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

geometry
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 in
Düsseldorf 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 ...

Düsseldorf
, 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 ...

Prussia
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 ...

University of Bonn
, 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 ...

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 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 ...

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 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 ...

Berlin
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 ...

Alexander von Brill
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 ...

Adolf Hurwitz
,
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 ...

Carl Runge
,
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 ...

Max Planck
,
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 ...

Luigi Bianchi
, 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 ...

geometry
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 ...

Leipzig
. 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 ...

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 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 ...

geometry
. 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 ...

mechanics
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 ...

algebraic geometry
, 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 ...

real analysis
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 ...

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 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 ...

calculus
, 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 ...

World War I
. He died in Göttingen in 1925.


Work

Klein's dissertation, on
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 ...

line geometry
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 ...

mechanics
, 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 ...

Weierstrass
'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 ...

Sophus Lie
, 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 ...

Kummer surface
. 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 ...

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 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 ...

Möbius strip
, 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 ...

Möbius strip
s. During the 1890s, Klein began studying
mathematical 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 (). ...

gyroscope
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 considered
metric 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 ...

Arthur Cayley
never accepted Klein's argument, believing it to be circular. Klein's synthesis 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 ...

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

Number theory
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 ...

Geometry
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 ...

tessellate
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''3''y'' + ''y''3''z'' + ''z''3''x'' = 0, and that its group of
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 ...

Leopold Kronecker
, 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 ...

icosahedron
, 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 (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