**Felix Klein** was a German mathematician whose 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 Programm*, profoundly influenced mathematical development.

- The revolution against the Prussians, which resulted in such a dramatic birth for Felix Klein, was completely crushed by the summer of 1849.
- Klein attended the Gymnasium in Düsseldorf.
- Plücker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plücker's interests had become very firmly rooted in geometry.
- Klein received his doctorate, which was supervised by Plücker, from the University of Bonn in 1868, with a dissertation Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische Form Ⓣ(On the transformation of the general equation of the second degree between line-coordinates in a canonical form) on line geometry and its applications to mechanics.
- In his dissertation Klein classified second degree line complexes using Weierstrass's theory of elementary divisors.
- However in the year Klein received his doctorate Plücker died leaving his major work on the foundations of line geometry incomplete.
- Klein was the obvious person to complete the second part of Plücker's Neue Géometrie des Raumes Ⓣ(New space geometry) and this work led him to become acquainted with Clebsch.
- Clebsch had moved to Göttingen in 1868 and, during 1869, Klein made visits to Berlin and Paris and Göttingen.
- In July 1870 Klein was in Paris when Bismarck, the Prussian chancellor, published a provocative message aimed at infuriating the French government.
- France declared war on Prussia on the 19th of July and Klein felt he could no longer remain in Paris and returned.
- Klein was appointed professor at Erlangen, in Bavaria in southern Germany, in 1872.
- He was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day, and so Klein held a chair from the remarkably early age of 23.
- However Klein did not build a school at Erlangen where there were only a few students, so he was pleased to be offered a chair at the Technische Hochschule at Munich in 1875.
- There he, and his colleague Brill, taught advanced courses to large numbers of excellent students and Klein's great talent at teaching was fully expressed.
- Among the students that Klein taught while at Munich were Hurwitz, von Dyck, Rohn, Runge, Planck, Bianchi and Ricci-Curbastro.
- After five years at the Technische Hochschule at Munich, Klein was appointed to a chair of geometry at Leipzig.
- The years 1880 to 1886 that Klein spent at Leipzig were in many ways to fundamentally change his life.
- His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886.
- Klein established a research centre at Göttingen which was to serve as a model for the best mathematical research centres throughout the world.
- Klein brought Hilbert from Königsberg to join his research team at Göttingen in 1895.
- The fame of the journal Mathematische Annalen is based on Klein's mathematical and management abilities.
- The journal was originally founded by Clebsch but only under Klein's management did it first rival, and then surpass in importance, Crelle's journal.
- Klein set up a small team of editors who met regularly and made democratic decisions.
- Klein retired due to ill health in 1913.
- It is a little hard to understand the significance of Klein's contributions to geometry.
- Klein's first important mathematical discoveries were made in 1870 in collaboration with Lie.
- Lie played an important role in Klein's development, introducing him to the group concept which played a major role in his later work.
- It is fair to add that Camille Jordan also played a part in teaching Klein about groups.
- During his time at Göttingen in 1871 Klein made major discoveries regarding geometry.
- Cayley never accepted Klein's ideas believing his arguments to be circular.
- 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 Programm (1872), profoundly influenced mathematical development.
- This was written for the occasion of Klein's inaugural address when he was appointed professor at Erlangen in 1872 although it was not actually the speech he gave on that occasion.
- 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.
- However Klein himself saw his work on function theory as his major contribution to mathematics.
- By considering the action of the modular group on the complex plane, Klein showed that the fundamental region is moved around to tessellate the plane.
- Klein considered equations of degree greater than 4 and was particularly interested in using transcendental methods to solve the general equation of the fifth degree.
- Working under great stress, Klein succeeded in formulating such a theorem and in sketching a strategy for proving it.
- However it was during this work that Klein's health collapsed as mentioned above.
- With Robert Fricke who came to Leipzig in 1884, Klein wrote a major four volume classic on automorphic and elliptic modular functions produced over the following 20 years.
- We should also mention the Klein bottle, a one-sided closed surface named after Klein.
- A Klein bottle cannot be constructed in Euclidean space.
- It is possible to construct a Klein bottle in non-Euclidean space.
- In the 1890s Klein became interested in mathematical physics, although throughout his career he showed he was never far from this area in attitude.
- Later in his career Klein became interested in teaching at school level.
- Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress of 1908.
- Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912.

Born 25 April 1849, Düsseldorf, Prussia (now Germany). Died 22 June 1925, Göttingen, Germany.

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Algebra, Geometry, Group Theory, Origin Germany, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive