Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with

.
During the 1890s, Klein began studying

geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

as the study of the properties of a space that is invariant under a given group of transformations, known as the ''

^{3}''y'' + ''y''^{3}''z'' + ''z''^{3}''x'' = 0, and that its group of

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

''Über die hypergeometrische Funktion''

* 1894: ''Über lineare Differentialgleichungen der 2. Ordnung'' * 1897: (with

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

"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

L'oeuvre mathematique de Klein

in ''Scientia''.

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

group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

, complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

, non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...

, and on the associations between geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

. His 1872 Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"titl ...

, classifying geometries by their basic symmetry group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 ...

s, was an influential synthesis of much of the mathematics of the time.
Life

Felix Klein was born on 25 April 1849 inDüsseldorf
Düsseldorf ( , , ; often in English sources; Low Franconian
Low Franconian, Low Frankish, NetherlandicSarah Grey Thomason, Terrence Kaufman: ''Language Contact, Creolization, and Genetic Linguistics'', University of California Press, 199 ...

, to Prussia
Prussia, , Old Prussian
Distribution of the Baltic tribes, circa 1200 CE (boundaries are approximate).
Old Prussian was a Western Baltic language belonging to the Balto-Slavic branch of the Indo-European languages
The Indo-Europ ...

n parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province
The Rhine Province (german: Rheinprovinz), also known as Rhenish Prussia (''Rheinpreußen'') or synonymous with the Rhineland
The Rhineland (german: Rheinland; french: Rhénanie; nl, Rijnland; ksh, Rhingland; Latinised name: ''Rhenania'' ...

. His mother was Sophie Elise Klein (1819–1890, née __NOTOC__
A birth name is the name of the person given upon their birth. The term may be applied to the surname
In some cultures, a surname, family name, or last name is the portion of one's personal name
300px, First/given, middle and l ...

Kayser). He attended the Gymnasium
Gymnasium may refer to:
*Gymnasium (ancient Greece), educational and sporting institution
*Gymnasium (school), type of secondary school that prepares students for higher education
**Gymnasium (Denmark)
**Gymnasium (Germany)
**Gymnasium UNT, high ...

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
A public university or public college is a university
A university ( la, universitas, 'a whole') is ...

, 1865–1866, intending to become a physicist. At that time, Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (n ...

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
In geometry, line coordinates are used to specify the position of a Line (geometry), line just as point coordinates (or simply Coordinate system, coordinates) are used to specify the position of a point.
Lines in the plane
There are several possibl ...

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
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...

, who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to Berlin
Berlin (; ) is the Capital city, capital and List of cities in Germany by population, largest city of Germany by both area and population. Its 3,769,495 inhabitants, as of 31 December 2019 makes it the List of cities in the European Union by ...

and Paris. In July 1870, at the beginning of the Franco-Prussian War
The Franco-Prussian War or Franco-German War,, german: Deutsch-Französischer Krieg often referred to in France as the War of 1870, was a conflict between the Second French Empire (later the Third French Republic) and the North German Confeder ...

, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army
The Royal Prussian Army (1701–1919, german: Königlich Preußische Armee) served as the army of the Kingdom of Prussia. It became vital to the development of Brandenburg-Prussia as a European power.
The Prussian Army had its roots in the cor ...

before being appointed lecturer at Göttingen in early 1871.
Erlangen
Erlangen (; East Franconian
East Franconian (german: Ostfränkisch), usually referred to as Franconian (') in German, is a dialect which is spoken in Franconia
Franconia (german: Franken; in the Franconian dialect: ''Franggn'' rɑŋgŋ ...

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
Alexander Wilhelm von Brill (20 September 1842 – 18 June 1935) was a Germany, German mathematician.
Born in Darmstadt, Hesse, Brill was educated at the University of Giessen, where he earned his doctorate under supervision of Alfred Clebsch. He h ...

taught advanced courses to many excellent students, including Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

, Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck and later von, ennobled, was a Germany, German mathematician. He is credited with being the first to define a mathematical group (mathematics), group, in the modern sen ...

, Karl Rohn, Carl Runge
Carl David Tolmé Runge (; 30 August 1856 – 3 January 1927) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics ...

, Max Planck
Max Karl Ernst Ludwig Planck, (; ; 23 April 1858 – 4 October 1947) was a Germans, German theoretical physicist whose discovery of quantum mechanics, energy quanta won him the Nobel Prize in Physics in 1918.
Planck made many substantial co ...

, Luigi Bianchi
Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italians, Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous Italian school of algebraic geometry, geometric school which flo ...

, and Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity ( ...

.
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
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For cit ...

.
After spending five years at the Technische Hochschule, Klein was appointed to a chair of geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

at Leipzig
Leipzig (, ; Upper Saxon: ) is the most populous city in the Germany, German States of Germany, state of Saxony. With a population of 605,407 inhabitants as of 2021 (1.1 million residents in the larger urban zone), it surpasses the Saxon c ...

. There his colleagues included Walther von Dyck
Walther Franz Anton von Dyck (6 December 1856 – 5 November 1934), born Dyck and later von, ennobled, was a Germany, German mathematician. He is credited with being the first to define a mathematical group (mathematics), group, in the modern sen ...

, Rohn, Eduard Study
Eduard Study, more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) inc ...

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
The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...

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 (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

. He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular, mechanics
Mechanics (Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

and potential theoryIn 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 gravity ...

.
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
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician
This is a List of German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, G ...

from the University of Königsberg
The University of Königsberg (german: Albertus-Universität Königsberg) was the university
A university ( la, universitas, 'a whole') is an educational institution, institution of higher education, higher (or Tertiary education, tertiary) educ ...

. 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
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ance ...

'' 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
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augu ...

'', based at the University of Berlin
Humboldt University of Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a public
In public relations and communication science, publics are groups of individual people, and the public (a.k.a. the general public) ...

. 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
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

, algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...

, and invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

. It also provided an
important outlet for real analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

and the new group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

.
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 (the official shortened name for the World's Fair: 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 Columb ...

. 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
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

'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 i ...

.
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
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...

, 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 generalizations ...

, and the function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

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 InstructionThe International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union and is an internationally acting organization focussing on mathematics education. ICMI was founded in 1908 at the International C ...

at the Rome International Congress of Mathematicians
The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU).
The Fields Medals, the Nevanlinna Prize (to be rename ...

. 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
A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization
An organization, or organisatio ...

awarded Klein its De Morgan Medal
The De Morgan Medal is a prize for outstanding contribution to mathematics, awarded by the London Mathematical Society. The Society's most prestigious award, it is given in memory of Augustus De Morgan, who was the first President of the society. ...

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
A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exis ...

in 1885, and was awarded its Copley Medal
The Copley Medal is an award given by the Royal Society, for "outstanding achievements in research in any branch of science". It alternates between the physical sciences or mathematics and the biological sciences. Given every year, the medal is t ...

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, a document penned in support of the German invasion of Belgium in the early stages of World War I
World War I, often abbreviated as WWI or WW1, also known as the First World War or the Great War, was a global war
A world war is "a war engaged in by all or most of the principal nations of the world". The term is usually reserved for ...

.
He died in Göttingen in 1925.
Work

Klein's dissertation, online geometry
In geometry, line coordinates are used to specify the position of a Line (geometry), line just as point coordinates (or simply Coordinate system, coordinates) are used to specify the position of a point.
Lines in the plane
There are several possibl ...

and its applications to mechanics
Mechanics (Greek#REDIRECT Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece
Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximat ...

, classified second degree line complexes using Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematics, mathematician often cited as the "father of modern mathematical analysis, analysis". Despite leaving university withou ...

's theory of elementary divisors.
Klein's first important mathematical discoveries were made during 1870. In collaboration with Sophus Lie
Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norway, Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations.
Biography
Marius Sop ...

, he discovered the fundamental properties of the asymptotic lines on the Kummer surface
Image:Kummer surface.png, 400px, Plot of the real points
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible space, irreducible nodal surface of degree 4 in Projective_space#Definition_of_projective_space, \mathbb ...

. They later investigated W-curves, curves invariant under a group of projective transformation
In projective geometry, a homography is an isomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), a ...

s. 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
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...

.
Klein devised the "Klein bottle in three-dimensional space
In topology, a branch of mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; it is a two-dimensional manifold against which a system for determining a normal vec ...

" named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...

, 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, band, or loop ( , ; ), also spelled ''Mobius'' or ''Moebius'', is a Surface (topology), surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary (topology), boundary ...

, with one method of construction being the attachment of the edges of two Möbius strip
In mathematics, a Möbius strip, band, or loop ( , ; ), also spelled ''Mobius'' or ''Moebius'', is a Surface (topology), surface with only one side (when embedded in three-dimensional Euclidean space) and only one boundary (topology), boundary ...

smathematical physics
Mathematical physics refers to the development of 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 the developme ...

more intensively, writing on the gyroscope
A gyroscope (from Ancient Greek
Ancient Greek includes the forms of the used in and the from around 1500 BC to 300 BC. It is often roughly divided into the following periods: (), Dark Ages (), the period (), and the period ().
...

with Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German people, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students f ...

. 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 consideredmetric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s determined by a Cayley–Klein metric
In mathematics, a Cayley–Klein metric is a metric (mathematics), metric on the complement of a fixed quadric in a projective space which is defined using a cross-ratio. The construction originated with Arthur Cayley's essay "On the theory of dist ...

. This insight had the corollary that non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geome ...

was consistent if and only if Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandria
Alexandria ( or ; ar, الإسكندرية ; arz, اسكندرية ; Coptic
Coptic may refer to:
Afro-Asia
* Copts, an ethnoreligious group mainly in the area of modern ...

was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

never accepted Klein's argument, believing it to be circular.
Klein's synthesis of Erlangen program
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"titl ...

'' (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 transformation
Transformation may refer to:
Science and mathematics
In biology and medicine
* Metamorphosis, the biological process of changing physical form after birth or hatching
* Malignant transformation, the process of cells becoming cancerous
* Transf ...

s 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 oncomplex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis
Analysis is the branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

as his major contribution to mathematics, specifically his work on:
*The link between certain ideas of Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...

and invariant theory
Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descr ...

,
*Number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen ...

and abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...

;
*Group theory
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

;
*Geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

in more than 3 dimensions and differential equations
In mathematics, a differential equation is an equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...

, especially equations he invented, satisfied by elliptic modular functions and automorphic functionIn mathematics, an automorphic function is a function on a space that is invariant under the action
ACTION is a bus operator in Canberra
Canberra ( )
is the capital city of Australia. Founded following the Federation of Australia, federa ...

s.
Klein showed that the modular group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

moves the fundamental region of the complex plane
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

so as to tessellate
A tiling or tessellation of a flat surface is the covering of a plane (mathematics), plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to high-dimensional sp ...

the plane. In 1879, he examined the action of PSL(2,7)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

, considered as an image of the modular group
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and obtained an explicit representation of a Riemann surface
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

now termed the Klein quartic
In hyperbolic geometry
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathema ...

. He showed that it was a complex curve in projective space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

, that its equation was ''x''symmetries
Symmetry (from Ancient Greek, Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" ...

was PSL(2,7)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

of order
Order, ORDER or Orders may refer to:
* Orderliness
Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...

168. His ''Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale'' (1882) treats complex analysis in a geometric way, connecting potential theoryIn 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 gravity ...

and conformal mapping
Conformal may refer to:
* Conformal (software), in ASIC Software
* Conformal coating in electronics
* Conformal cooling channel, in injection or blow moulding
* Conformal field theory in physics, such as:
** Boundary conformal field theory
** Co ...

s. 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) and ...

.
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
Charles Hermite () FRS
FRS may also refer to:
Government and politics
* Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States
* Family Reso ...

and Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

, he produced similar results to those of Brioschi and later completely solved the problem by means of the icosahedral group
A regular icosahedron has 60 rotational (or orientation-preserving) symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation. A regular dodecahedron has the same set of symmetries, since it is th ...

. This work enabled him to write a series of papers on elliptic modular functions.
In his 1884 book on the icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non-similarity (geometry), similar shapes of icosahedra, some of them ...

, Klein established a theory of automorphic functionIn mathematics, an automorphic function is a function on a space that is invariant under the action
ACTION is a bus operator in Canberra
Canberra ( )
is the capital city of Australia. Founded following the Federation of Australia, federa ...

s, associating algebra and geometry. Poincaré 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
In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that ...

that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it.
Klein summarized his work on automorphic and elliptic modular functions in a four volume treatise, written with Robert Fricke
Karl Emanuel Robert Fricke (24 September 1861 – 18 July 1930) was a German mathematician, known for his work in complex analysis, especially on elliptic functions, elliptic, modular function, modular and automorphic form, automorphic functio ...

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
A digital library, also called an online library, an internet library, a digital repository, or a digital collection is an online databaseAn online database is a database
In computing ...

* 1886: ''Über hyperelliptische Sigmafunktionen'' Erster Aufsatz p. 323–356, Mathematische Annalen
''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ance ...

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
Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German people, German theoretical physicist who pioneered developments in atomic physics, atomic and quantum physics, and also educated and mentored many students f ...

) ''Theorie des Kreisels'' (later volumes: 1898, 1903, 1910)
* 1890: (with Robert Fricke
Karl Emanuel Robert Fricke (24 September 1861 – 18 July 1930) was a German mathematician, known for his work in complex analysis, especially on elliptic functions, elliptic, modular function, modular and automorphic form, automorphic functio ...

) ''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 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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Line complex
* Grünbaum–Rigby configuration
* Homomorphism
* Ping-pong lemma
* Prime form
* W-curve
* Uniformization theorem
* Felix Klein Protocols
* List of things named after Felix Klein
References

Further reading

* David Mumford, Caroline Series, and David Wright ''Indra's Pearls (book), Indra's Pearls: The Vision of Felix Klein''. Cambridge Univ. Press. 2002. * Renate Tobies, Tobies, Renate (with Fritz König) ''Felix Klein''. Teubner Verlag, Leipzig 1981. * David E. Rowe, Rowe, David "Felix Klein, David Hilbert, and the Göttingen Mathematical Tradition", in Science in Germany: The Intersection of Institutional and Intellectual Issues, Kathryn Olesko, ed., Osiris, 5 (1989), 186–213. * Federigo Enriques (1921L'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