# Klein, Christian Felix

### geometry theory euclidean mathematics

(1849–1925) German mathematician: the founder of modern geometry unifying Euclidean and non-Euclidean geometry.

Klein studied at Bonn, Göttingen, Berlin and Paris, and began research in geometry, although he had at first wished to do physics. Work on transformation groups with followed in 1870, and he became professor of mathematics at Erlangen at 23, having just finished service as a medical orderly in the Franco-Prussian war. In his inaugural lecture of 1872 at Erlangen he put forward the audacious ‘ *Erlanger Programm* ’, a unification of mathematics to be achieved by considering each branch of geometry as the theory of invariants of a particular tranformation group. This was well received and influenced his colleagues to unify geometry. During most of his career he held a professorship at Göttingen, and helped to make it a centre for all the exact sciences, as well as mathematics.

Euclidean geometry comes from the metrical transformation, projective geometry from linear transformations, topology from continuous transformations and non-Euclidean geometries from their particular metrics.

Klein developed projective geometry, taking it from three to *n* dimensions and applied group theory widely, for example to the symmetries of regular solids (1884). He invented the Klein bottle in topology, which is a one-sided closed surface with no boundaries (it has no ‘inside’).

Klein also added to number theory and the theory of differential equations, and recast geometry as a part of function theory. In the 1890s Klein and worked out the theory of the gyroscope and produced a standard textbook on it; he was against the tendency of mathematics to become highly abstract and liked engineering applications. He was also uninterested in detailed calculations, which he left to his students.

## User Comments