Felix Klein 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. He was born on 25/4/1849 and delighted in pointing out that each of the day (52), month (22), and year (432) was the square of a prime.
Klein's father was secretary to the head of the government. There is a colourful description of Felix's birth in his obituary in the Proceedings of the Royal Society:-
Without, the cannon thundered on the barricades raised by the insurgent Rhinelanders against their hated Prussian rulers. Within, although all had been prepared for flight, there was no thought of departure; on that night was born a son to the stern Prussian secretary. That son was Felix Klein.
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. After graduating, he entered the University of Bonn and studied mathematics and physics there during 1865-1866. He started out on his career with the intention of becoming a physicist. While still studying at the University of Bonn, he was appointed to the post of laboratory assistant to Plücker held a chair of mathematics and experimental physics at Bonn but, by the time Klein became his assistant, Plücker, from the University of Bonn in 1868, with a dissertation Uber die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien- Koordinaten auf eine kanonische Form on line geometry and its applications to mechanics. In his dissertation Klein classified second degree line complexes using Weierstrass died leaving his major work on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker. 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. Then, for a short period, he did military service as a medical orderly before being appointed as a lecturer at Göttingen in early 1871.
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, Rohn, Planck and Ricci-Curbastro. Also in 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. Grace Chisholm Young recalls in :-
... the sunny days when the tall handsome young professor wooed and won the lovely granddaughter of the philosopher Hegel.
After five years at the Technische Hochschule at Munich, Klein was appointed to a chair of geometry at Leipzig. There he had as colleagues a number of talented young lecturers, including von Dyck and Engel. The years 1880 to 1886 that Klein spent at Leipzig were in many ways to fundamentally change his life. As D E Rowe writes in :-
Leipzig seemed to be a superb outpost for building the kind of school he now had in mind: one that would draw heavily on the abundant riches offered by Riemann's geometric approach to function theory. But unforeseen events and his always delicate health conspired against this plan. .. [In him were] two souls ... one longing for the tranquil scholar's life, the other for the active life of an editor, teacher, and scientific organiser. ... It was during the autumn of 1882 that the first of these two worlds came crashing down upon him ... his health collapsed completely, and throughout the years 1883-1884 he was plagued by depression.
His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886. He taught at Göttingen until he retired in 1913 but he now sought to re-establish Göttingen as the foremost mathematics research centre in the world. His own role as the leader of a geometrical school at Leipzig was never transferred to Göttingen. At Göttingen he taught a wide variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory.
Klein established a research centre at Göttingen which was to serve as a model for the best mathematical research centres throughout the world. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. 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 and the followers of Clebsch who supported the Mathematische Annalen. Klein set up a small team of editors who met regularly and made democratic decisions. The journal specialised in complex analysis, algebraic geometry and invariant theory. It also provided an important outlet for real analysis and the new area of group theory.
Klein retired due to ill health in 1913. However he continued to teach mathematics at his home during the years of World War I.
It is a little hard to understand the significance of Klein's contributions to geometry. This is not because it is strange to us today, quite the reverse, it has become so much a part of our present mathematical thinking that it is hard for us to realise the novelty of his results and also the fact that they were not universally accepted by all his contemporaries.
Klein's first important mathematical discoveries were made in 1870 in collaboration with Lie surface. Further collaboration with 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. He published two papers On the So-called Non-Euclidean Geometry in which he showed that it was possible to consider euclidean geometry and non-euclidean geometry as special cases a projective surface with a specific conic section adjoined. This had the remarkable corollary that non-euclidean geometry was consistent if and only if euclidean geometry was consistent. The fact that non-euclidean geometry was at the time still a controversial topic now vanished. Its status was put on an identical footing to euclidean 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 Programme (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. The Erlanger Programme gave a unified approach to geometry which is now the standard accepted view.
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.
However Klein himself saw his work on function theory as his major contribution to mathematics. As W Burau and B Schoenberg write in :-
Klein considered his work in function theory to be the summit of his work in mathematics. He owed some of his greatest successes to his development of Riemann's ideas and to the intimate alliance he forged between the later and the conception of invariant theory, of number theory and algebra, of group theory, and of multidimensional geometry and the theory of differential equations, especially in his own fields, elliptic modular functions and automorphic functions.
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. In 1879 he looked at the action of PSL(2,7), thought of as an image of the modular group, and obtained an explicit representation of a Riemann surface. He showed it had equation x3y + y3z + z3x = 0 as a curve in projective space and its group of symmetries was PSL(2,7) of order 168. He wrote Riemanns Theorie der algebraischen Funktionen und ihre Integrals in 1882 which treats function theory in a geometric way connecting potential theory and conformal mappings. He also used physical ideas in this work, especially those of fluid dynamics.
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. After building on methods due to Hermite, producing similar results to Brioschi, he went on to completely solve the problem using the group of the icosahedron. This work led him to consider elliptic modular functions which he studied in a series of papers.
He developed a theory of automorphic functions, connecting algebraic and geometric results in his important 1884 book on the icosahedron. However Poincaré began publishing an outline of his theory of automorphic functions in 1881 and, as explained in , this led to a competition between the two:-
Klein initiated a correspondence with Poincaré, and soon a friendly rivalry ensued as both sought to formulate and prove a grand uniformization theorem that would serve as a capstone to this theory. 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 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.
In the 1890s Klein became interested in mathematical physics, although throughout his career he showed he was never far from this area in attitude. Following from this interest, he wrote an important work on the gyroscope with A Sommerfeld.
Later in his career Klein became interested in teaching at school level. W Burau and B Schoenberg write in :-
Starting in 1900 he began to take a lively interest in mathematical instruction below university level while continuing to pursue his academic functions. An advocate of modernizing mathematics instruction in Germany, in 1905 he played a decisive role in formulating the "Meraner Lehrplanestwürfe". The essential change recommended was the introduction in secondary schools of the rudiments of differential and integral calculus and the function concept.
Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress of 1908. Under his guidance the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.
Another project he worked on around the turn of the century was the Encyklopädie der Mathematischen Wissenschaften. He took an active part in this project, editing with K Müller the four volume section on mechanics.
Grace Chisholm Young writes in  of Klein's efforts on behalf of women in mathematics:-
When in  he and others succeeded in opening the doors of the University of Göttingen to women, it was, I think, a real pleasure to him that the first woman to take the degree of D.Phil. should do so under his auspices, and should be a Girton girl who had sat at the feet of his revered friend Cayley.
Klein was elected a member of the Royal Society in 1885 and received the Copley medal of the Society in 1912. The London Mathematical Society awarded him their De Morgan Medal in 1893.
Chisholm Young writes in :-
[His mind] teemed with ideas and brilliant reflections, but it is true that his work lacks the stern aspects required by mathematical exactitude. It was in personal contact that this was corrected, at least in so far as his students were concerned. His favourite maxim was, "Never be dull".