Felix Klein
Felix Christian Klein
Born	April 25, 1849 Düsseldorf, Germany
Died	June 22, 1925 Göttingen, Germany

Felix Christian Klein (April 25, 1849, Düsseldorf, Germany – June 22, 1925, Göttingen) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Image File history File links Felix_Klein. ... April 25 is the 115th day of the year in the Gregorian Calendar (116th in leap years). ... 1849 was a common year starting on Monday (see link for calendar). ... Image File history File links Flag_of_Germany. ... Düsseldorf is the capital city of the German Federal State of North Rhine-Westphalia and (together with Cologne and the Ruhr Area) the economic center of Western Germany. ... June 22 is the 173rd day of the year (174th in leap years) in the Gregorian Calendar, with 192 days remaining. ... 1925 (MCMXXV) was a common year starting on Thursday (link will take you to calendar). ... Image File history File links Flag_of_Germany. ... Göttingen ( ) is a city in Lower Saxony, Germany. ... April 25 is the 115th day of the year in the Gregorian Calendar (116th in leap years). ... 1849 was a common year starting on Monday (see link for calendar). ... Düsseldorf is the capital city of the German Federal State of North Rhine-Westphalia and (together with Cologne and the Ruhr Area) the economic center of Western Germany. ... June 22 is the 173rd day of the year (174th in leap years) in the Gregorian Calendar, with 192 days remaining. ... 1925 (MCMXXV) was a common year starting on Thursday (link will take you to calendar). ... Göttingen ( ) is a city in Lower Saxony, Germany. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Group theory is that branch of mathematics concerned with the study of groups. ... Complex analysis is the branch of mathematics investigating holomorphic functions, i. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Table of Geometry, from the 1728 Cyclopaedia. ... Group theory is that branch of mathematics concerned with the study of groups. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

Life

Klein's parents were Prussian; his father was a Prussian government official stationed in the Rheinland. 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 held Bonn's chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868, Coat of Arms of the Kingdom of Prussia, 1701-1918 Prussia (German: ; Latin: Borussia, Prutenia; Lithuanian: ; Polish: ; Old Prussian: PrÅ«sa) was, most recently, a historic state originating in East Prussia, an area which for centuries had substantial influence on German and European history. ... The Rhineland (Rheinland in German) is the general name for the land on both sides of the river Rhine in the west of Germany. ... The main building, viewed from the Hofgarten. ... Julius Plücker. ...

Julius Plücker died in 1868, leaving his book on the foundations 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 Clebsch, who had moved to Göttingen in 1868. Klein visited Clebsch the following year, along with visits to Berlin and Paris. Klein was in Paris when (July 1870) Bismarck published a message intended to provoke France into declaring war on Prussia. Klein promptly left Paris when France did so. For a short time, he served as a medical orderly in the German army before being appointed lecturer at Göttingen in early 1871. Julius Plücker. ... In mathematics Plücker co-ordinates are a way to assign to each line in projective 3-space a point in projective 5-space. ... Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. ... Berlin is the capital city and one of the sixteen states of the Federal Republic of Germany. ... City flag City coat of arms Motto: Fluctuat nec mergitur (Latin: Tossed by the waves, she does not sink) Location Coordinates Time Zone CET (GMT +1) Administration Country France Région ÃŽle-de-France Département Paris (75) Subdivisions 20 arrondissements Mayor Bertrand Delanoë  (PS) (since 2001) City Statistics Land... Alternate meanings: See Bismarck (disambiguation). ... Coat of Arms of the Kingdom of Prussia, 1701-1918 Prussia (German: ; Latin: Borussia, Prutenia; Lithuanian: ; Polish: ; Old Prussian: PrÅ«sa) was, most recently, a historic state originating in East Prussia, an area which for centuries had substantial influence on German and European history. ...

Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, and so he was pleased to be offered a chair at Munich's Technische Hochschule in 1875. There he and Brill taught advanced courses to many excellent students, e.g., Hurwitz, van Dyck, Rohn, Carle Runge, Planck, Bianchi, and Gregorio Ricci-Curbastro. Erlangen around 1915 Erlangen is a German city in Middle Franconia. ... Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. ... Adolf Hurwitz Adolf Hurwitz (26 March 1859- 18 November 1919) was a German mathematician, and one of the most important figures in mathematics in the second half of the nineteenth century (according to Jean-Pierre Serre, always something good in Hurwitz). He was born in a Jewish family in Hildesheim... Self Portrait With a Sunflower Sir Anthony (Antoon) van Dyck (*March 22, 1599 - December 9, 1641) was a Flemish painter — mainly of portraits — who became the leading court painter in England. ... Carle David Tolmé Runge (August 30, 1856 – January 3, 1927) was a German mathematician, physicist, and spectroscopist. ... This article is about Planck, the German physicist. ... Luigi Bianchi born on the January 18, 1856 in Parma, Italy, and died on June 6, 1928 in Pisa, Italy. ... Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ...

In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Hegel. History teaches us that man learns nothing from history. ...

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. There his colleagues included van Dyck, Rohn, Study and Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883-1884, he was plagued by depression. Table of Geometry, from the 1728 Cyclopaedia. ... [] (Sorbian/Lusatian: Lipsk) is the largest city in the Federal State (Bundesland) of Saxony in Germany. ... Self Portrait With a Sunflower Sir Anthony (Antoon) van Dyck (*March 22, 1599 - December 9, 1641) was a Flemish painter — mainly of portraits — who became the leading court painter in England. ... Look up Study on Wiktionary, the free dictionary To study means to acquire knowledge, often by memorization or reading. ... In German and Dutch Engel means Angel. ...

His career as a research mathematician essentially over, Klein accepted a chair at the University of Göttingen in 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry. At Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics and potential theory. The Georg-August University of Göttingen (Georg-August-Universität Göttingen, often called the Georgia Augusta) was founded in 1734 by George II, King of Great Britain and Elector of Hanover, and opened in 1737. ... Table of Geometry, from the 1728 Cyclopaedia. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... Potential theory may be defined as the study of harmonic functions. ...

The research center Klein established at Göttingen served as a model for the best such centers throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired Hilbert away from Königsberg; this appointment proved fateful, because Hilbert continued Göttingen's glory until his own retirement in 1932.

Under Klein's editorship, Mathematische Annalen became one of the very best mathematics journals in the world. Founded by Clebsch, only under Klein's management did it first rival then surpass Crelle's journal based out of the University of Berlin. Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in complex analysis, algebraic geometry, and invariant theory (at least until Hilbert killed the subject). It also provided an important outlet for real analysis and the new group theory. The Mathematische Annalen is a German mathematical research journal published by Springer-Verlag. ... Alfred Clebsch (1832-1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ... There is no institution called the University of Berlin, but there are four universities in Berlin, Germany: Humboldt University of Berlin (Humboldt-Universität zu Berlin) Technical University of Berlin (Technische Universität Berlin) Free University of Berlin (Freie Universität Berlin) Berlin University of the Arts (Universität der... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ... Group theory is that branch of mathematics concerned with the study of groups. ...

Thanks in part to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman; she was an English student of Arthur Cayley's, whom Klein admired. Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ...

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that 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. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society awarded Klein its De Morgan Medal in 1893. He was elected a member of the Royal Society in 1885, and was awarded its Copley medal in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for some years more. The London Mathematical Society (LMS) is the leading mathematical society in England. ... This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... The premises of the Royal Society in London (first four properties only). ... The Copley Medal is a scientific award for work in any field of science, the highest award granted by the Royal Society of London. ...

Work

Klein's dissertation, on line geometry and its applications to mechanics, classified second degree line complexes using Weierstrass's theory of elementary divisors. In mathematics Plücker co-ordinates are a way to assign to each line in projective 3-space a point in projective 5-space. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ... Karl Theodor Wilhelm Weierstraß (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. (The letter ß may be transliterated as ss; one often writes Weierstrass. ...

Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan. Marius Sophus Lie (December 17, 1842 - February 18, 1899) was a Norwegian-born mathematician who largely created the theory of continuous symmetry, and applied it to the study of geometric structures and differential equations. ... A K3 manifold is a hyperkähler manifold of real dimension 4, i. ... Marie Ennemond Camille Jordan (January 5, 1838 – January 22, 1922) was a French mathematician, known both for his foundational work in group theory and for his influential Cours danalyse. ...

Klein devised the bottle named after him, a one-sided closed surface which cannot be constructed in Euclidean space. It is best pictured as a cylinder looped back through itself to join with its other end. This is not a continuous surface in 3-space as the surface cannot go through itself without a discontinuity. It is possible, however, to construct a Klein bottle in non-Euclidean space. The Klein bottle immersed in three-dimensional space. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

In the 1890s, Klein turned to mathematical physics, a subject from which he had never strayed far, writing on the gyroscope with Arnold Sommerfeld. In the same vein, he helped edit (with K Müller) the four volumes on mechanics of the Encyklopedie der Mathematischen Wissenschaften. Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... A gyroscope For other uses, see Gyroscope (disambiguation). ... Arnold Johannes Wilhelm Sommerfeld (December 5, 1868 in Königsberg, East Prussia – April 26, 1951 in Munich, Germany) was a German physicist who introduced the fine-structure constant in 1919. ... Mechanics (Greek ) is the branch of physics concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effect of the bodies on their environment. ...

Erlangen Program

In 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 special cases of 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, putting Euclidian and non-Euclidian geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry. Cayley never accepted Klein's argument, believing it to be circular. In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from the combinatorics of block designs. ... Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ... There have been several prominent people named Cayley: Arthur Cayley Mathematician Charles Bagot Cayley Linguist and friend of Christina Rossetti George Cayley Naturalist, Physical Scientist, Engineer, Inventor, Politician This is a disambiguation page — a list of articles associated with the same title. ...

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 Erlangen Program (1872), profoundly influenced the evolution of mathematics. This program was set out in 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 approach to geometry that became (and remains) the accepted view. 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. Table of Geometry, from the 1728 Cyclopaedia. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ...

Today the significance of Klein's contributions to geometry are less than evident, but not because those contributions are now seen as strange or wrong. On the contrary, those contributions have become so much a part of our present mathematical thinking that it is hard for us to appreciate their novelty, and the way in which they were not immediately accepted by all his contemporaries.

Function theory

Klein saw his work on function theory as his major contribution to mathematics, specifically his work on: Complex analysis is the branch of mathematics investigating holomorphic functions, i. ...

• The link between certain ideas of Riemann's and invariant theory,
• Number theory and abstract algebra;
• Group theory;
• Geometry with more than 3 dimensions and differential equations, especially equations he invented, namely elliptic modular functions and automorphic functions.

Klein showed that that the modular group moves the fundamental region of the complex plane so as to tessellate that 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 that that surface was a curve in projective space, that its equation was x³y + y³z + z³x = 0, and that its group of symmetries was PSL(2,7) of order 168. His Riemanns Theorie der algebraischen Funktionen und ihre Integrals (1882) treats function theory in a geometric way, connecting potential theory and conformal mappings. This work drew on notions from fluid dynamics. Bernhard Riemann. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Group theory is that branch of mathematics concerned with the study of groups. ... Table of Geometry, from the 1728 Cyclopaedia. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... In mathematics, the complex plane is a way of visualising the space of the complex numbers. ... A tessellated plane seen in street pavement. ... The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Pearson surface, is a one-dimensional complex manifold. ... In mathematics, a projective space is a fundamental construction from any vector space. ... The symmetry group of an object (e. ... The projective special linear group G = PSL(2,7) is a finite group in mathematics that has important applications in algebra, geometry, and number theory. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... Potential theory may be defined as the study of harmonic functions. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ...

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on the methods of Hermite and Kronecker, he produced similar results to those of Brioschi and went on to completely solve the problem by means of the icosahedral group. This work led him to write a series of papers on elliptic modular functions. Charles Hermite (pronounced in IPA, , or phonetically air-meet) (December 24, 1822 - January 14, 1901) was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... The symmetry group of a geometric figure is the group of congruencies under which it is invariant, with composition as the operation. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...

In his 1884 book on the icosahedron, Klein set out a theory of automorphic functions, connecting algebra and geometry. However Poincaré published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem that would serve as a capstone to the emerging theory. Klein succeeded in formulating such a theorem and in sketching a strategy for proving it. But while doing this work his health collapsed, as mentioned above. An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant, which has faces which are equilateral triangles. ... Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [][1]), generally known as Henri Poincaré, was one of Frances greatest mathematicians and theoretical physicists, and a philosopher of science. ... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...

Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke over a period of about 20 years. In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...

See also

• Erlangen program
• Klein bottle
• Klein four-group
• Klein quartic
• Kleinian group
• Klein geometry
• Klein quadric
• elliptic modular function

An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... The Klein bottle immersed in three-dimensional space. ... This article is about the mathematical group. ... The Klein quartic x3y + y3z + z3x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). ... In mathematics, a Kleinian group is a finitely generated discrete group Γ of conformal (i. ... In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. ... A Klein Quadratic set is defined as a hyperbolic quadratic set of a five dimensional projective space. ... In mathematics, the j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...

Bibliography

Primary:

• 1887. "The arithmetizing of mathematics" in Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press: 965-71.

Secondary

• Mumford, David, Carol Series, David Wright, 2002. Indra's Pearls: The Vision of Felix Klein. Cambridge Univ. Press.

External links

• O'Connor, John J., and Edmund F. Robertson. "Felix Klein". MacTutor History of Mathematics archive.
• Felix Klein at the Mathematics Genealogy Project
• Klein's Web Pages.
• Eric W. Weisstein, Klein, Felix (1849-1925) at ScienceWorld.