mathematician and mathematical physicist who unified the various Euclidean
and non-Euclidean geometries. He demonstrated that every individual geometry
could be constructed purely projectively; such projective models are now
called Klein models.
Klein was born in Düsseldorf and studied at Bonn, Göttingen, Berlin, and Paris. He was professor at Erlangen 1872-75, at the Technische Hochschule in Munich 1875-80, and then took up a similar post in Leipzig. From 1886 he was at Göttingen, and helped make it Germany's main centre for all the exact sciences.
Klein announced 1872 what he called the Erlangen Programm on the unification of geometries. He showed that each of the different geometries devised during the 19th century is associated with a separate 'collection', or 'group', of tranformations. Seen in this way, the geometries could all be treated as members of one family.
In his work on number theory, group theory, and the theory of differential equations, Klein was greatly influenced by German mathematician Bernhard Riemann, and redefined a Riemann surface so that it came to be regarded as an essential part of function theory.
Klein initiated 1895 the writing of an encyclopedia of mathematics. He published books on the historical development of mathematics in the 19th century, and a textbook (with his colleague Arnold Sommerfeld) on the theory of the gyroscope.