Veblen-Young axiom
The axiom of Veblen-Young (after Oswald Veblen and John Wesley Young ) is an axiom of projective geometry :
- If the lines AB and CD given by four points A , B , C and D intersect, then the lines AC and BD also intersect .
The Veblen-Young axiom has also been referred to in the literature as the Pasch axiom . The axiom usually named after Moritz Pasch is (e.g. in Hilbert's system of axioms of Euclidean geometry ) an arrangement axiom for affine spaces, whereas the axiom of Veblen-Young is a pure incidence axiom for projective spaces.
Together with the straight line axiom - ("Through two different points there is exactly one straight line.") - and two richness axioms ("Every straight line goes through at least 3 points." And "There are at least 2 different straight lines.") Characterizes the axiom of Veblen -Young a projective incidence geometry of any dimension greater than or equal to 2 in the sense of synthetic geometry .
A consequence of this system of four axioms is that an intersecting pair of lines defines a plane in which the axioms of a projective plane then apply:
- (PE1) For every two points there is exactly one straight line that cuts with both.
- (PE2) For every two straight lines there is exactly one point that intersects with both.
- (PE3) There is a complete quadrilateral , i.e. H. four points, no three of which incise with the same straight line.
A projective geometry that fulfills the following richness condition is at least three-dimensional:
- (D) There is a pair of lines that do not intersect.
It has now been shown that projective planes, which are subspaces of at least three-dimensional projective space, always fulfill the Desargues theorem ( i.e. are Desargues planes) and are therefore isomorphic to a coordinate plane over an inclined body . Therefore, in synthetic geometry, the two- dimensional projective spaces, ie planes , for which numerous non-Desarguessian examples are known, are examined almost exclusively . The historically important axiom of Veblen and Young is hardly used anymore, because the three- and higher-dimensional spaces can essentially be regarded as understood due to their classified oblique coordinates. An axiom system for planes equivalent to the axiom system (PE1), (PE2), (PE3) mentioned is obtained if one uses an at least two-dimensional projective geometry in place of the axiom of Veblen-Young (PE2) for the axioms mentioned, if it is required of any pair of straight lines, excludes that the statement of (D) is valid for the geometry.
literature
- Rudolf Fritzsch: Synthetic embedding of Desarguessian levels in rooms . Mathematical-Physical Semester Reports, No. 21. 1974, p. 237-249 . - (still without full text)
- Jeremy Gray : Worlds out of nothing: a course of the history of geometry of the 19th century . 1st edition. Springer, Berlin, Heidelberg, New York 2007, ISBN 978-0-85729-059-5 .
- Marshall Hall : Projective Planes . In: Transactions of the American Mathematical Society . tape 54 , no. 2 . American Mathematical Society, September 1943, pp. 229-277 , JSTOR : 1990331 .
- David Hilbert : Fundamentals of Geometry . new edition. Teubner, Stuttgart 1999, ISBN 3-519-00237-X . Online copy of the 1903 edition
- David Hilbert : Fundamentals of Geometry . In: Michael Toepell (ed.): Ed. and with appendices and comments. by Michael Toepell . 14th edition. Teubner, Stuttgart 1999, ISBN 3-519-00237-X ( archive.org ).