# Minkowski plane

In mathematics, a **Minkowski plane** (named after Hermann Minkowski) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.

## The classical real Minkowski plane

Applying the pseudo-euclidean distance
on two points
(instead of the euclidean one) we get the geometry of *hyperbolas*, because
a pseudoeuclidean circle is a
hyperbola with midpoint . By a suitable coordinate transformation we can
rewrite the pseudo-euclidean distance as
. Now the hyperbolas have asymptotes parallel
to the coordinate axes. The following completion (see Moebius and
Laguerre planes) *homogenizes* the geometry of hyperbolas:

The incidence structure is called **classical real Minkowski plane.**

The set of points cosists of and two copies of and point .

Any line is completed by point , any hyperbola
by the two points (see figure).

Two points can not be connected by a cycle if and only if
or . We define:

Two points are **(+)-parallel** () if and **(-)-parallel** () if .

Both these relations are equivalence relations on the set of points.

Two points are called **parallel** () if
or .

From the definition above we find:

**Lemma:**

- For any pair of non parallel points there is exactly one point with .
- For any point and any cycle there are exactly two points with .
- For any three points , pairwise non parallel, there is exactly one cycle which contains .
- For any cycle , any point and any point and there exists exactly one cycle such that , i.e.
**touches**at point P.

Like the classical Moebius 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 be an incidence structure with the set of points, the set
of cycles and two equivalence relations ((+)-parallel) and
((-)-parallel) on set .
For we define:
and
.
An equivalence class or is called **(+)-generator**
and **(-)-generator**, respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.)

Two points are called **parallel** () if or .

An incidence structure is called **Minkowski plane** if the following axioms hold:

**C1:**For any pair of non parallel points there is exactly one point with .**C2:**For any point and any cycle there are exactly two points with .**C3:**For any three points , pairwise non parallel, there is exactly one cycle which contains .**C4:**For any cycle , any point and any point and there exists exactly one cycle such that , i.e.**touches**at point P.**C5:**Any cycle contains at least 3 points. There is at least one cycle and a point not in .

For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.

First consequences of the axioms are

**Lemma:** For a Minkowski plane the following is true

- a) Any point is contained in at least one cycle.
- b) Any generator contains at least 3 points.
- c) Two points can be connected by a cycle if and only if they are non parallel.

Analogously to Moebius and Laguerre planes we get the connection to the linear geometry via the residues.

For a Minkowski plane and we define the local structure

and call it the **residue at point P**.

For the classical Minkowski plane is the real affine plane .

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

**Theorem:** For a Minkowski plane any residue is an affine plane.

**Theorem:**
Let be an incidence structure with two equivalence relations and on the set of points (see above).

The **minimal model** of a Minkowski plane can be established over the set
of three elements:

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

**Lemma:**
Let be a finite Minkowski plane, i.e. . For any pair
of cycles and any pair of generators we have:
.

This gives rise of the **definition:**

For a finite Minkowski plane and a cycle of we call the integer the **order** of .

Simple combinatorial considerations yield

**Lemma:**
For a finite Minkowski plane the following is true:

## Miquelian Minkowski planes

We get the most important examples of Minkowski planes by generalizing the
classical real model: Just replace by an arbitrary field
then we get **in any case** a Minkowski plane .

Analogously to Moebius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane .

**Theorem (MIQUEL):** For the Minkowski plane the following is true:

- If for any 8 pairwise not parallel points 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 satisfies the theorem of Miquel.

Because of the last Theorem is called a **miquelian Minkowski plane**.

**Remark:** The **minimal model** of a Minkowski plane is miquelian.

An astonishing result is

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

**Remark:** A suitable stereographic projection shows: is isomorphic
to the geometry of the plane sections on a hyperboloid of one sheet (quadric of index 2) in projective 3-space over field .

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

## References

- W. Benz,
*Vorlesungen über Geomerie der Algebren*, Springer (1973) - F. Buekenhout (ed.),
*Handbook of Incidence Geometry*, Elsevier (1995) ISBN 0-444-88355-X