Möbius level

A Möbius plane , named after August Ferdinand Möbius , is in the classic case an incidence structure that essentially describes the geometry of the straight lines and circles in the real viewing plane: A straight line is clearly defined by 2 points, a circle by 3 points. A straight line or circle intersects / touches a circle in 0, 1 or 2 points. In order to cancel the special role of the straight line, a common new point is added to each straight line and called circles and the straight lines expanded in this way . The new incidence structure now has the simpler properties: ${\displaystyle \infty }$

Möbius level: contact relation (A2)

• (A1): For every 3 points there is exactly one cycle that contains them. (If the points lie on a straight line , then the extended straight line is the cycle sought; if they do not lie on a straight line, there is exactly one circle through .) ${\displaystyle A,B,C}$ ${\displaystyle z}$ ${\displaystyle g}$ ${\displaystyle \infty }$ ${\displaystyle A,B,C}$
• (A2): To a Cycles , a point on and a point not on , there is exactly one Cycles through and which in touching. (In order to prove this property, you have to play through the various possibilities for and , which is possible without any effort (see picture).) ${\displaystyle z}$ ${\displaystyle P}$ ${\displaystyle z}$ ${\displaystyle Q}$ ${\displaystyle z}$ ${\displaystyle z^{\prime }}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle z}$ ${\displaystyle P}$ ${\displaystyle z,P}$ ${\displaystyle Q}$

However, it is not to be expected that the geometry of the extended straight lines and circles described here is the only incidence structure that has the properties (A1), (A2). If the real numbers are replaced by the rational numbers , (A1), (A2) remain valid. However, the validity of (A1), (A2) is lost when using the complex numbers (instead of the real ones). That means, only the use of certain number fields (see below) receives the properties (A1), (A2). ${\displaystyle \mathbb {Q} }$

In addition to the formally inhomogeneous model (there are straight lines and circles), a homogeneous spatial model is obtained with the help of the inversion of a suitable stereographic projection : The points of the new incidence structure are the points on the surface of the sphere and the cycles are the circles on the sphere. The classic real Möbius plane can also be understood as the geometry of the plane cuts (circles) on a sphere. The proof of (A1) and (A2) does not require any annoying case distinctions in the spatial model.

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

The axioms of a Möbius plane

Based on the incidence properties (A1), (A2) of the classical real Möbius plane, one defines:

Möbius plane: axioms (A1), (A2)

An incidence structure with the set of points and the set of cycles is called a Möbius plane if the following axioms are fulfilled: ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {Z}}}$

(A1): For every 3 points there is exactly one cycle that contains. ${\displaystyle A,B,C}$ ${\displaystyle z}$ ${\displaystyle A,B,C}$

(A2): ( Axiom of touch ) For any cycle and any two points , there is exactly one cycle through which in touches, i.e. H. with . ${\displaystyle z}$ ${\displaystyle P\in z}$ ${\displaystyle Q\notin z}$ ${\displaystyle z'}$ ${\displaystyle P,Q,}$ ${\displaystyle z}$ ${\displaystyle P}$ ${\displaystyle z\cap z'=\{P\}}$

(A3): Each cycle contains at least 3 points. There is at least one cycle.

Four points are called concyclic if there is a cycle that contains. ${\displaystyle A,B,C,D}$ ${\displaystyle z}$ ${\displaystyle A,B,C,D}$

Möbius level: minimal model (only the cycles through are drawn. 3 points each form a cycle.) ${\displaystyle \infty }$

As already mentioned above, not only does the classical real Möbius plane fulfill the axioms (A1), (A2), (A3). There are many examples of Möbius levels that differ from the classic model (see below). Similar to the minimal model of an affine or projective plane, there is also a minimal model of a Möbius plane. It consists of 5 points:

${\displaystyle {\mathcal {P}}:=\{A,B,C,D,\infty \},\quad {\mathcal {Z}}:=\{z\mid z\subset {\mathcal {P}},|z|=3\}}$

So is . ${\displaystyle |{\mathcal {Z}}|={5 \choose 3}=10}$

The close relationship between the classic Möbius plane and the real affine plane can also be seen between the minimal model of a Möbius plane and the minimal model of an affine plane. This close relationship is even typical of Möbius levels:

For a Möbius plane and we define the incidence structure and call it derivative at P . ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathfrak {A}}_{P}:=({\mathcal {P}}\setminus \{P\},\{z\setminus \{P\}|P\in z\in {\mathcal {Z}}\},\in )}$

In the classical model, the derivative at the point is the underlying real affine plane (see below). The great importance of a derivation in one point consists in the statement that is easy to prove: ${\displaystyle {\mathfrak {A}}_{\infty }}$ ${\displaystyle \infty }$

• Every derivative of a Möbius plane is an affine plane.

This property allows the use of many results about affine planes and is also the reason for an alternative definition of a Möbius plane:

Theorem: An incidence structure is a Möbius plane if and only if: ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

A ': For each point the derivative is an affine plane. ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathfrak {A}}_{P}}$

For finite Möbius planes, i.e. H. , applies (similar to affine planes): ${\displaystyle |{\mathcal {P}}|<\infty }$

• Every two cycles contain the same number of points.

This property gives rise to the following definition:

• For a finite Möbius plane and a cycle , the natural number is called the order of . ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle z\in {\mathcal {Z}}}$ ${\displaystyle n:=|z|-1}$ ${\displaystyle {\mathfrak {M}}}$

From combinatorial considerations we get:

• For a finite Möbius plane of order, the following applies: ${\displaystyle {\mathfrak {M}}=({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle n}$

a) Every derivative is an affine plane of order ${\displaystyle {\mathfrak {A}}_{P}}$ ${\displaystyle n.}$

b) ${\displaystyle |{\mathcal {P}}|=n^{2}+1}$

c) ${\displaystyle |{\mathcal {Z}}|=n(n^{2}+1)}$

The classic real Möbius plane

Classic Möbius level: 2D / 3D model

We start from the real affine plane and get the real Euclidean plane with the square form : is the set of points, straight lines are described by equations or and a circle is a set of points that form an equation ${\displaystyle {\mathfrak {A}}(\mathbb {R} )}$ ${\displaystyle \rho (x,y)=x^{2}+y^{2}}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle y=mx+b}$ ${\displaystyle x=c}$

${\displaystyle \rho (x-x_{0},y-y_{0})=(x-x_{0})^{2}+(y-y_{0})^{2}=r^{2},\ r>0}$

Fulfills.

The geometry of straight lines and circles may be homogenized (similar to the extension of an affine plane to a projective plane) by embedding in the incidence structure it with ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

${\displaystyle {\mathcal {P}}:=\mathbb {R} ^{2}\cup \{\infty \},\infty \notin \mathbb {R} }$as a set of points and

${\displaystyle {\mathcal {Z}}:=\{g\cup \{\infty \}\mid g{\text{ Gerade von }}{\mathfrak {A}}(\mathbb {R} )\}\cup \{k\mid k{\text{ Kreis von }}{\mathfrak {A}}(\mathbb {R} )\}}$as a set of cycles.

${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$is called the classic real Möbius plane.

Within the new incidence structure, the extended straight lines no longer play a special role geometrically and fulfill axioms (A1) and (A2). ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

The usual description of the real plane by complex numbers (z now does not denote cycles!)

${\displaystyle z=x+iy\ \leftrightarrow \ (x,y)\in \mathbb {R} ^{2}}$

provides the following description of ( is the complex number to be conjugated): ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle {\overline {z}}=x-iy}$ ${\displaystyle z}$

${\displaystyle {\mathcal {P}}:=\mathbb {C} \cup \{\infty \}}$

${\displaystyle {\mathcal {Z}}:=\{\{z\in \mathbb {C} \mid az+{\overline {az}}+b=0\}\ \cup \ \{\infty \}\mid 0\neq a\in \mathbb {C} ,b\in \mathbb {R} \}\ \cup \ \{\{z\in \mathbb {C} \mid (z-z_{0}){\overline {(z-z_{0})}}=d\}\mid z_{0}\in \mathbb {C} ,d\in \mathbb {R} ,d>0\}}$

The great advantage of this description is the simple possibility of specifying automorphisms (permutations of that map cycles to cycles). The following figures are automorphisms of ${\displaystyle {\mathcal {P}}}$ ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in ):}$

(1) with (twist extension) ${\displaystyle \ z\mapsto rz,\ \infty \mapsto \infty ,\ }$ ${\displaystyle r\in \mathbb {C} \ }$

(2) with (Translation) ${\displaystyle \ z\mapsto z+s,\ \infty \mapsto \infty ,\ }$ ${\displaystyle s\in \mathbb {C} \ }$

(3) (reflection on ) ${\displaystyle \ z\mapsto \displaystyle {\tfrac {1}{z}},\ z\neq 0,\ 0\mapsto \infty ,\ \infty \mapsto 0\ }$ ${\displaystyle \pm 1}$

(4) (reflection on the real axis) ${\displaystyle \ z\mapsto {\overline {z}},\ \infty \mapsto \infty \ }$

If one considers the projective straight line over the complex numbers , one recognizes that the images (1) - (3) generate the group of Möbius transformations . So the geometry is a very homogeneous structure. For example, the real axis can be mapped onto any other cycle with an automorphism. Together with the figure (4) results: ${\displaystyle \mathbb {C} \cup \{\infty \}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle PGL(2,\mathbb {C} )}$ ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

At every cycle there is a reflection, also called an inversion . For example: the inversion is on the unit circle . This property justifies the name inversive plane, which is commonly used in English literature . ${\displaystyle z\mapsto 1/{\overline {z}}}$ ${\displaystyle z{\overline {z}}=1}$

Stereographic projection

Similar to the spatial model of a projective plane, there is also a spatial model of the classic Möbius plane that eliminates the formal difference between circles and extended straight lines: the geometry is isomorphic to the geometry of the circles on a sphere. The associated isomorphism is provided by a suitable stereographic projection . For example: ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$ ${\displaystyle ({\mathcal {P}},{\mathcal {Z}},\in )}$

${\displaystyle \Phi :(x,y)\mapsto \left({\frac {x}{1+x^{2}+y^{2}}},{\frac {y}{1+x^{2}+y^{2}}},{\frac {x^{2}+y^{2}}{1+x^{2}+y^{2}}}\right)=(u,v,w)}$

${\displaystyle \Phi }$projected from the point of ${\displaystyle (0,0,1)}$

• the xy plane on the sphere with the equation . This sphere has the center and the radius . ${\displaystyle u^{2}+v^{2}+w^{2}-w=0}$ ${\displaystyle (0,0,{\tfrac {1}{2}})}$ ${\displaystyle r={\tfrac {1}{2}}}$
• the circle with the equation in the plane . This means that the image of the circle is a plane section with the sphere and thus again a circle (on the sphere). So the figure is true to the circle. The county levels all go not through the center of projection . ${\displaystyle x^{2}+y^{2}-ax-by-c=0}$ ${\displaystyle au+bv-(1+c)w+c=0}$ ${\displaystyle \Phi }$ ${\displaystyle (0,0,1)}$
• the straight line into the plane . That is, a straight line is mapped onto a spherical circle reduced by the point in a plane through the projection center . ${\displaystyle ax+by+c=0}$ ${\displaystyle au+bv-cw+c=0}$ ${\displaystyle (0,0,1)}$ ${\displaystyle (0,0,1)}$

For the inverse mapping (from the sphere dotted in N to the xy plane) applies:

${\displaystyle \Phi ^{-1}:(u,v,w)\mapsto ({\frac {u}{1-w}},{\frac {v}{1-w}})=(x,y)}$

Miquelsche Möbius planes

When looking for other examples of a Möbius plane it is worth to generalize the classical model: We go from one affine plane over a body and a suitable square shape on out to define circles. But simply replacing the real numbers with an arbitrary field and keeping the classic square shape to describe the circles doesn't always work. Möbius planes are only obtained for suitable pairs of solids and square shapes . These are characterized (like the classical model) by a high degree of homogeneity (many automorphisms) and the following Miquel theorem . ${\displaystyle {\mathfrak {A}}(K)}$ ${\displaystyle K}$ ${\displaystyle \rho }$ ${\displaystyle {\mathfrak {A}}(K)}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle K}$ ${\displaystyle x^{2}+y^{2}}$ ${\displaystyle {\mathfrak {M}}(K,\rho )}$

Miquel's theorem

Sentence ( MIQUEL ):

The following applies to a Möbius level : ${\displaystyle {\mathfrak {M}}(K,\rho )}$

If for any 8 points that can be assigned to the corners of a cube that 4 points with 5 faces each lie on a circle, this is also the case for the 4 points on the 6th face (see picture). ${\displaystyle P_{1},\ldots ,P_{8}}$

The strength of this closing figure is shown in the validity of the reverse of Miquel's theorem:

Sentence ( CHEN ):

Only Möbius planes of the form fulfill Miquel's theorem. ${\displaystyle {\mathfrak {M}}(K,\rho )}$

Because of the last sentence, a Möbius plane is called Miquelsch. ${\displaystyle {\mathfrak {M}}(K,\rho )}$

Note: The minimal model of a Möbius plane is miquelian. It is isomorphic to the Möbius plane

${\displaystyle {\mathfrak {M}}(K,\rho )}$with (body ) and ${\displaystyle K=GF(2)}$ ${\displaystyle \{0,1\}}$ ${\displaystyle \rho (x,y)=x^{2}+xy+y^{2}.}$

(E.g. describes the point set .) ${\displaystyle x^{2}+xy+y^{2}=1}$ ${\displaystyle \{(0,1),(1,0),(1,1)\}}$

Comment:

In the case is also suitable. The “circles” here are ellipses . ${\displaystyle K=\mathbb {R} }$ ${\displaystyle \rho (x,y)=x^{2}+2y^{2}}$

In the case (the field of rational numbers) is suitable. ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle \rho (x,y)=x^{2}+y^{2}}$

In the case is also suitable. ${\displaystyle K=\mathbb {Q} }$ ${\displaystyle \rho (x,y)=x^{2}-2y^{2}}$

If the field is complex numbers, then there is no suitable quadratic form at all . ${\displaystyle K=\mathbb {C} }$

Comment:

1. A stereographic projection shows: is isomorphic to the geometry of the plane sections of a sphere (projective quadric of index 1) in the 3-dimensional projective space above the body ${\displaystyle {\mathfrak {M}}(K,\rho )}$ ${\displaystyle K.}$
2. A Miquelian Möbius plane can, analogously to the classical real case, always be described as a projective straight line over an extension field of . ${\displaystyle {\mathfrak {M}}(K,\rho )}$ ${\displaystyle K}$

Remark: In the classical case, Miquel's theorem can be proven with elementary means (circular square), see Sect. Miquel's theorem .

Ovoid Möbius planes

Möbius level: set of tufts

There are many Möbius levels that are not miquelian (see web link). A large class of Möbius planes that includes the Miquelian form the ovoid Möbius planes. An ovoid Möbius plane is the geometry of the plane cuts on an ovoid . An ovoid is a square set and has the same geometric properties as a sphere in real 3-dimensional space: 1) A straight line meets an ovoid in 0, 1 or 2 points. 2) The set of tangents at a point covers a plane (the tangent plane at this point). In real 3-dimensional space one can e.g. B. glue a hemisphere in a suitable manner smoothly with one half of an ellipsoid in order to obtain an ovoid that is not a quadric. Even in the finite case there are ovoids that are not quadrics (see quadratic set). For the class of ovoidalen Mobius levels, there is a set of the Miquel similar lock set, the bundle theorem (ger .: Bundle Theorem ). It characterizes the ovoid Möbius planes. Miquel's theorem and the Büschels theorem have a similar meaning for Möbius planes as the Pappos and Desargues theorems have for projective planes.

Classification of the Möbius planes

In 1965 Christoph Hering published a classification of the Möbius planes analogous to the classification of the projective planes ( Lenz-Barlotti classification ), which is based on the richness of the respective automorphism group.

Individual evidence

1. ^ P. Dembowski: Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 , p. 252.
2. ↑ Erich Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 60.
3. ↑ Erich Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 50.
4. ↑ Yi Chen: Miquel's theorem in the Möbius plane. Math. Annalen 186, (1970), pp. 81-100.
5. ↑ Erich Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 59.
6. ↑ Erich Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 51.
7. ↑ ^{a b} Erich Hartmann: Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. Script, TH Darmstadt (PDF; 891 kB), p. 61.
8. ^ CH Hering: A classification of the Möbius levels. Math. Z., 87 (1965), pp. 252-262.
9. ^ P. Dembowski: Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 , p. 261.

literature

• W. Benz: Lectures on the geometry of algebras. Springer , 1973, p. 1.
• Albrecht Beutelspacher : Introduction to Finite Geometry. Vol. 2: Projective rooms. Bibliographisches Institut, Mannheim u. a. 1983, ISBN 3-411-01648-5 , p. 116.
• F. Buekenhout (Ed.): Handbook of Incidence Geometry. Elsevier , 1995, ISBN 0-444-88355-X , p. 1336.
• P. Dembowski: Finite Geometries. Springer-Verlag, 1968, ISBN 3-540-61786-8 , pp. 252, 261.

Web links

• Entry in Springer's Encyclopedia of Mathematics