Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
,
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
,
non-Euclidean geometry, and on the associations between
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. His 1872
Erlangen program, classifying geometries by their basic
symmetry groups, was an influential synthesis of much of the mathematics of the time.
Life

Felix Klein was born on 25 April 1849 in
Düsseldorf
Düsseldorf ( , , ; often in English sources; Low Franconian and Ripuarian: ''Düsseldörp'' ; archaic nl, Dusseldorp ) is the capital city of North Rhine-Westphalia, the most populous state of Germany. It is the second-largest city in ...
, to
Prussia
Prussia, , Old Prussian: ''Prūsa'' or ''Prūsija'' was a German state on the southeast coast of the Baltic Sea. It formed the German Empire under Prussian rule when it united the German states in 1871. It was ''de facto'' dissolved by an ...
n parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the
Rhine Province. His 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
The Rhenish Friedrich Wilhelm University of Bonn (german: Rheinische Friedrich-Wilhelms-Universität Bonn) is a public research university located in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the ( en, Rhine ...
, 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, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.
Plücker died in 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 in 1868. Klein visited Clebsch the next year, along with visits to
Berlin
Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...
and Paris. In 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 in early 1871.
Erlangen
Erlangen (; East Franconian: ''Erlang'', Bavarian: ''Erlanga'') is a Middle Franconian city in Bavaria, Germany. It is the seat of the administrative district Erlangen-Höchstadt (former administrative district Erlangen), and with 116,062 inhabi ...
appointed Klein professor in 1872, when he was only 23 years old. For this, he was endorsed by Clebsch, who regarded him as likely to become the best mathematician of his time. Klein did not wish to remain in Erlangen, where there were very few students, and was pleased to be offered a professorship at the
Technische Hochschule München in 1875. There he and
Alexander von Brill taught advanced courses to many excellent students, including
Adolf Hurwitz,
Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundation ...
,
Karl Rohn,
Carl Runge,
Max Planck,
Luigi Bianchi, and
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus.
With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on t ...
.
In 1875, Klein married Anne Hegel, granddaughter of the philosopher
Georg Wilhelm Friedrich Hegel
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends ...
.
After spending five years at the Technische Hochschule, Klein was appointed to a chair of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
at
Leipzig
Leipzig ( , ; Upper Saxon: ) is the most populous city in the German state of Saxony. Leipzig's population of 605,407 inhabitants (1.1 million in the larger urban zone) as of 2021 places the city as Germany's eighth most populous, as ...
. There his colleagues included
Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck () and later ennobled, was a German mathematician. He is credited with being the first to define a mathematical group, in the modern sense in . He laid the foundation ...
, Rohn,
Eduard Study and
Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883–1884, he was afflicted with depression. Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period.
Klein accepted a professorship at the
University of Göttingen in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer of
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
. He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular,
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
and
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
.
The research facility Klein established at Göttingen served as model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruited
David Hilbert from the
University of Königsberg. This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932.
Under Klein's editorship, ''
Mathematische Annalen'' became one of the best mathematical journals in the world. Founded by Clebsch, it grew under Klein's management, to rival, and eventually surpass ''
Crelle's Journal'', based at the
University of Berlin. Klein established a small team of editors who met regularly, making decisions in a democratic spirit. The journal first specialized in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
,
algebraic geometry, and
invariant theory. It also provided an
important outlet for
real analysis and the new
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
.
In 1893, Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the
World's Columbian Exposition
The World's Columbian Exposition (also known as the Chicago World's Fair) was a world's fair held in Chicago in 1893 to celebrate the 400th anniversary of Christopher Columbus's arrival in the New World in 1492. The centerpiece of the Fair, h ...
. Due partly 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, by
Grace Chisholm Young, an English student of
Arthur Cayley's, whom Klein admired. In 1897, Klein became a foreign member of the
Royal Netherlands Academy of Arts and Sciences
The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ...
.
Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending that
analytic geometry, the rudiments of differential and integral
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, 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 president of the
International Commission on Mathematical Instruction at the Rome
International Congress of Mathematicians. Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany.
The
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...
awarded Klein its
De Morgan Medal in 1893. He was elected a member of the
Royal Society
The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, r ...
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 several further years.
Klein was one of ninety-three signatories of the
Manifesto of the Ninety-Three
The "Manifesto of the Ninety-Three" (originally "To the Civilized World" by "Professors of Germany") is a 4 October 1914 proclamation by 93 prominent Germans supporting Germany in the start of World War I. The Manifesto galvanized support for the w ...
, a document penned in support of the German invasion of Belgium in the early stages of
World War I
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was List of wars and anthropogenic disasters by death toll, one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, ...
.
He died in Göttingen in 1925.
Work
Klein's dissertation, on
line geometry and its applications to
mechanics
Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects ...
, classified second degree line complexes using
Weierstrass's theory of elementary divisors.
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.
Klein devised the "
Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, 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
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
, with one method of construction being the attachment of the edges of two
Möbius strip
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
s
.
During the 1890s, Klein began studying
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
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
''Encyklopädie der mathematischen Wissenschaften''. This enterprise, which endured until 1935, provided an important standard reference of enduring value.
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
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general sett ...
s determined by a
Cayley–Klein metric. This insight had the corollary that
non-Euclidean geometry was consistent if and only if
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
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
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
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.
Currently, the significance of Klein's contributions to geometry is evident. They have become so much part of mathematical thinking that it is difficult to appreciate their novelty when first presented, and understand the fact that they were not immediately accepted by all his contemporaries.
Complex analysis
Klein saw his work on
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
as his major contribution to mathematics, specifically his work on:
*The link between certain ideas of
Riemann and
invariant theory,
*
Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
and
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
;
*
Group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
;
*
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
in more than 3 dimensions and
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
, especially equations he invented, satisfied by
elliptic modular functions and
automorphic functions.
Klein showed that the
modular group moves the fundamental region of the
complex plane so as to
tessellate the plane. In 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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
now termed the
Klein quartic. He showed that it was a complex curve in
projective space, that its equation was ''x''
3''y'' + ''y''
3''z'' + ''z''
3''x'' = 0, and that its group of
symmetries was
PSL(2,7) of
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
168. His ''Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale'' (1882) treats complex analysis in a geometric way, connecting
potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
and
conformal mappings. This work drew on notions from
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other 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 methods of
Charles Hermite and
Leopold Kronecker, he produced similar results to those of Brioschi and later completely solved the problem by means of the
icosahedral group. This work enabled him to write a series of papers on
elliptic modular functions.
In his 1884 book on the
icosahedron, Klein established a theory of
automorphic functions, associating algebra and geometry.
Poincaré
Poincaré is a French surname. Notable people with the surname include:
* Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science
* Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré
* Luci ...
had published an outline of his theory of automorphic functions in 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. He came up with his proof during an
asthma attack at 2:30 A.M. on March 23, 1882.
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.
Selected works
* 1882: ''Über Riemann's Theorie der Algebraischen Functionen und ihre Integrale''
**
also available from Cornell* 1884:''Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade''
** English translation by G. G. Morrice (1888) ''Lectures on the Ikosahedron; and the Solution of Equations of the Fifth Degree'' via
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music ...
* 1886: ''Über hyperelliptische Sigmafunktionen'' Erster Aufsatz p. 323–356,
Mathematische Annalen Bd. 27,
* 1888: ''Über hyperelliptische Sigmafunktionen'' Zweiter Aufsatz p. 357–387, Math. Annalen, Bd. 32,
* 1894
''Über die hypergeometrische Funktion''* 1894: ''Über lineare Differentialgleichungen der 2. Ordnung''
* 1897: (with
Arnold Sommerfeld) ''Theorie des Kreisels'' (later volumes: 1898, 1903, 1910)
* 1890: (with
Robert Fricke) ''Vorlesungen über die Theorie der elliptischen Modulfunktionen'' (2 volumes) and 1892)
* 1894: ''Evanston Colloquium'' (1893) reported and published by Ziwet (New York, 1894)
*
Zweiter Band. 1901.
[
* 1901:
*
* 1897: ''Mathematical Theory of the Top'' (Princeton address, New York)
* 1895: ''Vorträge über ausgewählte Fragen der Elementargeometrie''
** 1897: English translation by W. W. Beman and ]D. E. Smith
David Eugene Smith (January 21, 1860 – July 29, 1944) was an American mathematician, educator, and editor.
Education and career
David Eugene Smith is considered one of the founders of the field of mathematics education. Smith was born in Cort ...
Famous Problems of Elementary Geometry
' via Internet Archive
* 1908: ''Elementarmathematik vom höheren Standpunkte aus'' (Leipzig)
* 1926: ''Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert'' (2 Bände), Julius Springer Verlag, Berlin & 1927. S
Felix Klein ''Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert''
* 1928: ''Vorlesungen über nichteuklidische Geometrie'', Grundlehren der mathematischen Wissenschaften, Springer Verlag
* 1933: ''Vorlesungen über die hypergeometrische Funktion'', Grundlehren der mathematischen Wissenschaften, Springer Verlag
Bibliography
*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.
*1921. "Felix Klein gesammelte mathematische Abhandlungen" R. Fricke and A. Ostrowski (eds.) Berlin, Springer. 3 volumes. (online copy a
GDZ
* 1890.
Nicht-Euklidische Geometrie
See also
* Dianalytic manifold
* j-invariant
* Line complex
* Grünbaum–Rigby configuration
* Homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
* Ping-pong lemma
* Prime form
* W-curve
* Uniformization theorem
* Felix Klein Protocols
"The Felix Klein Protocols" is a collection of handwritten records of the Göttingen seminar lectures of Felix Klein and his school. They span over 8000 pages in 29 volumes, and are regarded as one of the richest records of mathematical activity ...
* List of things named after Felix Klein {{Short description, none
These are things named after Felix Klein (1849 – 1925), a German mathematician.
Mathematics
*Klein bottle
**Solid Klein bottle
* Klein configuration
* Klein cubic threefold
* Klein four-group
* Klein geometry
* Klein gr ...
References
Further reading
* David Mumford, Caroline Series, and David Wright '' Indra's Pearls: The Vision of Felix Klein''. Cambridge Univ. Press. 2002.
* Tobies, Renate (with Fritz König) ''Felix Klein''. Teubner Verlag, Leipzig 1981.
* Rowe, David "Felix Klein, David Hilbert, and the Göttingen Mathematical Tradition", in Science in Germany: The Intersection of Institutional and Intellectual Issues, Kathryn Olesko
Kathryn Mary Olesko (born 1951) is an American historian of science. She is an associate professor at Georgetown University, where she is affiliated with the Science, Technology and International Affairs program in the School of Foreign Service, t ...
, ed., Osiris, 5 (1989), 186–213.
* Federigo Enriques (1921
L'oeuvre mathematique de Klein
in ''Scientia''.
External links
*
*
*
*
*
Felix Klein, Klein Protokolle
Felix Klein (Encyclopædia Britannica)
F. Klein, "On the theory of line complexes of first and second order"
F. Klein, "On line geometry and metric geometry"
F. Klein, "On the transformation of the general second-degree equation in line coordinates into canonical coordinates"
{{DEFAULTSORT:Klein, Felix
1849 births
1925 deaths
Scientists from Düsseldorf
19th-century German mathematicians
20th-century German mathematicians
Differential geometers
German military personnel of the Franco-Prussian War
Group theorists
Members of the Prussian House of Lords
People from the Rhine Province
Recipients of the Copley Medal
University of Bonn alumni
Humboldt University of Berlin alumni
University of Göttingen faculty
University of Erlangen-Nuremberg faculty
Technical University of Munich faculty
Leipzig University faculty
Foreign associates of the National Academy of Sciences
Foreign Members of the Royal Society
Members of the Royal Netherlands Academy of Arts and Sciences
Recipients of the Pour le Mérite (civil class)
De Morgan Medallists
Prussian Army personnel
Scientists from North Rhine-Westphalia