Hjelmslev's theorem

The set of Hjelmslev (also Hjelmslevscher centerline set called, in the English literature as Hjelmslev's theorem known) is a set of plane geometry , which is based on the Danish mathematician Johannes Hjelmslev back (1873-1950). Hjelmslev formulates this proposition in the context of his famous treatise on a New Justification of Plane Geometry , in which he shows that plane geometry can be built up using plane axioms only, without considering continuity, completely independently of the question of parallels . The investigations into the plane congruence mappings made in § 2 of the treatise ( congruence and symmetry ) culminate in Hjelmslev's theorem , which deals with a fundamental property of these congruence maps.

Formulation of the sentence

The connected red points are image-archetype pairs of a congruence, the green center points lie on a straight line.

In the Euclidean plane a congruence mapping and two straight lines and with are given . ${\displaystyle \phi }$ ${\displaystyle g}$ ${\displaystyle g'}$ ${\displaystyle g'=\phi (g)}$

For each point and its image point, let it be the center of the line . ${\displaystyle P\in g}$ ${\displaystyle P'=\phi (P)\in g'}$ ${\displaystyle M(P)}$ ${\displaystyle {\overline {PP'}}}$

Then:

Either

the centers are all different in pairs and form a single straight line ${\displaystyle M(P)}$

or

the centers coincide into a single point. ${\displaystyle M(P)}$

literature

Original work

• J. Hjelmslev : New justification of the plane geometry . In: Math. Ann . 64, 1907, pp. 449-474.
• Frank Löbell : Hjelmslev's middle line sentence and related sentences . In: Monthly books for mathematics . 65, 1961, pp. 249-251.

Monographs

• Friedrich Bachmann : plane reflection geometry. A lecture on Hjelmslev groups . BI-Wissenschafts-Verlag, Mannheim [u. a.] 1989, ISBN 3-411-03219-7 .
• HSM Coxeter : Immortal Geometry . Translated into German by JJ Burckhardt (=  Science and Culture . Volume 17 ). Birkhäuser Verlag, Basel / Stuttgart 1963, p. 54 ( MR0692941 ).
• Helmut Karzel , Hans-Joachim Kroll: History of Geometry since Hilbert . Scientific Book Society, Darmstadt 1988, ISBN 3-534-08524-8 .
• D. Pedoe : A Course of Geometry for Colleges and Universities . Cambridge University Press, Cambridge 1970, ISBN 0-521-07638-2 .

Web links

• Jay Warendorff: Animation for the sentence by Hjelmslev
• Frank Löbell: Link to the treatise

Individual evidence

1. ↑ Bachmann: p. 79.
2. ^ Coxeter: p. 69.
3. ↑ Löbell: The Hjelmslev's middle line sentence and related sentences. In: monthly Math. Band 65 , p. 249 ff .
4. ↑ Pedoe: p. 195.
5. ↑ Hjelmslev: New justification of the plane geometry . In: Math. Ann. tape 64 , p. 449 ff .
6. ↑ In modern terminology, for example in Karzel, Kroll: History of Geometry since Hilbert. Pp. 160 ff., We are talking about Hjelmslev's justification of plane absolute geometry with half turns . Karzel / Kroll raise with regard to this treatise by Hjelmslev shows that the Hjelmslev rule methods for the development of geometry were of great significance
7. ↑ Hjelmslev: New justification of the plane geometry. In: Math. Ann. tape 64 , p. 459 .