Johannes Hjelmslev

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Johannes Trolle Hjelmslev , born Johannes Petersen, (born April 7, 1873 in Hørning ; † February 16, 1950 in Copenhagen ) was a Danish mathematician who dealt with geometry.

He first published under his maiden name J. Petersen and only later called himself Hjelmslev.

life and work

Hjelmslev studied in Copenhagen, where he graduated in 1894 and received his doctorate in 1897. From 1903 he was a private lecturer and from 1905 professor for descriptive geometry at the Technical University (Polyteknisk Laereanstalt) Copenhagen and from 1917 professor at the University of Copenhagen , where he was rector in the academic year 1928/29.

Hjelmslev introduced new geometric construction problems (described in his book "Geometric Experiments") and dealt with the special geometries that result from loosening the axioms of Euclidean geometry.

He is considered the founder of reflection geometry : in 1907 he developed plane absolute geometry from reflection axioms. This was later taken up and further developed by Friedrich Bachmann and others. In 1926, Max Dehn called this axiomatic justification of absolute geometry by Hjelmslev the highest point that modern mathematics has reached beyond Euclid in the justification of elementary geometry.

In work from 1929 he developed a general theory of congruence in which he relaxed further axioms such as the arrangement and incidence. One motive was that he wanted to take into account the vivid "blurring" of points. Along with Dan Barbilian, he became one of the founders of ring geometry (geometry about rings), in Hjelmslev's case using the ring of dual numbers .

With the Hjelmslev transformation named after him, the hyperbolic plane can be mapped into a circle. Hjelmslev groups and planes (in which more than one straight line can pass through two points) are named after him.

In the history of mathematics he researched Georg Mohr (1640–1697), the author of "Euclides Danicus" (Amsterdam 1672), and published it again in 1928.

He is the father of the linguist Louis Hjelmslev (1899–1965), professor in Copenhagen.


See also


  • Gottwald, Ilgauds, Schlote: Lexicon of important mathematicians. Harri Deutsch Verlag, Leipzig 1990, ISBN 3323003195

Individual evidence

  1. For example J. Petersen Géométrie des droites dans l´espace non euclidien , Copenhagen Academy 1900, pp. 305-330
  2. ^ List of rectors on the University of Copenhagen website
  3. ^ Hjelmslev New Justification of Planar Geometry , Mathematische Annalen, Volume 64, 1907, pp. 449–474
  4. Quoted from Friedrich Bachmann Structure of Geometry on the Concept of Reflection , Springer Verlag, 2nd edition 1973, p. XI (foreword)
  5. Hjelmslev Introduction to the General Congruence Theory , Kgl. Dan. Vid. Selsk. Mat. Fys. Medd., 1929-1949
  6. ^ Walter Benz Fundamentals of Geometry , in Gerd Fischer a. a. A Century of Mathematics 1890–1990. Festschrift for the anniversary of the DMV , Vieweg 1990