Felix Klein

Encyclopedia

Christian Felix Klein was a German

mathematician

, known for his work in group theory

, function theory, non-Euclidean geometry

, and on the connections between geometry

and group theory

. His 1872 Erlangen Program

, classifying geometries by their underlying symmetry groups

, was a hugely influential synthesis of much of the mathematics of the day.

, to Prussia

n parents; his father was a Prussian government official's secretary stationed in the Rhine Province

. He attended the Gymnasium

in Düsseldorf, then studied mathematics and physics at the University of Bonn

, 1865–1866, intending to become a physicist. At that time, Julius Plücker

held Bonn's chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.

Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch

, who had moved to Göttingen in 1868. Klein visited Clebsch the following year, along with visits to Berlin

and Paris

. In July 1870, at the outbreak of the Franco-Prussian War

, he was in Paris and had to leave the country. For a short time, he served as a medical orderly in the Prussian army

before being appointed lecturer at Göttingen in early 1871.

Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, and so he was pleased to be offered a chair at Munich

's Technische Hochschule in 1875. There he and Alexander von Brill

taught advanced courses to many excellent students, e.g., Adolf Hurwitz

, Walther von Dyck

, Karl Rohn, Carl Runge, Max Planck

, Luigi Bianchi

, and Gregorio Ricci-Curbastro

.

In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel

.

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry

at Leipzig

. There his colleagues included Walther von Dyck

, 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 plagued by depression. Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published in 1886 and 1888.

Klein accepted a chair at the University of Göttingen in 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry

. At Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics

and potential theory

.

The research center Klein established at Göttingen served as a model for the best such centers throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired David Hilbert

away from Königsberg

; this appointment proved fateful, because Hilbert continued Göttingen's glory until his own retirement in 1932.

Under Klein's editorship, Mathematische Annalen

became one of the very best mathematics journals in the world. Founded by Clebsch, only under Klein's management did it first rival then surpass Crelle's Journal

based out of the University of Berlin. Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in complex analysis

, algebraic geometry

, and invariant theory

(at least until Hilbert killed the subject). It also provided an important outlet for real analysis

and the new group theory

.

Thanks in part to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman; she was Grace Chisholm Young

, an English student of Arthur Cayley

's, whom Klein admired.

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that the rudiments of differential and integral calculus and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society

awarded Klein its De Morgan Medal

in 1893. He was elected a member of the Royal Society

in 1885, and was awarded its Copley medal

in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for some years more.

Klein bore the title of Geheimrat

.

He died in Göttingen in 1925.

, classified second degree line complexes using Weierstrass's theory of elementary divisors.

Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie

, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curve

s, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan

.

Klein devised the bottle

named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space

, 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 Euclidean space of dimensions 4 and higher.

In the 1890s, Klein turned to mathematical physics

, a subject from which he had never strayed far, writing on the gyroscope

with Arnold Sommerfeld

. In 1894 he launched the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften

. This enterprise, which ran until 1935, provided an important standard reference of enduring value.

adjoined. This had the remarkable corollary that non-Euclidean geometry

was consistent if and only if Euclidean geometry

was, putting Euclidean and non-Euclidean geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry. Cayley

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

Klein's synthesis of geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

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 set out in Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The Program proposed a unified approach to geometry that became (and remains) the accepted view. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the Programs definition of geometry encompassed both Euclidean and non-Euclidean geometry.

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

Klein showed that the modular group

moves the fundamental region of the complex plane

so as to tessellate

that plane. In 1879, he looked at the action of PSL(2,7)

, thought of as an image of the modular group

, and obtained an explicit representation of a Riemann surface

today called the Klein quartic

. He showed that that surface was a curve in projective space

, that its equation was x³y + y³z + z³x = 0, and that its group of symmetries

was PSL(2,7)

of order

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

and conformal mappings. This work drew on notions from fluid dynamics

.

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on the methods of Hermite

and Kronecker

, he produced similar results to those of Brioschi and went on to completely solve the problem by means of the icosahedral group. This work led him to write a series of papers on elliptic modular functions.

In his 1884 book on the icosahedron

, Klein set out a theory of automorphic function

s, connecting algebra and geometry. However Poincaré

published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem

that would serve as a capstone to the emerging theory. Klein succeeded in formulating such a theorem and in sketching a strategy for proving it. But while doing this work his health collapsed, as mentioned above.

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.

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

mathematician

Mathematician

A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

, known for his work in group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, function theory, non-Euclidean geometry

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

, and on the connections between geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

and group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

. His 1872 Erlangen Program

Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

, classifying geometries by their underlying symmetry groups

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

, was a hugely influential synthesis of much of the mathematics of the day.

## Life

Klein was born in DüsseldorfDüsseldorf

Düsseldorf is the capital city of the German state of North Rhine-Westphalia and centre of the Rhine-Ruhr metropolitan region.Düsseldorf is an important international business and financial centre and renowned for its fashion and trade fairs. Located centrally within the European Megalopolis, the...

, to Prussia

Prussia

Prussia was a German kingdom and historic state originating out of the Duchy of Prussia and the Margraviate of Brandenburg. For centuries, the House of Hohenzollern ruled Prussia, successfully expanding its size by way of an unusually well-organized and effective army. Prussia shaped the history...

n parents; his father was a Prussian government official's secretary stationed in the Rhine Province

Rhine Province

The Rhine Province , also known as Rhenish Prussia or synonymous to the Rhineland , was the westernmost province of the Kingdom of Prussia and the Free State of Prussia, within the German Reich, from 1822-1946. It was created from the provinces of the Lower Rhine and Jülich-Cleves-Berg...

. He attended the Gymnasium

Gymnasium (school)

A gymnasium is a type of school providing secondary education in some parts of Europe, comparable to English grammar schools or sixth form colleges and U.S. college preparatory high schools. The word γυμνάσιον was used in Ancient Greece, meaning a locality for both physical and intellectual...

in Düsseldorf, then studied mathematics and physics at the University of Bonn

University of Bonn

The University of Bonn is a public research university located in Bonn, Germany. Founded in its present form in 1818, as the linear successor of earlier academic institutions, the University of Bonn is today one of the leading universities in Germany. The University of Bonn offers a large number...

, 1865–1866, intending to become a physicist. At that time, Julius Plücker

Julius Plücker

Julius Plücker was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves.- Early...

held Bonn's chair of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.

Plücker died in 1868, leaving his book on the foundations of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch

Alfred Clebsch

Rudolf Friedrich Alfred Clebsch was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. He subsequently taught in Berlin and Karlsruhe...

, who had moved to Göttingen in 1868. Klein visited Clebsch the following year, along with visits to Berlin

Berlin

Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...

and Paris

Paris

Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...

. In July 1870, at the outbreak of the Franco-Prussian War

Franco-Prussian War

The Franco-Prussian War or Franco-German War, often referred to in France as the 1870 War was a conflict between the Second French Empire and the Kingdom of Prussia. Prussia was aided by the North German Confederation, of which it was a member, and the South German states of Baden, Württemberg and...

, he was in Paris and had to leave the country. For a short time, he served as a medical orderly in the Prussian army

Prussian Army

The Royal Prussian Army was the army of the Kingdom of Prussia. It was vital to the development of Brandenburg-Prussia as a European power.The Prussian Army had its roots in the meager mercenary forces of Brandenburg during the Thirty Years' War...

before being appointed lecturer at Göttingen in early 1871.

Erlangen appointed Klein professor in 1872, when he was only 23. In this, he was strongly supported by Clebsch, who regarded him as likely to become the leading mathematician of his day. Klein did not build a school at Erlangen where there were few students, and so he was pleased to be offered a chair at Munich

Munich

Munich The city's motto is "" . Before 2006, it was "Weltstadt mit Herz" . Its native name, , is derived from the Old High German Munichen, meaning "by the monks' place". The city's name derives from the monks of the Benedictine order who founded the city; hence the monk depicted on the city's coat...

's Technische Hochschule in 1875. There he and Alexander von Brill

Alexander von Brill

Alexander Wilhelm von Brill was a German mathematician.Born in Darmstadt, Hesse, he attended University of Giessen where he earned his doctorate under supervision of Alfred Clebsch. He held a chair at the University of Tübingen, where Max Planck was among his students.-External links:...

taught advanced courses to many excellent students, e.g., Adolf Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

, Walther von Dyck

Walther von Dyck

Walther Franz Anton von Dyck , born Dyck and later ennobled, was a German mathematician...

, Karl Rohn, Carl Runge, Max Planck

Max Planck

Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

, Luigi Bianchi

Luigi Bianchi

- External links :* offers translations of some of Bianchi's papers, plus a biography of Bianchi.* PDF copy at * * * *...

, and Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro

Gregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....

.

In 1875 Klein married Anne Hegel, the granddaughter of the philosopher Georg Wilhelm Friedrich Hegel

Georg Wilhelm Friedrich Hegel

Georg Wilhelm Friedrich Hegel was a German philosopher, one of the creators of German Idealism. His historicist and idealist account of reality as a whole revolutionized European philosophy and was an important precursor to Continental philosophy and Marxism.Hegel developed a comprehensive...

.

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

at Leipzig

Leipzig

Leipzig Leipzig has always been a trade city, situated during the time of the Holy Roman Empire at the intersection of the Via Regia and Via Imperii, two important trade routes. At one time, Leipzig was one of the major European centres of learning and culture in fields such as music and publishing...

. There his colleagues included Walther von Dyck

Walther von Dyck

Walther Franz Anton von Dyck , born Dyck and later ennobled, was a German mathematician...

, Rohn, Eduard Study

Eduard Study

Eduard Study was a German mathematician known for work on invariant theory of ternary forms and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.Study was born in Coburg in the Duchy of...

and Friedrich Engel

Friedrich Engel (mathematician)

Friedrich Engel was a German mathematician.Engel was born in Lugau, Saxony, as the son of a Lutheran pastor. He attended the Universities of both Leipzig and Berlin, before receiving his doctorate from Leipzig in 1883.Engel studied under Felix Klein at Leipzig, and collaborated with Sophus Lie for...

. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed; in 1883–1884, he was plagued by depression. Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published in 1886 and 1888.

Klein accepted a chair at the University of Göttingen in 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's leading mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own role as the leader of a school of geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. At Göttingen, he taught a variety of courses, mainly on the interface between mathematics and physics, such as mechanics

Mechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

and potential theory

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

.

The research center Klein established at Göttingen served as a model for the best such centers throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein hired David Hilbert

David Hilbert

David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...

away from Königsberg

University of Königsberg

The University of Königsberg was the university of Königsberg in East Prussia. It was founded in 1544 as second Protestant academy by Duke Albert of Prussia, and was commonly known as the Albertina....

; this appointment proved fateful, because Hilbert continued Göttingen's glory until his own retirement in 1932.

Under Klein's editorship, Mathematische Annalen

Mathematische Annalen

Mathematische Annalen is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann...

became one of the very best mathematics journals in the world. Founded by Clebsch, only under Klein's management did it first rival then surpass Crelle's Journal

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 .- History :...

based out of the University of Berlin. Klein set up a small team of editors who met regularly, making democratic decisions. The journal specialized in complex analysis

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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...

, algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, and invariant theory

Invariant theory

Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

(at least until Hilbert killed the subject). It also provided an important outlet for real analysis

Real analysis

Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...

and the new group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

.

Thanks in part to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman; she was Grace Chisholm Young

Grace Chisholm Young

Grace Chisholm Young was an English mathematician. She was educated at Girton College, Cambridge, England and continued her studies at Göttingen University in Germany, where in 1895 she became the first woman to receive a doctorate in any field in that country...

, an English student of Arthur Cayley

Arthur Cayley

Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

's, whom Klein admired.

Around 1900, Klein began to take an interest in mathematical instruction in schools. In 1905, he played a decisive role in formulating a plan recommending that the rudiments of differential and integral calculus and the function concept be taught in secondary schools. This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected chairman of the International Commission on Mathematical Instruction at the Rome International Mathematical Congress. Under his guidance, the German branch of the Commission published many volumes on the teaching of mathematics at all levels in Germany.

The London Mathematical Society

London Mathematical Society

-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...

awarded Klein its De Morgan Medal

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

Royal Society

The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...

in 1885, and was awarded its Copley medal

Copley Medal

The Copley Medal is an award given by the Royal Society of London for "outstanding achievements in research in any branch of science, and alternates between the physical sciences and the biological sciences"...

in 1912. He retired the following year due to ill health, but continued to teach mathematics at his home for some years more.

Klein bore the title of Geheimrat

Geheimrat

Geheimrat was the title of the highest advising officials at the Imperial, royal or principal courts of the Holy Roman Empire, who jointly formed the Geheimer Rat reporting to the ruler...

.

He died in Göttingen in 1925.

## Work

Klein's dissertation, on line geometry and its applications to mechanicsMechanics

Mechanics is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment....

, classified second degree line complexes using Weierstrass's theory of elementary divisors.

Klein's first important mathematical discoveries were made in 1870. In collaboration with Sophus Lie

Sophus Lie

Marius Sophus Lie was a Norwegian mathematician. He largely created the theory of continuous symmetry, and applied it to the study of geometry and differential equations.- Biography :...

, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They went on to investigate W-curve

W-curve

In geometry, a W-curve is a curve in projective n-space that is invariant under a 1-parameter group of projective transformations. W-curves were first investigated by Felix Klein and Sophus Lie in 1871, who also named them. W-curves in the real projective plane can be constructed with straightedge...

s, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to play a major role in his later work. Klein also learned about groups from Camille Jordan

Camille Jordan

Marie Ennemond Camille Jordan was a French mathematician, known both for his foundational work in group theory and for his influential Cours d'analyse. He was born in Lyon and educated at the École polytechnique...

.

Klein devised the bottle

Klein bottle

In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...

named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space

Euclidean space

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...

, 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 Euclidean space of dimensions 4 and higher.

In the 1890s, Klein turned to mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

, a subject from which he had never strayed far, writing on the gyroscope

Gyroscope

A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. In essence, a mechanical gyroscope is a spinning wheel or disk whose axle is free to take any orientation...

with Arnold Sommerfeld

Arnold Sommerfeld

Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...

. In 1894 he launched the idea of an encyclopedia of mathematics including its applications, which became the Encyklopädie der mathematischen Wissenschaften

Klein's encyclopedia

In mathematics, Klein’s encyclopedia refers to a German mathematical encyclopedia published in six volumes from 1898 to 1933. Felix Klein and Wilhelm Meyer were organizers of the encyclopedia. Its title in English is "Encyclopedia of mathematical sciences including their applications", which is...

. This enterprise, which ran 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 special cases of a projective surface with a specific conic sectionConic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

adjoined. This had the remarkable corollary that non-Euclidean geometry

Non-Euclidean geometry

Non-Euclidean geometry is the term used to refer to two specific geometries which are, loosely speaking, obtained by negating the Euclidean parallel postulate, namely hyperbolic and elliptic geometry. This is one term which, for historical reasons, has a meaning in mathematics which is much...

was consistent if and only if Euclidean geometry

Euclidean geometry

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...

was, putting Euclidean and non-Euclidean geometries on the same footing, and ending all controversy surrounding non-Euclidean geometry. Cayley

Arthur Cayley

Arthur Cayley F.R.S. was a British mathematician. He helped found the modern British school of pure mathematics....

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

Klein's synthesis of geometry

Geometry

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

Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...

(1872), profoundly influenced the evolution of mathematics. This program was set out in Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The Program proposed a unified approach to geometry that became (and remains) the accepted view. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the Programs definition of geometry encompassed both Euclidean and non-Euclidean geometry.

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

### Function theory

Klein saw his work on function theory as his major contribution to mathematics, specifically his work on:- The link between certain ideas of Riemann's and invariant theoryInvariant theoryInvariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...

, - Number theoryNumber theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...

and abstract algebraAbstract algebraAbstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

; - Group theoryGroup theory

; - GeometryGeometry

with more than 3 dimensions and differential equations, especially equations he invented, namely elliptic modular functions and automorphic functionAutomorphic functionIn mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....

s.

Klein showed that the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

moves the fundamental region of the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

so as to tessellate

Tessellation

A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...

that plane. In 1879, he looked at the action of PSL(2,7)

PSL(2,7)

In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...

, thought of as an image of the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

, and obtained an explicit representation of a Riemann surface

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...

today called the Klein quartic

Klein quartic

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...

. He showed that that surface was a curve in projective space

Projective space

In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....

, that its equation was x³y + y³z + z³x = 0, and that its group of symmetries

Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

was PSL(2,7)

PSL(2,7)

In mathematics, the projective special linear group PSL is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane...

of order

Order (group theory)

In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

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

Potential theory

In mathematics and mathematical physics, potential theory may be defined as the study of harmonic functions.- Definition and comments :The term "potential theory" was coined in 19th-century physics, when it was realized that the fundamental forces of nature could be modeled using potentials which...

and conformal mappings. This work drew on notions from fluid dynamics

Fluid dynamics

In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

.

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on the methods of Hermite

Charles Hermite

Charles Hermite was a French mathematician who did research on number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra....

and Kronecker

Leopold Kronecker

Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...

, he produced similar results to those of Brioschi and went on to completely solve the problem by means of the icosahedral group. This work led him to write a series of papers on elliptic modular functions.

In his 1884 book on the icosahedron

Icosahedron

In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

, Klein set out a theory of automorphic function

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....

s, connecting algebra and geometry. However Poincaré

Henri Poincaré

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...

published an outline of his theory of automorphic functions in 1881, which led to a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem

Uniformization theorem

In mathematics, the uniformization theorem says that any simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular it admits a Riemannian metric of constant curvature...

that would serve as a capstone to the emerging theory. Klein succeeded in formulating such a theorem and in sketching a strategy for proving it. But while doing this work his health collapsed, as mentioned above.

Klein summarized his work on automorphic

Automorphic function

In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group....

and elliptic modular functions in a four volume treatise, written with Robert Fricke

Robert Fricke

Karl Emmanuel Robert Fricke was a German mathematician, known for his work in function theory, especially on elliptic, modular and automorphic functions...

over a period of about 20 years.

## Some of his important works

- Ueber Riemann's Theorie der Algebraischen Functionen und ihre Integrale (1882) , , also available from Cornell
- Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade (1884); English translation by G. G. Morrice, Lectures on the Icosahedron; and the Solution of Equations of the Fifth Degree, (2nd revised edition, New York, 1914)
- Über hyperelliptische Sigmafunktionen Erster Aufsatz p. 323-356, Math. Annalen, Bd. 27, (1886)
- Über hyperelliptische Sigmafunktionen Zweiter Aufsatz p. 357-387, Math. Annalen, Bd. 32, (1888)
- Über die hypergeometrische Funktion (1894)
- Über lineare Differentialgleichungen der 2. Ordnung (1894)
- Theorie des Kreisels, joint with Arnold Sommerfeld (4 volumes: 1897, 1898, 1903, 1910)
- Vorlesungen über die Theorie der elliptischen Modulfunktionen, joint with Robert Fricke (2 volumes: 1890 and 1892)
- Mathematical Theory of the Top (Princeton address, New York, 1897)
- Vorträge über ausgewählte Fragen der Elementargeometrie (1895; English translation by W. W. Beman and D. E. Smith, Famous Problems of Elementary Geometry, Boston, 1897)
- Evanston Colloquium (1893) before the Congress of Mathematics, reported and published by Ziwet (New York, 1894)
- Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908)