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
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.
Biographies of mathematicians are from the
Mathematics archive at the University of St. Andrews, and are
used with permission.