Moulton Plain

Straight lines in the Moulton Plain

The Moulton plane is a frequently used example of an affine plane in which Desargues' theorem does not apply, i.e. a non- Desargue plane . Their coordinates thus provide an example of a ternary body that is not a sloping body. It was first described in 1902 by the American astronomer Forest Ray Moulton and later named after him.

The points of the Moulton plane are the normal points of the real plane and the straight lines are the normal straight lines of the real plane with the exception that straight lines with a negative slope have a kink on the Y-axis, i.e. H. when passing the Y-axis, its slope changes: In the right half-plane it is twice as large as in the left half-plane.

Formal definition

We define the incidence structure as follows , where the set of points, the set of straight lines and the incidence relation denote "lies on": ${\displaystyle {\mathfrak {M}}=\langle P,G,{\textrm {I}}\rangle }$ ${\displaystyle P}$ ${\displaystyle G}$ ${\displaystyle {\textrm {I}}}$

${\displaystyle P:=\mathbb {R} ^{2}}$

${\displaystyle G:=(\mathbb {R} \cup \{\infty \})\times \mathbb {R} ,}$

where is just a formal symbol . ${\displaystyle \infty }$ ${\displaystyle \not \in \mathbb {R} }$

The incidence relation is defined for and (see straight line equation ) by ${\displaystyle p=(x,y)\in P}$ ${\displaystyle g=(m,b)\in G}$

${\displaystyle p\,{\textrm {I}}\,g\iff {\begin{cases}x=b&{\textrm {falls}}\,m=\infty \\y=mx+b&{\textrm {falls}}\,m\geq 0\\y=mx+b&{\textrm {falls}}\,m\leq 0,x\leq 0\\y=2mx+b&{\textrm {falls}}\,m\leq 0,x\geq 0.\end{cases}}}$

One can easily prove that this incidence structure fulfills the axioms of an affine plane, in particular that exactly one straight line passes through two different points and that there is exactly one parallel to a straight line through a given point.

Invalidity of Desargues theorem

Desargues constellation in the ordinary plane

One starts with a Desargues constellation of ten points and ten straight lines in the usual Euclidean plane as in the adjacent figure and places it in such a way that the only one of the ten points has a negative coordinate and only one of the three straight lines has a negative slope (In the picture: the straight line ). If one now goes over to the Moulton Plain, all incidences are retained except for at , i. H. the (Moulton) straight lines , and do not all intersect at one point. Thus Desargues' theorem has no general validity in the Moulton plane. ${\displaystyle X}$ ${\displaystyle x}$ ${\displaystyle X}$ ${\displaystyle XBC}$ ${\displaystyle X}$ ${\displaystyle BC}$ ${\displaystyle B'C'}$ ${\displaystyle YZ}$

Applications

The existence of the Moulton plane proves that there are non-Desarguean affine planes and even that there are affine planes that are not affine translation planes . Since one can construct an associated projective plane for each affine plane (the projective closure ), the existence of non-disargue projective planes is secured and even the existence of projective planes that are not moufang planes . Since Desargues' theorem applies to the following: Not all projective planes can be described with the help of the canonical construction of 3-dimensional (left) vector spaces over a (oblique) body . ${\displaystyle PG(2,K)}$ ${\displaystyle K}$

Generalizations

Real type moulton planes

Analogous to the real Moulton plane described in this article , one can define affine planes starting from any ordered body by modifying the multiplication as for the Moulton plane. This generalization is described in the article Cartesian Group .

Finite Moulton Plains

A quasi-body can be obtained from certain finite fields by modifying the multiplication . The affine plane over such a quasi-body is called the finite Moulton plane according to Pierce and Pickert . They are always finite affine translation planes . The algebraic structure of their coordinate areas is described in more detail in the article Quasi -Bodies.

literature

• Forest Ray Moulton : A simple non-desarguesian plane geometry . In: Trans. Amer. Math. Soc . tape 3 , 1902, pp. 192-195 .
• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry . From the basics to the applications (=  Vieweg course; 41: advanced course in mathematics ). 1st edition. Vieweg, Wiesbaden 1992, ISBN 3-528-07241-5 , pp. 70-71 .
• WA Pierce: Moulton Planes . In: Canadian J. Math. Band 13 , 1961, pp. 427-436 .
• Günter Pickert : Geometric identification of a class of finite Moulton planes . In: Journal for Pure and Applied Mathematics (Crelles Journal) . tape 1964 , no. 214-215 , 1964, ISSN  1435-5345 , pp. 405-411 , doi : 10.1515 / crll.1964.214-215.405 .

Individual evidence

1. ^ Moulton (1902)
2. ↑ ^{a b} Pierce (1961)
3. ↑ Pickert (1964)