Tuft set

The bundle theorem (ger .: bundle theorem) is the simplest case, a statement about 6 circles and 8 points in the Euclidean plane . In its general form, it describes a characteristic characteristic of the ovoid Möbius planes , i.e. i.e., only the ovoid planes under the Möbius planes satisfy this proposition.

One should not confuse the bunch sentence with Miquel's sentence .

As an example of an ovoid Möbius plane, the geometry of the plane sections of an egg-shaped surface (e.g .: sphere, ellipsoid , a surface composed of a hemisphere and a suitable half-ellipsoid, the surface with the equation , …) imagine. If the surface of the egg is a sphere, the result is the spatial model of the classic Möbius plane , the geometry of the circles on the sphere. ${\displaystyle x^{4}+y^{4}+z^{4}=1}$

The essence of an ovoid Möbius plane is the existence of a spatial model by means of an ovoid . An ovoid in a 3-dimensional projective space is a set of points which a) is intersected by a straight line in 0, 1 or 2 points and b) whose tangents cover a plane (tangential plane) at any point. The geometry of the plane sections of an ovoid in a 3-dimensional projective space is called the ovoid Möbius plane. The points of the geometry are the points of the ovoid and the connecting curves (blocks) are the plane sections of the ovoid. A suitable stereographic projection shows that every ovoid Möbius plane has a flat model. In the classic case, this is the geometry of the circles and the straight lines extended by the point . The tuft sentence can thus be interpreted spatially as well as flat. The simple proof is given in the spatial model. ${\displaystyle \infty }$

Möbius level: set of tufts

The applies to every ovoid Möbius plane ${\displaystyle {\mathfrak {M}}}$

Tuft set:

• If there are different points and 5 of the 6 quadruples are concyclic (lie on one cycle) on at least 4 cycles , then the 6th quadruple is also cyclic . ${\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},B_{3},B_{4}}$ ${\displaystyle Q_{ij}:=\{A_{i},B_{i},A_{j},B_{j}\},\ i<j,}$ ${\displaystyle c_{ij}}$

The proof results from the following considerations, which essentially make use of the fact that 3 planes in a 3-dimensional projective space always intersect at one point:

1. The planes through the cycles intersect at one point . So is the intersection of the straight lines (in space!) . ${\displaystyle c_{23},c_{34},c_{24}}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle A_{2}B_{2},\ A_{4}B_{4}}$
2. The planes through the cycles intersect at one point . is also the intersection of the straight lines . ${\displaystyle c_{12},c_{14},c_{24}}$ ${\displaystyle P'}$ ${\displaystyle P'}$ ${\displaystyle A_{2}B_{2},\ A_{4}B_{4}}$

This results in a) and b) also intersect in . The latter means: lie concyclically. The levels involved all contain the point ; i.e., they are in a tuft . ${\displaystyle P=P'}$ ${\displaystyle A_{1}B_{1},\ A_{3}B_{3}}$ ${\displaystyle P}$ ${\displaystyle A_{1},B_{1},A_{3},B_{3}}$ ${\displaystyle P}$

The Büschelsatz derives its great significance from the following statement, which was assumed for a long time and proved by Jeff Kahn in 1980 :

Kahn's theorem: A Möbius plane is ovoidal if and only if it satisfies the tuft theorem.

The Büschels theorem has the same meaning for Möbius planes as the Desargues theorem for projective planes . With the help of the tuft theorem, a) an oblique body and b) an ovoid can be constructed. If the stricter Miquel theorem applies, the oblique body is even commutative (body) and the ovoid a quadric. So: the bunch sentence follows from Miquel's sentence but not the other way around.

Note: There are Möbius planes that are not ovoid.

Note: There is also a tuft sentence with the same meaning for ovoid Laguerre planes .

Individual evidence

1. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 63.
2. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 61.
3. ↑ Inversive plan satisfying the bundle theorem , Journal Combinatorial Theory, Series A, Vol. 29, 1980, pp 1-19
4. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 62.
5. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 64.
6. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 78.

literature

• W. Benz, Lectures on the Geometry of Algebras , Springer (1973)
• P. Dembowski, Finite Geometries , Springer-Verlag (1968) ISBN 3-540-61786-8 , p. 256