Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents; his father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. Klein's mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a Physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, during 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn during 1868.
Plücker died during 1868, leaving his book concerning the basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had relocated to Göttingen during 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. During July 1870, at the beginning of the Franco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen during early 1871.
Klein's first important mathematical discoveries were made during 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan.
During 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley-Klein metric. This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular.
Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen Program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.
During 1875 Klein married Anne Hegel, the granddaughter of the Philosopher Georg Wilhelm Friedrich Hegel.
Klein showed that the modular group moves the fundamental region of the complex plane so as to tessellate that plane. During 1879, he examined the action of PSL(2,7), considered as an image of the modular group, and obtained an explicit representation of a Riemann surface now termed the Klein quartic. He showed that that surface was a curve in projective space, that its equation was xy + yz + zx = 0, and that its group of symmetries was PSL(2,7) of order 168. His Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale (1882) treats complex analysis in a geometric way, connecting potential theory and conformal mappings. This work drew on notions from fluid dynamics.
After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. There his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. During 1882, his health collapsed; during 1883–1884, he was plagued by depression. Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published during 1886 and 1888.
In his 1884 book on the icosahedron, Klein established a theory of automorphic functions, associating algebra and geometry. However Poincaré published an outline of his theory of automorphic functions during 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. But while doing this work his health decreased, as mentioned above.
Klein accepted a professorship at the University of Göttingen during 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's main mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own primacy as a developer of geometry. At Göttingen, he taught a variety of courses, mainly concerning the interface between mathematics and physics, such as mechanics and potential theory.
During the 1890s, Klein began studying mathematical physics more intensively, writing on the gyroscope with Arnold Sommerfeld. During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the Enzyklopädie der mathematischen Wissenschaften. This enterprise, which endured until 1935, provided an important standard reference of enduring value.
The London Mathematical Society awarded Klein its De Morgan Medal during 1893. He was elected a member of the Royal Society during 1885, and was awarded its Copley Medal during 1912. He retired the next year due to ill health, but continued to teach mathematics at his home for some years more.
The research facility Klein established at Göttingen served as a model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. During 1895, Klein hired David Hilbert away from the University of Königsberg; this appointment proved fateful, because Hilbert continued Göttingen's good reputation until his own retirement during 1932.
About 1900, Klein began to become interested in mathematical instruction in schools. During 1905, he was decisive in formulating a plan recommending that analytic geometry, the rudiments of differential and integral calculus, and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. During 1908, Klein was elected President of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians. With his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany.
