c}$

Individual evidence

1. ^ Walter Benz : Lectures on the geometry of algebras . Reprint from 1973. Springer , Heidelberg 2013, ISBN 978-3-642-88671-3 , p. 45 .
2. ^ Walter Benz : Lectures on the geometry of algebras . Reprint from 1973. Springer , Heidelberg 2013, ISBN 978-3-642-88671-3 , p. 42 .
3. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 93.
4. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 95.
5. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 110.
6. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 97.
7. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 76.
8. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 100.
9. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 113.

literature

• W. Benz, Lectures on the Geometry of Algebras , Springer , pp. 42–81 (1973)
• F. Buekenhout (ed.), Handbook of Incidence Geometry , Elsevier (1995) ISBN 0-444-88355-X , p. 1339

Web links

• Benz plane at SpringerLink
• Lecture Note Planar Circle Geometries , an Introduction to Moebius-, Laguerre- and Minkowski Planes