Klein

Klein received his doctorate from the University of Bonn in 1868, where he had studied mathematics and physics. After teaching at a number of universities he was appointed to a chair at the University of Gvttingen in 1886. He taught at Gvttingen until he retired in 1913.

Klein established a research establishment at Gvttingen which served as a model for the best mathematical research centres. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. Klein brought Hilbert Kvnigsberg to join his research team at Gvttingen.

The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities. He set up a small team of editors who met regularly and made democratic decisions.

Klein's synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programme, profoundly influenced mathematical development.

He is best known for his work in non-euclidean geometry, for his work on the connections between geometry and group theory, and for results in function theory.

Transformations play a major role in modern mathematics and Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. In this way the Erlanger Programme defined geometry so that it included both Euclidean geometry and non-Euclidean geometry.

A Klein bottle is an one-sided closed surface named after Klein. A Klein bottle cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. However this is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible to construct a Klein bottle in non-Euclidean space.

Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912.