Background
Felix Klein was born on April 25, 1849, in Dusseldorf, Germany, to Prussian parents. His father, Caspar Klein, was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Kayser.
1885
Royal Society, London, England
Klein was a member of the Royal Society.
1893
Klein was awarded the De Morgan Medal in 1893.
1912
Klein was awarded the Copley Medal for his researches in mathematics in 1912.
University of Bonn, Bonn, Germany
Beginning in the winter semester of 1865-1866 Klein studied mathematics and physics at the University of Bonn, where he received his doctorate, supervised by Julius Plucker, in December 1868.
Saxon Academy of Sciences, Leipzig, Germany
Klein was a member of the Saxon Academy of Sciences.
German Academy of Sciences Leopoldina, Halle, Germany
Klein was a member of the German Academy of Sciences Leopoldina.
Gottingen Academy of Sciences, Gottingen, Germany
Klein was a member of the Gottingen Academy of Sciences.
Saint Petersburg Academy of Sciences, Saint Petersburg, Russia
Klein was a member of the Saint Petersburg Academy of Sciences.
Hungarian Academy of Sciences, Budapest, Hungary
Klein was a member of the Hungarian Academy of Sciences.
Bavarian Academy of Sciences and Humanities, Munich, Germany
Klein was a member of the Bavarian Academy of Sciences and Humanities.
Lincean Academy, Rome, Italy
Klein was a member of the Lincean Academy.
American Academy of Arts and Sciences, Cambridge, Massachusetts, United States
Klein was a member of the American Academy of Arts and Sciences.
Russian Academy of Sciences, Moscow, Russia
Klein was a member of the Russian Academy of Sciences.
Royal Prussian Academy of Sciences, Berlin, Germany
Klein was a member of the Royal Prussian Academy of Sciences.
Accademia Nazionale delle Scienze detta dei XL, Verona, Italy
Klein was a member of the Accademia Nazionale delle Scienze detta dei XL.
Academy of Sciences of Turin, Turin, Italy
Klein was a member of the Academy of Sciences of Turin.
Royal Netherlands Academy of Arts and Sciences, Amsterdam, Netherlands
Klein was a member of the Royal Netherlands Academy of Arts and Sciences.
A photo of Klein.
A photo of Klein.
A photo of Klein.
A photo of Klein.
educator mathematician scientist author
Felix Klein was born on April 25, 1849, in Dusseldorf, Germany, to Prussian parents. His father, Caspar Klein, was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Kayser.
Klein graduated from the Gymnasium in Dusseldorf. Beginning in the winter semester of 1865-1866 he studied mathematics and physics at the University of Bonn, where he received his doctorate, supervised by Julius Plucker, in December 1868. In order to further his education he went at the start of 1869 to Gottingen, Berlin, and Paris, spending several months in each city. The Franco-Prussian War forced him to leave Paris in 1870. After a short period of military service as a medical orderly, Klein qualified as a lecturer at Gottingen at the beginning of 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, 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. There he had as colleagues a number of talented young lecturers, including von Dyck, Rohn, Study and Engel. 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 Gottingen in 1886. He taught at Gottingen until he retired in 1913, but he now sought to re-establish Gottingen as the foremost mathematics research center in the world. His own role as the leader of a geometrical school at Leipzig was never transferred to Gottingen. There 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 Gottingen which was to serve as a model for the best mathematical research centers throughout the world. He introduced weekly discussion meetings, a mathematical reading room with a mathematical library. Klein brought Hilbert from Konigsberg to join his research team at Gottingen 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. In a sense these journals represented the rival teams of the Berlin school of mathematics who ran 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 specialized in complex analysis, algebraic geometry and invariant theory. It also provided an important outlet for real analysis and the new area of group theory.
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. With his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany.
During 1908, he was elected president of the International Commission on Mathematical Instruction at the Rome International Congress of Mathematicians.
Klein also devised the "Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside." It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Mobius strip, with one method of construction being the attachment of the edges of two Mobius strips.
Klein retired due to ill health in 1913. However, he continued to teach mathematics at his home during the years of World War I. He was also one of the 93 signatories of the Manifesto of the Ninety-Three, a document penned in support of the German invasion of Belgium in the early stages of World War I.
Klein was less interested in work requiring subtle and detailed calculations, which he gladly left to his students. In his later years Klein’s great organizational skill came to the fore, enabling him to initiate and supervise large-scale encyclopedic works devoted to many areas of mathematics, to their applications, and to their teaching.
Klein’s extraordinarily rapid development as a mathematician was characteristic. At first he wanted to be a physicist, and while still a student he assisted J. Plucker in his physics lectures at Bonn. At that time Plucker, who had returned to mathematics after a long period devoted to physics, was working on a book entitled Neue Geometrie des Raumes, gegrundet auf der geraden Linie als Raumelement. His sudden death in 1868 prevented him from completing it, and the young Klein took over this task. Klein’s dissertation and his first subsequent works also dealt with topics in line geometry. The new aspects of his efforts were that he worked with homogeneous coordinates, which Plucker did only occasionally; that he understood how to apply the theory of elementary divisors, developed by Weierstrass a short time before, to the classification of quadratic straight line complexes; and that he early viewed the line geometry of P3 as point geometry on a quadric of P5, which was a completely new conception.
In 1870 Klein and S. Lie discovered the fundamental properties of the asymptotic lines of the famous Kummer surface, which, as the surface of singularity of a general quadratic straight-line complex, occupied a place in algebraic line geometry. Here and in his simultaneous investigations of cubic surfaces there is evidence of Klein’s special concern for geometric intuition, whether regarding the forms of plane curves or the models of spatial constructions. A further result of his collaboration with Lie was the investigation, in a joint work, of the so-called W-curves. These are curves that admit a group of projective transformations into themselves.
Hyperbolic geometry, it is true, had already been discovered by Lobachevsky (1829) and J. Bolyai (1832); and in 1868, shortly before Klein, E. Beltrami had recognized that it was valid on surfaces of constant negative curvature. Nevertheless, the non-Euclidean geometries had not yet become common knowledge among mathematicians when, in 1871 and 1873, Klein published two works entitled Vber die sogenannte nicht-euklidische Geometrie. His essential contribution here was to furnish so-called projective models for three types of geometry: hyperbolic, stemming from Bolyai and Lobachevsky; elliptic, valid on a sphere on which antipodal points have been taken as identical; and Euclidean. Klein based his work on the projective geometry that C. Staudt had earlier established without the use of the metric concepts of distance and angle, merely adding a continuity postulate to Staudt’s construction. Then he explained, for example, plane hyperbolic geometry as a geometry valid in the interior of a real conic section and reduced the lines and angles to cross ratios. This had already been done for the Euclidean angle by Laguerre in 1853 and, more generally, by A. Cayley in 1860; but Klein was the first to recognize clearly that in this way the geometries in question can be constructed purely projectively. Thus one speaks of Klein models with Cayley-Klein metric.
The conceptions grouped together under the name “Erlanger Programm” were presented in 1872 in “Vergleichende Betrachtungen liber neuere geometrische Forschungen.” This work reveals the early familiarity with the concept of group that Klein acquired chiefly through his contact with Lie and from C. Jordan. The essence of the “Erlanger Programm” is that every geometry known so far is based on a certain group, and the task of the geometry in question consists in setting up the invariants of this group. The geometry with the most general group, which was already known, was topology; it is the geometry of the invariants of the group of all continuous transformations - for example, of the plane. Klein then successively distinguished the projective, the affine, and the equiaffine or principal group of the particular dimension; in certain cases the succeeding group is a subgroup of the previous one. To these groups belong the projective, affine, and equiaffine geometries with their invariants, whereby the equiaffine geometry is the same as the Euclidean elementary geometry.
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 latter and the conceptions 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.
For Klein the Riemann surface is no longer necessarily a multisheeted covering surface with isolated branch points on a plane, which is how Riemann presented it in his own publications. Rather, according to Klein, it loses its relationships to the complex plane and then, generally, to three-dimensional space. It is through Klein that the Riemann surface is regarded as an indispensable component of function theory and not only as a valuable means of representing multivalued functions.
Klein provided a comprehensive account of his conception of the Riemann surface in 1882 in Riemanns Theorie der algebraischen Funktionen und ihre Integrate. In this book he treated function theory as geometric function theory in connection with potential theory and conformal mapping - as Riemann had done. Moreover, in his efforts to grasp the actual relationships and to generate new results, Klein deliberately worked with spatial intuition and with concepts that were borrowed from physics, especially from fluid dynamics. He repeatedly stressed that he was much concerned about the deficiencies of this method of demonstration and that he expected them to be eliminated in the future. A portion of the existence theorems employed by Klein had already been proved, before the appearance of the book by Klein, by H. A. Schwarz and C. Neumann. Klein did not incorporate their results in his own work: He opposed the spirit of the reigning school of Berlin mathematicians led by Weierstrass, with its abstract-critical, arithmetizing tendency; Riemann’s approach, which inclined more toward geometry and spatial representation, he considered more fruitful. The rigorous foundation of his own theorems and the fusion of Riemann’s and Weierstrass’ concepts that Klein hoped for and expected found its expression - still valid today - in 1913 in H. Weyl’s Die Idee der Riemannschen Flache.
A problem that greatly interested Klein was the solution of fifth-degree equations, for its treatment involved the simultaneous consideration of algebraic equations, group theory, geometry, differential equations, and function theory. Hermite, Kronecker, and Brioschi had already employed transcendental methods in the solution of the general algebraic equation of the fifth degree. Klein succeeded in deriving the complete theory of this equation from a consideration of the icosahedron, one of the regular polyhedra known since antiquity. These bodies sometimes can be transformed into themselves through a finite group of rotations. The icosahedron in particular allows sixty such rotations into itself. If one circumscribes a sphere about a regular polyhedron and maps it onto a plane by stereographic projection, then to the group of rotations of the polyhedron into itself there corresponds a group of linear transformations of the plane into itself. Klein demonstrated that in this way all finite groups of linear transformations are obtained, if the so-called dihedral group is added. By a dihedron Klein meant a regular polygon with n sides, considered as a rigid body of null volume.
In the 1890s Klein was especially interested in mathematical physics and engineering. One of the first results of this shift in interest was the textbook he composed with A. Sommerfeld on the theory of the gyroscope.
Klein was not pleased with the increasingly abstract nature of contemporary mathematics. His longstanding concern with applications was further strengthened by the impressions he received during two visits to the United States. He sought, on the one hand, to awaken a greater feeling for applications among pure mathematicians and, on the other, to lead engineers to a greater appreciation of mathematics as a fundamental science. The first goal was advanced by the founding, largely through Klein’s initiative, of the Gottingen Institute for Aeronautical and Hydrodynamical Research; at that time such institutions were still uncommon in university towns. Moreover, at the turn of the century he took an active part in the major publishing project Encyklo-padie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. He himself was editor, along with Konrad Muller, of the four-volume section on mechanics.
Klein was elected a member of the Royal Society during 1885. He was also a member of the Saxon Academy of Sciences, the German Academy of Sciences Leopoldina, the Gottingen Academy of Sciences, the Saint Petersburg Academy of Sciences, the Hungarian Academy of Sciences, the Bavarian Academy of Sciences and Humanities, the Lincean Academy, the American Academy of Arts and Sciences, the Russian Academy of Sciences, the Royal Prussian Academy of Sciences, the Accademia Nazionale delle Scienze detta dei XL, the Academy of Sciences of Turin, and the Royal Netherlands Academy of Arts and Sciences.
Royal Society , United Kingdom
1885
Saxon Academy of Sciences , Germany
German Academy of Sciences Leopoldina , Germany
Gottingen Academy of Sciences , Germany
Saint Petersburg Academy of Sciences , Russia
Hungarian Academy of Sciences , Hungary
Bavarian Academy of Sciences and Humanities , Germany
Lincean Academy , Italy
American Academy of Arts and Sciences , United States
Russian Academy of Sciences , Russia
Royal Prussian Academy of Sciences , Germany
Accademia Nazionale delle Scienze detta dei XL , Italy
Academy of Sciences of Turin , Italy
Royal Netherlands Academy of Arts and Sciences , Netherlands
Klein possessed an extraordinary ability to discover quickly relationships between different areas of research and to exploit them fruitfully.
In August 1875 Klein married Anne Hegel, a granddaughter of the philosopher Georg Wilhelm Friedrich Hegel; they had one son and three daughters.