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{{Short description|Type of Benz planes}}

In mathematics, a '''Minkowski plane''' (named after [[Biography:Hermann Minkowski|Hermann Minkowski]]) is one of the [[Benz plane]]s (the others being [[Möbius plane]] and [[Laguerre plane]]).

==Classical real Minkowski plane==
[[Image:Minkowski-2d3d-model.svg|300px|thumb|classical Minkowski plane: 2d/3d-model]]
Applying the pseudo-euclidean distance <math>d(P_1,P_2) = (x'_1-x'_2)^2 - (y'_1-y'_2)^2</math> on two points <math>P_i = (x'_i, y'_i)</math> (instead of the euclidean distance) we get the geometry of ''hyperbolas'', because a pseudo-euclidean circle <math>\{P\in \R^2 \mid d(P,M)=r\}</math> is a [[Hyperbola|hyperbola]] with midpoint <math>M</math>. 

By a transformation of coordinates <math>x_i = x'_i + y'_i</math>, <math>y_i = x'_i - y'_i</math>, the pseudo-euclidean distance can be rewritten as <math>d(P_1,P_2) = (x_1 - x_2) (y_1 - y_2)</math>. The hyperbolas then have asymptotes parallel to the non-primed coordinate axes.

The following completion (see Möbius and Laguerre planes) ''homogenizes'' the geometry of hyperbolas:

* the set of '''points''': <math display="block">\mathcal P := \left(\R \cup \left\{\infty\right\}\right)^2 =
\R^2 \cup \left(\left\{\infty\right\} \times \R\right) \cup \left(\R \times \left\{\infty\right\}\right) \ 
     \cup \left\{\left(\infty,\infty\right)\right\} \ ,
 \ \infty \notin \R,</math>
* the set of '''cycles''' <math display="block">\begin{align}
\mathcal Z :={} & \left\{\left\{\left(x,y\right) \in \R^2 \mid y = ax + b\right\} \cup \left\{\left(\infty,\infty\right)\right\} \mid a,b \in \R, a\ne 0\right\}\\
& \quad \cup \left\{\left\{\left(x,y\right) \in \R^2 \mid y=\frac{a}{x-b} + c, x \ne b\right\}
\cup \left\{\left(b,\infty\right),\left(\infty,c\right)\right\} \mid a,b,c \in \R, a \ne 0\right\}.
\end{align}</math>

The [[Incidence structure|incidence structure]] <math>({\mathcal P},{\mathcal Z},\in)</math> is called the '''classical real Minkowski plane'''.

The set of points consists of <math>\R^2</math>, two copies of <math>\R</math> and the point <math>(\infty,\infty)</math>.

Any line <math>y=ax+b ,a\ne0</math> is completed by point <math>(\infty,\infty)</math>, any hyperbola <math> y = \frac{a}{x-b} + c, a\ne0 </math> by the two points <math>(b,\infty),(\infty,c)</math> (see figure).

Two points <math>(x_1,y_1)\ne(x_2,y_2)</math> can not be connected by a cycle if and only if <math>x_1 = x_2</math> or <math>y_1 = y_2</math>. 

We define:
Two points <math>P_1,P_2</math> are '''(+)-parallel''' (<math>P_1 \parallel_+ P_2</math>) if <math>x_1 = x_2</math> and '''(−)-parallel''' (<math>P_1\parallel_- P_2</math>) if <math>y_1 = y_2</math>. <br />
Both these relations are equivalence relations on the set of points.

Two points <math>P_1,P_2</math> are called '''parallel''' (<math>P_1\parallel P_2</math>) if
<math>P_1 \parallel_+ P_2</math> or <math>P_1 \parallel_- P_2</math>.

From the definition above we find:

'''Lemma:'''
*For any pair of non parallel points <math>A,B</math> there is exactly one point <math>C</math> with <math>A\parallel_+ C \parallel_- B</math>.
*For any point <math>P</math> and any cycle <math>z</math> there are exactly two points <math>A,B \in z</math> with <math>A\parallel_+ P \parallel_- B</math>.
*For any three points <math>A</math>, <math>B</math>, <math>C</math>, pairwise non parallel, there is exactly one cycle <math>z</math> that contains <math>A,B,C</math>.
*For any cycle <math>z</math>, any point <math>P \in z</math> and any point <math>Q, P \not\parallel Q</math> and <math>Q \notin z</math> there exists exactly one cycle <math>z'</math> such that <math>z \cap z' = \{P\}</math>, i.e. <math>z</math> '''touches''' <math>z'</math> at point P.

Like the classical Möbius and Laguerre planes Minkowski planes can be described as the geometry of plane sections of a suitable quadric. But in this case the quadric lives in '''projective''' 3-space: The classical real Minkowski plane is isomorphic to the geometry of plane sections of a hyperboloid of one sheet (not degenerated quadric of index 2).

==The axioms of a Minkowski plane==

Let <math> \left( {\mathcal P} , {\mathcal Z} ; \parallel_+ , \parallel_- , \in \right) </math> be an incidence structure with the set <math>\mathcal P</math> of points, the set <math>\mathcal Z</math> of cycles and two equivalence relations <math>\parallel_+</math> ((+)-parallel) and <math>\parallel_-</math> ((−)-parallel) on set <math>\mathcal P</math>.  For <math>P\in \mathcal P</math> we define:
<math> \overline{P}_+ := \left\{ Q \in \mathcal P \mid Q\parallel_+ P \right\} </math> and <math> \overline{P}_- := \left\{ Q \in \mathcal P \mid Q\parallel_- P \right\} </math>.
An equivalence class <math>\overline{P}_+</math> or <math>\overline{P}_-</math> is called '''(+)-generator''' and '''(−)-generator''', respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.)<br />
Two points <math> A , B </math> are called '''parallel''' (<math> A \parallel B </math>) if <math> A \parallel_+ B </math> or <math> A \parallel_- B</math>.

An incidence structure <math>\mathfrak M := ( \mathcal P , \mathcal Z ; \parallel_+ , \parallel_- , \in ) </math> is called '''Minkowski plane''' if the following axioms hold:<br />
[[File:Minkowski-axioms-c1-c2.svg|thumb|Minkowski-axioms-c1-c2]]
[[File:Minkowski-axioms-c3-c4.svg|thumb|Minkowski-axioms-c3-c4]]
* '''C1''': For any pair of non parallel points <math>A,B</math> there is exactly one point <math>C</math> with <math>A\parallel_+ C \parallel_- B </math>.
* '''C2''': For any point <math>P</math> and any cycle <math>z</math> there are exactly two points <math> A , B \in z</math> with <math>A\parallel_+ P \parallel_- B </math>.
* '''C3''': For any three points <math>A,B,C</math>, pairwise non parallel, there is exactly one cycle <math>z</math> which contains <math> A , B , C </math>.
* '''C4''': For any cycle <math>z</math>, any point <math>P\in z</math> and any point <math> Q, P \not\parallel Q </math> and <math>Q\notin z</math> there exists exactly one cycle <math>z'</math> such that <math> z \cap z' = \{ P \} </math>, i.e., <math>z</math> '''touches''' <math>z'</math> at point <math> P </math>.
* '''C5''': Any cycle contains at least 3 points. There is at least one cycle <math>z</math> and a point <math>P</math> not in <math>z</math>.

For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.
*'''C1′''': For any two points <math> A , B </math> we have {{nowrap|<math> \left| \overline{A}_+ \cap\overline{B}_- \right| = 1 </math>.}}
*'''C2′''': For any point <math> P </math> and any cycle <math> z </math> we have: {{nowrap|<math> \left| \overline{P}_+ \cap z \right | = 1 = \left| \overline{P}_- \cap z \right| </math>.}}

First consequences of the axioms are
{{math theorem | name = Lemma | math_statement = For a Minkowski plane <math>{\mathfrak M}</math> the following is true
{{ordered list | list-style-type = lower-alpha
| Any point is contained in at least one cycle.
| Any generator contains at least 3 points.
| Two points can be connected by a cycle if and only if they are non parallel.
}}
}}
Analogously to Möbius and Laguerre planes we get the connection to the linear
geometry via the residues.

For a Minkowski plane <math>\mathfrak M = (\mathcal P, \mathcal Z; \parallel_+, \parallel_-, \in)</math> and <math>P \in \mathcal P</math> we define the local structure
<math display="block">\mathfrak A_P:= (\mathcal P \setminus \overline{P},\{z \setminus\{\overline{P}\} \mid P \in z \in \mathcal Z\}
\cup \{E\setminus \overline{P} \mid E \in \mathcal E \setminus \{\overline{P}_+,\overline{P}_-\}\}, \in)</math>
and call it the '''residue at point P'''.

For the classical Minkowski plane <math>\mathfrak A_{(\infty,\infty)}</math> is the real affine plane <math>\R^2</math>.

An immediate consequence of axioms C1 to C4 and C1′, C2′ are the following two theorems.

{{math theorem | For a Minkowski plane <math> \mathfrak M = ( \mathcal P , \mathcal Z ; \parallel_+ , \parallel , \in ) </math> any residue is an affine plane.}}

{{math theorem | Let be <math>\mathfrak M = (\mathcal P, \mathcal Z;\parallel_+,\parallel_-,\in)</math> an incidence structure with two equivalence relations <math>\parallel_+</math> and <math>\parallel_-</math> on the set <math>\mathcal P</math> of points (see above).

Then, <math>\mathfrak M</math> is a Minkowski plane if and only if for any point <math>P</math> the residue <math>\mathfrak A_P</math> is an affine plane.}}

===Minimal model===
[[File:Minkowski-minimal-model.svg|300px|thumb|Minkowski plane: minimal model]]

The '''minimal model''' of a Minkowski plane can be established over the set
<math>\overline{K}:=\{0,1,\infty\}</math> of three elements:

<math display="block">\mathcal {P} := \overline{K}^2 </math>

<math display="block">\begin{align}
\mathcal Z :\!&= \left\{ \{ (a_1,b_1),(a_2,b_2),(a_3,b_3) \} \mid \{a_1,a_2,a_3\} = \{b_1,b_2,b_3\} = \overline{K} \right\} \\
&= \{
  \{ (0,0), (1,1), (\infty,\infty) \} , \; \{ (0,0), (1,\infty), (\infty,1) \}, \\
 &\qquad\{ (0,1), (1,0), (\infty,\infty) \}, \; \{ (0,1), (1,\infty), (\infty,0) \}, \\
 &\qquad\{ (0,\infty), (1,1), (\infty,0) \}, \; \{ (0,\infty), (1,0), (\infty,1) \} \}
\end{align}</math> 

Parallel points:
*<math> (x_1,y_1) \parallel_+ (x_2,y_2) </math> if and only if <math> x_1 = x_2 </math> 
*<math>(x_1,y_1)\parallel_- (x_2,y_2)</math> if and only if <math> y_1 = y_2 </math>.

Hence <math> \left| \mathcal P \right| = 9 </math> and <math> \left| \mathcal Z \right| = 6 </math>.

===Finite Minkowski-planes===
For finite Minkowski-planes we get from C1′, C2′:

{{math theorem | name = Lemma | math_statement = Let be <math>\mathfrak M =(\mathcal P, \mathcal Z; \parallel_+, \parallel_-,\in)</math> a finite Minkowski plane, i.e. <math> \left| \mathcal P \right| < \infty </math>. For any pair of cycles <math> z_1 , z_2 </math> and any pair of generators <math> e_1 , e_2 </math> we have:
<math> \left| z_1 \right| = \left| z_2 \right| = \left| e_1 \right| = \left| e_2 \right| </math>.}}

This gives rise of the '''definition''':<br />
For a finite Minkowski plane <math>\mathfrak M</math> and a cycle <math>z</math> of <math>\mathfrak M</math> we call the integer <math> n = \left| z \right| - 1 </math> the '''order''' of <math>{\mathfrak M}</math>.

Simple combinatorial considerations yield

{{math theorem | name = Lemma | math_statement = For a finite Minkowski plane <math> \mathfrak M = (\mathcal P , \mathcal Z ; \parallel_+ , \parallel_- , \in ) </math> the following is true:
{{ordered list | list-style-type = lower-alpha
| Any residue (affine plane) has order <math>n</math>.
| <math> \left| \mathcal P \right| = ( n + 1 ) ^2 </math>,  
| <math> \left| \mathcal Z \right| = ( n + 1) n ( n - 1 ) </math>.
}}}}

==Miquelian Minkowski planes==

We get the most important examples of Minkowski planes by generalizing the classical real model: Just replace <math>\R</math> by an arbitrary [[Field (mathematics)|field]] <math>K</math> then we get ''in any case'' a Minkowski plane <math>{\mathfrak M}(K)=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math>.

Analogously to Möbius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane <math>\mathfrak M (K)</math>.

[[File:Theorem-of-miquel.svg|300px|thumb|Theorem of Miquel]]
'''Theorem (Miquel):''' For the Minkowski plane <math>\mathfrak M (K)</math> the following is true:
: If for any 8 pairwise not parallel points <math>P_1,...,P_8 </math> which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples than the sixth quadruple of points is concyclical, too.

(For a better overview in the figure there are circles drawn instead of hyperbolas.)

'''Theorem (Chen):''' Only a Minkowski plane <math>\mathfrak M (K)</math> satisfies the theorem of Miquel.

Because of the last theorem <math>\mathfrak M(K) </math> is called a '''miquelian Minkowski plane'''.

'''Remark:''' The '''minimal model''' of a Minkowski plane is miquelian.
: It is isomorphic to the Minkowski plane <math>\mathfrak M(K) </math> with <math> K = \operatorname{GF}(2)</math> (field <math>\{0,1\}</math>).

An astonishing result is

'''Theorem (Heise):'''  Any Minkowski plane of ''even'' order is miquelian.

'''Remark:''' A suitable [[Stereographic projection|stereographic projection]] shows: <math>\mathfrak M(K) </math> is isomorphic
to the geometry of the plane sections on a hyperboloid of one sheet ([[Quadric|quadric]] of index 2) in projective 3-space over field <math> K </math>.

'''Remark:''' There are a lot of Minkowski planes that are '''not miquelian''' (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because any [[Quadratic set|quadratic set]] of index 2 in projective 3-space is a quadric (see [[Quadratic set|quadratic set]]).

==See also==
* [[Conformal geometry]]

==References==
<references/>
* W. Benz, ''Vorlesungen über Geomerie der Algebren'', [[Company:Springer Science+Business Media|Springer]] (1973)
* F. Buekenhout (ed.), ''Handbook of [[Incidence (geometry)|Incidence Geometry]]'', [[Company:Elsevier|Elsevier]] (1995) {{isbn|0-444-88355-X}}

==External links==
* [https://encyclopediaofmath.org/wiki/Benz_plane Benz plane] in the ''[[Encyclopedia of Mathematics]]''
*[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note '''''Planar Circle Geometries''''', an Introduction to Moebius-, Laguerre- and Minkowski Planes]

[[Category:Geometry]]

{{Sourceattribution|Minkowski plane|1}}