Tamaschke axiom

In affine geometry , one of the branches of mathematics , the Tamaschke axiom (or triangle axiom ) is one of those statements with the help of which the incidence geometries occurring there can be determined axiomatically . The axiom is named after the mathematician Olaf Tamaschke from Tübingen , who was the first to recognize its importance for geometry.

Formulation of the axiom

The Tamaschke axiom requires the following additional property for incidence geometries that satisfy the connection axiom and the parallel axiom : ${\displaystyle {\mathcal {X}}}$

If there are five spatial points , which should not lie on a common straight line , and if the straight lines and are parallel here , the parallel to through and the parallel to through meet at a common point of intersection . ${\displaystyle {\mathcal {X}}}$ ${\displaystyle A,B,C,A^{*},B^{*}\in {\mathcal {X}}}$ ${\displaystyle A,B,C}$ ${\displaystyle AB}$ ${\displaystyle A^{*}B^{*}}$ ${\displaystyle AC}$ ${\displaystyle A^{*}}$ ${\displaystyle BC}$ ${\displaystyle B^{*}}$ ${\displaystyle C^{*}}$

Axiomatics of affine spaces

According to Albrecht Beutelspacher's presentation , the affine spaces are precisely those incidence geometries in which both

• the axiom of connection

as well as

• the axiom of parallels

as well as

• the Tamaschke axiom

are fulfilled.

Notes and explanations

• The above condition that they should not lie on a common straight line means - clearly! - nothing more than that the points form a triangle . This explains why the Tamaschke axiom is also called the triangle axiom. ${\displaystyle A,B,C}$ ${\displaystyle A,B,C}$
• If one goes to the Analytical Geometry usual way the affine spaces from the corresponding vector spaces of connection vectors to define, then the Tamaschke axiom arises in this context as a theorem .
• The above axioms are not sufficient for an axiomatic justification of affine spatial geometry in the narrower sense. Here one must - not least because of the incidences between planes and straight lines as well as planes and spatial points - create an extended axiomatic.

literature

• Albrecht Beutelspacher: Linear Algebra . An introduction to the science of vectors, maps, and matrices. 8th, updated edition. Springer Spectrum , Wiesbaden 2014, ISBN 978-3-658-02412-3 , doi : 10.1007 / 978-3-658-02413-0 .
• Gerd Fischer : Analytical Geometry . An introduction for new students (=  Vieweg Studium. Basic course in mathematics ). 7th, revised edition. Vieweg Verlag , Braunschweig, Wiesbaden 2001, ISBN 978-3-322-88921-8 , doi : 10.1007 / 978-3-322-88921-8 .
• H. Lenz : Fundamentals of elementary mathematics . 3rd, revised edition. Hanser Verlag , Munich ( inter alia ) 1976, ISBN 3-446-12160-9 ( MR0460009 ).
• Olaf Tamaschke: Projective Geometry II (=  BI university scripts. 838 / a / b). Bibliographisches Institut , Mannheim, Vienna, Zurich 1972 ( MR0338893 ).

Individual evidence

1. ^ Albrecht Beutelspacher: Lineare Algebra. 2014, p. 123ff
2. ↑ ^{a b} Beutelspacher, op.cit., P. 123
3. ↑ Gerd Fischer: Analytical Geometry. 2001, p. 1ff
4. ↑ Beutelspacher, op.cit., P. 126
5. ↑ Hanfried Lenz: Fundamentals of elementary mathematics. 1976, p. 148ff