Felix Klein

How to use this course

Felix Klein had two careers. In the first, which lasted barely fifteen years, he was one of the most inventive geometers of the nineteenth century: he unified Euclidean and non-Euclidean geometry, classified geometries by their groups, solved the quintic with the icosahedron and helped found the theory of automorphic functions. In the second, after a collapse in his health in 1882, he became an organizer. He built the Göttingen institute, recruited David Hilbert, opened German mathematics to women, launched an encyclopedia and led an international movement to reform school mathematics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows Klein’s own path: line geometry (1868), projective models of non-Euclidean geometry (1871), the Erlangen Program (1872), the icosahedron (1877–1884), modular functions (1878–1879), Riemann surfaces (1882), and the late work on mechanics and relativity.

Timeline

Düsseldorf and Bonn, 1849–1868

Christian Felix Klein was born in Düsseldorf on 25 April 1849. His father, Caspar Klein, was secretary to the head of the Prussian government in the Rhine Province; his mother was Sophie Elise Kayser. Klein liked to point out that his birth date was made of three squares: 25 April 1849, or \(5^2\), \(2^2\), \(43^2\). He attended the Gymnasium in Düsseldorf, a classical school, and entered the University of Bonn in 1865 intending to become a physicist.

In 1866 he became assistant to Julius Plücker, who held the chair of mathematics and experimental physics. Plücker had made his name in geometry decades earlier, then turned to physics, and in his last years returned to geometry with a new idea: to treat straight lines, not points, as the basic elements of space. Klein helped with the physics demonstrations and was drawn into the geometry.

The doctorate

Klein received his doctorate in December 1868, aged nineteen, with a dissertation on the transformation of the general second-degree equation in line coordinates into canonical form. He used Karl Weierstrass’s new theory of elementary divisors to classify quadratic line complexes, a direct extension of the classification of quadrics. Plücker died in May 1868 before finishing his book Neue Geometrie des Raumes. Alfred Clebsch, who had just moved to Göttingen, asked the young Klein to complete and edit it. Lesson 04 explains the geometry he inherited.

Wanderjahre, 1868–1871

Editing Plücker’s book took Klein to Göttingen and into the circle of Clebsch, then the leading algebraic geometer in Germany and founding editor, with Carl Neumann, of Mathematische Annalen. Clebsch became his patron. In 1869 Klein went on to Berlin, where he met a Norwegian visitor seven years older than himself, Sophus Lie. Both were outsiders to the Berlin school of Weierstrass and Kronecker, and both thought geometrically. They became close friends.

In the spring of 1870 the two went to Paris. There they met Camille Jordan, whose treatise on substitution groups had just appeared, and Gaston Darboux. Jordan and Lie gave Klein the concept that would organize his life’s work: the group. Klein and Lie wrote joint papers on curves invariant under one-parameter families of projective transformations (their “W-curves”) and on the asymptotic lines of Kummer’s surface.

When France declared war on Prussia on 19 July 1870, Klein, a Prussian, had to leave at once. Lie stayed, was arrested at Fontainebleau as a suspected spy, and was released after about a month through Darboux’s intervention. Klein served briefly as a medical orderly. Early in 1871 he habilitated at Göttingen and became a Privatdozent. In the same year he published the first of two papers Über die sogenannte Nicht-Euklidische Geometrie, which showed that hyperbolic, elliptic and Euclidean geometry could all be modelled inside projective geometry (Lesson 05).

Erlangen, Munich and Leipzig, 1872–1886

In 1872, on Clebsch’s recommendation, the University of Erlangen in Bavaria appointed Klein full professor. He was 23. On taking up the chair he published a programme, Vergleichende Betrachtungen über neuere geometrische Forschungen, now called the Erlangen Program. It was printed for the occasion and distributed, but it was not the inaugural address he actually delivered. The pamphlet drew little attention at first. Its fame came later, after Mellen Haskell’s English translation (1892–93) and republication in Mathematische Annalen in 1893. Clebsch died suddenly in November 1872, and Klein later became the leading editor of his journal, the Annalen.

In 1875 Klein moved to the Technische Hochschule in Munich, where he and Alexander von Brill taught a remarkable group of students: Adolf Hurwitz, Walther von Dyck, Carl Runge, Luigi Bianchi and Gregorio Ricci-Curbastro among them, with Max Planck in his lectures. Ferdinand von Lindemann, who in 1882 proved that \(\pi\) is transcendental, had already taken his doctorate under Klein at Erlangen. In August 1875 Klein married Anne Hegel, a granddaughter of the philosopher Georg Wilhelm Friedrich Hegel. They had a son and three daughters.

Leipzig and the contest with Poincaré

In 1880 Klein took the chair of geometry at Leipzig. These were his most productive years. Building on the icosahedron and on modular functions (Lessons 07 and 08), he turned to groups of Möbius transformations acting discontinuously and to the functions invariant under them. In 1881 a young Frenchman, Henri Poincaré, began publishing short notes in the Comptes Rendus on what he called Fuchsian functions. Klein wrote to him in June 1881. A courteous, competitive correspondence followed, in which Klein objected to the name Fuchsian and Poincaré answered by naming another class of groups Kleinian.

Both men were reaching for the same prize, a uniformization theorem for all algebraic curves. Klein, suffering from asthma, had gone to the North Sea island of Norderney to recover. According to his own later account, the key idea came to him at 2:30 in the morning of 23 March 1882, during an asthma attack. He left the island almost at once and wrote it up. In the autumn of 1882 his health collapsed. He suffered from exhaustion and depression through 1883 and 1884. He continued to work and publish, and his Vorlesungen über das Ikosaeder appeared in 1884, but he later said that his creative period in research ended there.

Line geometry and the Klein quadric

For the general reader. We usually think of space as made of points. Plücker suggested building it from straight lines instead. Lines in three-dimensional space form a four-dimensional family, and Klein, as a student, found that this family is itself a beautiful geometric object: a single quadratic equation in five-dimensional projective space.

Plücker coordinates

Let \(x=(x_0:x_1:x_2:x_3)\) and \(y=(y_0:y_1:y_2:y_3)\) be distinct points of projective space \(\mathbb P^3\) over a field \(k\). The line through them has Plücker coordinates

These are the \(2\times2\) minors of the \(2\times4\) matrix with rows \(x\) and \(y\). Replacing \(x,y\) by two other points of the same line multiplies the matrix on the left by an invertible \(2\times2\) matrix, which multiplies every minor by its determinant. So the six numbers \(p_{ij}\) define a point of \(\mathbb P^5\) depending only on the line.

Proof sketch

In the language of exterior algebra, \(p=x\wedge y\in\Lambda^2k^4\), and the quadratic form is \(p\wedge p\in\Lambda^4k^4\cong k\). A bivector is decomposable, of the form \(x\wedge y\), if and only if \(p\wedge p=0\); this gives the relation and its converse. Two lines \(x\wedge y\) and \(u\wedge v\) meet exactly when \(x,y,u,v\) are linearly dependent, that is when \(x\wedge y\wedge u\wedge v=0\), and this 4-vector is \(\langle p,q\rangle\) times a basis vector. The quadratic form has signature \((3,3)\) over \(\mathbb R\), and its symmetry group, \(\mathrm{SO}(3,3)\) up to sign, contains the projective group of \(\mathbb P^3\) as the subgroup preserving the two families of planes on \(Q\).

Line complexes

A line complex is a three-dimensional family of lines, the intersection of \(Q\) with another hypersurface. A quadratic complex is \(Q\cap Q'\) with \(Q'\) a second quadric. Klein’s thesis classified quadratic complexes by putting the pencil \(\lambda Q+\mu Q'\) into normal form using Weierstrass’s elementary divisors. In 1870 Klein showed that the singular surface of a general quadratic complex is Kummer’s quartic surface with 16 nodes, which led him and Lie to Kummer surfaces.

Non-Euclidean geometry from projective geometry

For the general reader. In hyperbolic geometry, through a point outside a line there pass infinitely many parallels. Klein showed how to draw this world inside an ordinary disk with a ruler: lines are just straight chords, and only the rule for measuring distance changes. Since the model is built from ordinary geometry, hyperbolic geometry is consistent if Euclidean geometry is.

Cayley’s idea and Klein’s turn

Arthur Cayley had shown in 1859 that Euclidean measurement can be expressed projectively by singling out a fixed conic, the absolute, and defining distance via cross-ratios. Cayley thought of this as a reinterpretation of Euclidean geometry. In 1871 Klein saw that a real non-degenerate conic as absolute gives the hyperbolic geometry of Lobachevsky and Bolyai, an imaginary conic gives elliptic geometry, and a degenerate conic gives Euclidean geometry. He introduced the names hyperbolic, elliptic and parabolic for the three cases. Eugenio Beltrami had found the disk model in 1868 by a different route; Klein’s contribution was the projective explanation.

The cross-ratio of four collinear points \(A,B,P,Q\) is

with signed lengths along the line. It is invariant under all projective transformations of the line.

Why it works

Projective maps preserving \(C\) preserve cross-ratios and map chords to chords, so they are isometries. Additivity follows from a cross-ratio identity: for \(P,Q,R\) in order on a chord, \((A,B;P,Q)\,(A,B;Q,R)=(A,B;P,R)\). Lift to \(\mathbb R^{2,1}\) with form \(x^2+y^2-z^2\): the disk is the set of negative lines, and the map \((x,y)\mapsto(x,y,1)/\sqrt{1-x^2-y^2}\) sends it to the hyperboloid model, where \(\cosh d(P,Q)=-\langle \hat P,\hat Q\rangle\). In formulas,

The price of straight lines is that angles are distorted: the model is not conformal, unlike Poincaré’s disk. The two disk models are related by a vertical projection through the hemisphere.

The Erlangen Program

For the general reader. By 1870 there were many geometries: Euclidean, projective, affine, non-Euclidean, inversive, and more. Klein proposed one principle to organize them all. A geometry is a space together with a group of transformations, and its theorems are exactly the statements that the group leaves unchanged. Fewer symmetries mean more theorems.

Homogeneous spaces

In modern terms Klein’s geometries are homogeneous spaces. If a group \(G\) acts transitively on a set \(X\), fix \(x_0\in X\) with stabilizer \(H=\{g: gx_0=x_0\}\). Then \(gH\mapsto gx_0\) is a \(G\)-equivariant bijection \(G/H\to X\). So a Klein geometry is just a pair \((G,H)\). Euclidean plane geometry is \(\mathrm{E}(2)/\mathrm{O}(2)\); the hyperbolic plane is \(\mathrm{PSL}_2(\mathbb R)/\mathrm{PSO}(2)\); the round sphere is \(\mathrm{SO}(3)/\mathrm{SO}(2)\).

The hierarchy

Klein placed projective geometry at the top. The projective group \(\mathrm{PGL}_{n+1}\) preserves incidence and cross-ratio. Fixing a hyperplane at infinity gives the affine group, which also preserves parallelism and ratios along a line. Fixing in addition an imaginary conic in that hyperplane (the circle points) gives the similarity group, which preserves angles; adding a unit of length gives the Euclidean group. Fixing instead a real conic gives hyperbolic geometry (Lesson 05). Klein also treated inversive geometry, where Möbius transformations preserve circles, and showed that the same group can appear in different spaces: \(\mathrm{PSL}_2(\mathbb C)\) acts on the Riemann sphere and, as the Lorentz group, on hyperbolic 3-space.

The Program had limits. Riemannian manifolds generally have no symmetries at all, so they fall outside it. Élie Cartan resolved this in the 1920s with Cartan connections, which model a space infinitesimally on a Klein geometry \(G/H\). Samuel Eilenberg and Saunders Mac Lane described their 1945 paper founding category theory as a continuation of Klein’s idea.

The icosahedron and the quintic

For the general reader. The general equation of degree five cannot be solved by radicals. Klein showed that its true symmetry is the symmetry of the icosahedron. Place an icosahedron in a sphere and the sphere in the complex plane: solving a quintic becomes the problem of finding a point on the sphere from a symmetric function of its 60 images.

Finite groups of rotations

Stereographic projection identifies the sphere with \(\hat{\mathbb C}=\mathbb C\cup\{\infty\}\), and rotations become Möbius transformations \(z\mapsto\frac{\alpha z+\beta}{-\bar\beta z+\bar\alpha}\) (Lesson 10). Klein classified the finite groups of rotations: cyclic, dihedral, and the rotation groups of the tetrahedron (\(A_4\), order 12), octahedron (\(S_4\), order 24) and icosahedron (\(A_5\), order 60). Each lifts to a “binary” group of twice the order in \(\mathrm{SL}_2(\mathbb C)\).

Orient the icosahedron with vertices at \(0\), \(\infty\) and the ten roots of \(z^{10}+11z^5-1=0\). The binary forms vanishing at the 12 vertices, 20 face centres and 30 edge midpoints are

The icosahedral equation

The rational function \(Z(z)=\dfrac{H(z,1)^3}{1728\,f(z,1)^5}\) has degree 60 and is invariant under the icosahedral group, so it identifies \(\hat{\mathbb C}/A_5\) with \(\hat{\mathbb C}\). By the relation, \(Z-1=-T^2/(1728f^5)\): the function takes the value \(\infty\) at vertices, 0 at face centres and 1 at edge midpoints, each with the appropriate multiplicity. The icosahedral equation asks, given \(Z\), for the 60 solutions \(z\). Its Galois group over \(\mathbb C(Z)\) is \(A_5\).

The relation \(T^2+H^3-1728f^5=0\) defines, after rescaling, the surface \(x^2+y^3+z^5=0\), the \(E_8\) singularity. Klein’s icosahedron thus sits at the start of the McKay correspondence between finite subgroups of \(\mathrm{SL}_2(\mathbb C)\) and simply laced Dynkin diagrams.

Modular functions and the Klein quartic

For the general reader. Elliptic curves come in families described by a single complex number. Klein studied the function that tells two such curves apart, and the symmetry groups behind it. In doing so he found a curve of genus three with 168 symmetries, the largest number any curve of that genus can have.

The absolute invariant

For \(\tau\) in the upper half-plane \(\mathbb H\), the lattice \(\mathbb Z+\mathbb Z\tau\) has Eisenstein invariants \(g_2(\tau)\), \(g_3(\tau)\) and discriminant \(\Delta=g_2^3-27g_3^2\neq0\). Klein introduced the absolute invariant

The fundamental domain is the region \(|\operatorname{Re}\tau|\le\tfrac12,\ |\tau|\ge1\). Richard Dedekind found the same function independently in 1877. Klein’s originality was to see the quotient as a Riemann surface and to study its coverings.

Congruence subgroups and the level 7 curve

The principal congruence subgroup \(\Gamma(N)=\ker\bigl(\mathrm{SL}_2(\mathbb Z)\to\mathrm{SL}_2(\mathbb Z/N)\bigr)\) has finite index, and the compactified quotient \(X(N)\) is a compact Riemann surface covering the \(J\)-sphere with group \(\mathrm{PSL}_2(\mathbb Z/N)\). For \(N=5\) this is the icosahedron again: \(\mathrm{PSL}_2(\mathbb F_5)\cong A_5\) and \(X(5)\) is the sphere. In 1879, in a paper on the transformation of order seven of elliptic functions, Klein studied \(N=7\).

Klein visualized the curve as 24 regular heptagons, the quotient of the \((2,3,7)\) triangle tiling of the hyperbolic plane. A marble sculpture of this tiling, The Eightfold Way by Helaman Ferguson, stands at the Mathematical Sciences Research Institute in Berkeley.

Riemann surfaces and uniformization

For the general reader. Bernhard Riemann described multivalued functions by surfaces. Klein insisted that these surfaces are the real objects and the formulas secondary. He then guessed, in competition with Poincaré, that every such surface can be built by gluing pieces of the sphere, the plane or the hyperbolic disk. Along the way he described a surface with no inside or outside, the Klein bottle.

Riemann’s theory, Klein’s way

In his 1882 booklet Über Riemanns Theorie der algebraischen Functionen und ihre Integrale, Klein built Riemann surfaces from physical intuition: a harmonic function is a steady current of fluid on a conducting surface, and the existence of such flows with given singularities replaces Riemann’s Dirichlet principle. He also stressed that a Riemann surface need not be drawn over the plane; any closed surface with a conformal structure will do. In 1882 he also described a closed one-sided surface, obtained from a square by gluing one pair of sides directly and the other with a twist. In German it is the Kleinsche Fläche (surface); the familiar name “bottle” probably comes from a confusion with Flasche.

Groups and uniformization

A Fuchsian group is a discrete subgroup of \(\mathrm{PSL}_2(\mathbb R)\), acting on the disk or upper half-plane; a Kleinian group is a discrete subgroup of \(\mathrm{PSL}_2(\mathbb C)\), acting on the sphere and on hyperbolic 3-space. Friedrich Schottky had studied an important special class in 1877. Klein and Poincaré developed the general theory in 1881–1883.

Klein’s 1882 “limit-circle theorem” asserted that every algebraic curve of genus \(g\ge2\) can be uniformized by a Fuchsian group, with a proof by continuity: compare the \(6g-6\) real parameters of curves with those of groups and argue that the map between them is a bijection. The method was sound in outline but needed topological tools, such as invariance of domain, that did not yet exist. The rigorous proofs of 1907 used potential theory instead.

Rotations, the top and relativity

For the general reader. Klein’s groups reached into physics. With Arnold Sommerfeld he described a spinning top by complex numbers, the same numbers later used for electron spin. And he read Einstein’s relativity as a chapter of the Erlangen Program: special relativity is the geometry of the Lorentz group.

Cayley–Klein parameters

One proof: \(\mathrm{SU}(2)\) acts by conjugation on the 3-dimensional real space of traceless Hermitian matrices \(X=\begin{pmatrix}z&x-iy\\x+iy&-z\end{pmatrix}\), preserving \(\det X=-(x^2+y^2+z^2)\). This gives a homomorphism to \(\mathrm O(3)\), landing in \(\mathrm{SO}(3)\) by connectedness, with kernel the scalars \(\pm I\). Klein and Sommerfeld’s Über die Theorie des Kreisels (four volumes, 1897–1910) used \(\alpha,\beta\) to integrate the motion of the heavy symmetric top. The same parameters returned in 1927 as Pauli spinors.

The geometry of the Lorentz group

In 1910 Klein lectured on the geometric foundations of the Lorentz group, arguing that Hermann Minkowski’s spacetime was precisely what his Program predicted: the invariant theory of the group preserving \(x_0^2-x_1^2-x_2^2-x_3^2\). The same \(2\times2\) trick, with Hermitian matrices of determinant \(x_0^2-x_1^2-x_2^2-x_3^2\), gives the double cover \(\mathrm{SL}_2(\mathbb C)\to\mathrm{SO}^+(1,3)\), so the Lorentz group is the Möbius group of Lesson 06.

Energy in general relativity, 1918

In 1915–1918 Klein and Hilbert, with Einstein in correspondence, debated why energy conservation in general relativity seemed to hold identically. Klein brought Emmy Noether to Göttingen partly for this problem. In July 1918 he presented her Invariante Variationsprobleme to the Göttingen Society of Sciences and published two papers of his own, on Hilbert’s first note on the foundations of physics and on the differential laws for conservation of momentum and energy in Einstein’s theory. Klein showed that the total energy–momentum current, matter plus gravitational field, can be written on solutions as \(\partial_\lambda U^{\mu\lambda}{}_\nu\) with \(U\) antisymmetric in \(\mu,\lambda\). Its divergence therefore vanishes identically, and the total energy reduces to a surface integral. Noether’s second theorem explained the general reason.

Göttingen, 1886–1913

In 1886 Klein accepted a chair at Göttingen, arranged with the help of Friedrich Althoff of the Prussian Ministry of Culture. Lie succeeded him at Leipzig. Göttingen had been the university of Carl Friedrich Gauss, Lejeune Dirichlet and Riemann, but by the 1880s Berlin dominated German mathematics. Klein set out to change that, and with Althoff’s backing he largely succeeded.

Building an institution

Klein created a reading room and library, a collection of mathematical models, and a seminar system whose handwritten protocols, kept from 1872 to 1913, fill 29 volumes. In 1895 he brought David Hilbert from Königsberg, and later Hermann Minkowski joined them. He raised money from industry for applied institutes and in 1904 persuaded Ludwig Prandtl to come to Göttingen; Prandtl’s laboratory grew into a world centre of aerodynamics. With Georg Cantor and others Klein helped found the German Mathematical Society (DMV) in 1890. He edited Mathematische Annalen, which under his direction rivalled and then overtook Crelle’s journal. From the mid-1890s he organized the Encyklopädie der mathematischen Wissenschaften, a multi-volume survey of all of mathematics and its applications that was completed only in 1935.

He was also a determined advocate for women in mathematics. Göttingen admitted women to mathematics lectures from 1893, and in 1895 Grace Chisholm (later Grace Chisholm Young) received her doctorate under Klein. In 1915 he and Hilbert brought Emmy Noether to Göttingen.

Abroad and in the schools

In 1893 Klein represented the German government at the Mathematical Congress held with the World’s Columbian Exposition in Chicago, then gave a two-week colloquium at Northwestern University in Evanston. Many young Americans came to study with him, among them Maxime Bôcher, Frank Nelson Cole, Henry Fine, Edward Kasner, Virgil Snyder and Henry Seely White. By one count Klein supervised 48 doctorates over his career, among them Hurwitz, von Dyck, Robert Fricke, Philipp Furtwängler and Ludwig Bieberbach.

Klein argued that school mathematics should centre on the function concept and lead to the elements of calculus. His 1905 Meran proposals shaped the curricula of German secondary schools. In 1908, at the International Congress in Rome, he was elected the first president of the International Commission on Mathematical Instruction. His lectures for teachers, Elementarmathematik vom höheren Standpunkte aus (from 1908), remain in print.

Lie

The friendship with Lie ended badly. Klein had burned Lie’s letters up to 1877, against an agreement between them, and Lie came to feel that Klein claimed too much credit for ideas they had shared. In the preface to the third volume of his Theorie der Transformationsgruppen (1893), Lie stated publicly that he was no pupil of Klein.

Last years and legacy, 1913–1925

In October 1914 Klein was one of the 93 signatories of the manifesto An die Kulturwelt, which defended Germany’s conduct in the opening weeks of the war, including the invasion of Belgium. The episode damaged his standing with colleagues abroad.

Ill health forced him to retire in 1913, but he went on teaching students in his home through the war years. In 1918 he took an active part in the debate on general relativity (Lesson 10). He supervised the edition of his collected papers, the Gesammelte mathematische Abhandlungen (1921–23), adding commentaries on their origins, and lectured on the history of nineteenth-century mathematics; the lectures were published after his death as Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (1926–27). He died in Göttingen on 22 June 1925, aged 76.

Honours

Klein was elected a Fellow of the Royal Society in 1885 and received the De Morgan Medal of the London Mathematical Society in 1893, the Copley Medal of the Royal Society in 1912 and the Ackermann–Teubner Memorial Award in 1914. Since 2003 the International Commission on Mathematical Instruction has awarded the Felix Klein Medal for lifetime achievement in mathematics education.

Legacy

• Geometry: the Erlangen Program’s idea that symmetry defines geometry runs through Cartan’s geometry, Lie theory, and the gauge theories of physics.
• Hyperbolic geometry and groups: Kleinian groups became central again after the work of Lars Ahlfors, Lipman Bers and William Thurston on hyperbolic 3-manifolds.
• Invariants and singularities: the icosahedral relation is the first case of the ADE classification.
• Modular forms: Klein’s and Fricke’s treatises laid the groundwork for the theory of modular curves used in modern number theory.
• Institutions and teaching: Göttingen, the Encyklopädie, ICMI and the reform of school mathematics.

Connections

People linked to Felix Klein across this collection. A name without a link has no page here yet. See Felix Klein in the network of all 337 lives →

Teachers and advisors

• Julius Plücker 1866-1868

Students

• Adolf Hurwitz 1881
• Thomas Young 1895
• Ferdinand von Lindemann 1873
• Ludwig Bieberbach
• Maxime Bôcher
• Philipp Furtwängler
• Robert Fricke
• Walther von Dyck

Influenced by

• Arthur Cayley
• August Ferdinand Möbius
• Bernhard Riemann
• Camille Jordan 1870
• Eugenio Beltrami
• Hermann Grassmann
• Jean-Victor Poncelet
• János Bolyai
• Nikolai Lobachevsky
• Alfred Clebsch 1868-1872

Influenced

• David Mumford
• Gregorio Ricci-Curbastro 1877-1878
• Hermann Weyl
• Max Born
• Paul Ehrenfest 1901-1903
• Theodore von Kármán 1906-1912
• Élie Cartan

Colleagues and collaborators

• Albert Einstein 1917-1918
• Arnold Sommerfeld 1894-1910
• David Hilbert 1895-1925
• Emmy Noether 1915-1925
• Erich Hecke
• Hermann Minkowski 1902-1909
• Ludwig Prandtl 1904-1925
• Max Noether
• Paul Gordan 1874-1875
• Richard Courant
• Richard Dedekind
• Sophus Lie 1869-1893
• Tullio Levi-Civita
• Alexander von Brill 1875-1880

Rivals

• Georg Frobenius
• Henri Poincaré 1881-1882

Also mentioned in this life

• Lars Ahlfors
• Georg Cantor
• Gaston Darboux
• Augustus De Morgan
• Peter Gustav Lejeune Dirichlet
• Leonhard Euler
• Évariste Galois
• Carl Friedrich Gauss
• Charles Hermite
• Leopold Kronecker
• Ernst Eduard Kummer
• Hendrik Antoon Lorentz
• Wolfgang Pauli
• Max Planck
• William P. Thurston
• Karl Weierstrass

Fields

• geometry 32 lives
• group theory 9 lives
• complex analysis 13 lives

On CoursesHub

• Group Theory: From Galois to Gauge · History of Mathematics & Physics
• Group Theory · Physical Mathematics
• Complex Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Felix Christian Klein, MacTutor History of Mathematics.
2. Felix Klein, Wikipedia.
3. H. K. Strick, Felix Klein, Spektrum der Wissenschaft (2012), English version via MacTutor.
4. Klein, Christian Felix, Dictionary of Scientific Biography.
5. R. Tobies, Felix Klein: Visions for Mathematics, Applications, and Education, Birkhäuser, 2021.
6. D. E. Rowe, Felix Klein, Springer, 2025.
7. Y. Tschinkel, About the cover: the Felix Klein Protocols, Notices AMS (2007).
8. W. Abikoff, The uniformization theorem, Amer. Math. Monthly 88 (1981).
9. Marius Sophus Lie, MacTutor.

Mathematics

1. F. Klein, Über die sogenannte Nicht-Euklidische Geometrie, Math. Ann. 4 (1871) and 6 (1873).
2. F. Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen, Erlangen, 1872; Math. Ann. 43 (1893). English translation by M. W. Haskell, Bull. New York Math. Soc. 2 (1893); see Erlangen program.
3. F. Klein, Über die Transformation siebenter Ordnung der elliptischen Functionen, Math. Ann. 14 (1879). English translation in S. Levy (ed.), The Eightfold Way: The Beauty of Klein’s Quartic Curve, MSRI Publications 35, Cambridge, 1999.
4. F. Klein, Über Riemanns Theorie der algebraischen Functionen und ihre Integrale, Teubner, 1882.
5. F. Klein, Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade, Teubner, 1884 (English: Lectures on the Icosahedron, Dover).
6. F. Klein and R. Fricke, Vorlesungen über die Theorie der elliptischen Modulfunctionen, 2 vols., 1890–92.
7. F. Klein and A. Sommerfeld, Über die Theorie des Kreisels, 4 vols., 1897–1910.
8. F. Klein, Elementary Mathematics from a Higher Standpoint, 3 vols., Springer (new translation 2016).
9. F. Klein, On Hilbert’s first note on the foundations of physics and On the differential laws for the conservation of momentum and energy in Einstein’s theory of gravitation, English translations, arXiv:1912.12158 and arXiv:1912.12161.
10. J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, Birkhäuser, 2nd ed., 2000.
11. I. M. Yaglom, Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century, Birkhäuser, 1988.