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{{Short description|German mathematician (1849–1925)}}
{{about|the German mathematician|the French priest|Félix Klein}}
{{Use dmy dates|date=July 2025}}
{{Infobox scientist
| name              = Felix Klein
| image             = Felix Klein, ante 1897 - Accademia delle Scienze di Torino 0078 B.jpg
| birth_date        = {{birth date|df=y|1849|04|25}}
| birth_place       = [[Düsseldorf]], [[Rhine Province]], [[Kingdom of Prussia|Prussia]]
| death_date        = {{death date and age|df=y|1925|6|22|1849|4|25}}
| death_place       = [[Göttingen]], [[Province of Hanover]], [[Free State of Prussia (1918–1933)|Prussia]], [[Weimar Republic|Germany]]
| alma_mater        = [[University of Bonn]]
| doctoral_advisors = [[Julius Plücker]] and [[Rudolf Lipschitz]]
| doctoral_students = {{show|List|[[Ludwig Bieberbach]]<br>[[Maxime Bôcher]]<br>[[Oskar Bolza]]<br>[[Max Brückner]]<br>[[Frank Nelson Cole]]<br>[[Friedrich Dingeldey]]<br>[[Henry Burchard Fine|Henry B. Fine]]<br>[[Erwin Freundlich]]<br>[[Robert Fricke]]<br>[[Philipp Furtwängler]]<br>[[Carl Gustav Axel Harnack|Axel Harnack]]<br>[[Mellen Haskell]]<br>[[Adolf Hurwitz]]<br>[[Edward Kasner]]<br>[[Ferdinand von Lindemann]]<br>[[Alexander Ostrowski]]<br>[[Julio Rey Pastor]]<br>[[Hermann Rothe]]<br>[[Friedrich Schilling (mathematician)|Friedrich Schilling]]<br>[[Virgil Snyder]]<br>[[Edward Van Vleck]]<br>[[Walther von Dyck]]<br>[[Adolf Weiler]]<br>[[Henry Seely White]]<br>[[Alexander Witting]]<br>[[Grace Chisholm Young]]}}
| notable_students  = [[Edward Kasner]]
| known_for         = [[Erlangen program]]<br />[[Klein bottle]]<br>[[Beltrami–Klein model]]<br>[[Klein's Encyclopedia of Mathematical Sciences|Klein's ''Encyclopedia of Mathematical Sciences'']]<br />[[Kleinian group]]<br />[[Klein four-group]]<br />''[[Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert]]''<br />[[Klein polyhedron]]
| field             = Mathematics
| work_institution  = [[University of Erlangen|Universität Erlangen]]<br>[[Technical University of Munich|Technische Hochschule München]]<br>[[Universität Leipzig]]<br>[[Georg-August-Universität Göttingen]]
| prizes            = [[De Morgan Medal]] (1893)<br>[[Copley Medal]] (1912)<br>[[Ackermann–Teubner Memorial Award]] (1914)
}}

'''Felix Christian Klein''' ({{IPAc-en|k|l|aɪ|n}};<ref>{{Cite Dictionary.com|Klein}}</ref> {{IPA|de|klaɪn|lang}}; 25 April 1849&nbsp;– 22 June 1925) was a German [[mathematician]], [[Mathematics education|mathematics educator]] and [[History of mathematics|historian of mathematics]], known for his work in [[group theory]], [[complex analysis]], [[non-Euclidean geometry]], and the associations between [[geometry]] and [[group theory]]. His 1872 [[Erlangen program]] classified geometries by their basic [[symmetry group]]s and was an influential synthesis of much of the mathematics of the time.

During his tenure at the [[University of Göttingen]], Klein was able to turn it into a center for mathematical and scientific research through the establishment of new lectures, professorships, and institutes. His [[Felix Klein Protocols|seminars]] covered most areas of mathematics then known as well as their applications. Klein also devoted considerable time to mathematical instruction and promoted mathematics education reform at all grade levels in Germany and abroad. He became the first president of the [[International Commission on Mathematical Instruction]] in 1908 at the Fourth International Congress of Mathematicians in Rome.

==Life==
[[File:Felix Klein, Leipzig years.png|thumb|upright|Klein during his Leipzig period.]]
Felix Klein was born on 25 April 1849 in [[Düsseldorf]],<ref>{{cite journal|author=Snyder, Virgil|author-link=Virgil Snyder|title=Klein's Collected Works|journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc.]]|year=1922|volume=28|issue=3|pages=125–129|url=http://projecteuclid.org/euclid.bams/1183425933|doi=10.1090/S0002-9904-1922-03510-0|doi-access=free}}</ref> to [[Prussia]]n parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the [[Rhine Province]]. His mother was Sophie Elise Klein (1819–1890, [[Name at birth|née]] Kayser).<ref>{{cite book|author=Rüdiger Thiele|title=Felix Klein in Leipzig: mit F. Kleins Antrittsrede, Leipzig 1880|url=https://books.google.com/books?id=KznSxd-HcckC&pg=PA195|year=2011|isbn=978-3-937219-47-9|pages=195|publisher=Ed. am Gutenbergplatz |language=de}}</ref> He attended the [[Gymnasium (school)|Gymnasium]] in Düsseldorf, then studied mathematics and physics at the [[University of Bonn]],<ref>{{cite journal|author=Halsted, George Bruce|author-link=G. B. Halsted|title=Biography: Felix Klein|journal=[[The American Mathematical Monthly]]|year=1894|volume=1|issue=12|pages=416–420|jstor=2969034|doi=10.2307/2969034}}</ref> 1865–1866, intending to become a physicist. At that time, [[Julius Plücker]] had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest was mainly geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868.

Plücker died in 1868, leaving his book concerning the basis of [[line geometry]] incomplete. Klein was the obvious person to complete the second part of Plücker's ''Neue Geometrie des Raumes'', and thus became acquainted with [[Alfred Clebsch]], who had relocated to Göttingen in 1868. Klein visited Clebsch the next year, along with visits to [[Berlin]] and Paris. In July 1870, at the beginning of the [[Franco-Prussian War]], he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the [[Prussian army]] before being appointed ''Privatdozent'' (lecturer) at Göttingen in early 1871.

The [[University of Erlangen–Nuremberg|University of Erlangen]] appointed Klein professor in 1872, when he was only 23 years old.<ref>{{cite book|editor=Ivor Grattan-Guinness|editor-link=Ivor Grattan-Guinness|title=Landmark Writings in Western Mathematics 1640–1940|url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA546|year=2005|publisher=Elsevier|isbn=978-0-08-045744-4|page=546}}</ref> 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 [[Technical University of Munich|Technische Hochschule München]] in 1875. There he and [[Alexander von Brill]] taught advanced courses to many excellent students, including [[Adolf Hurwitz]], [[Walther von Dyck]], [[Karl Rohn]], [[Carl Runge]], [[Max Planck]], [[Luigi Bianchi]], and [[Gregorio Ricci-Curbastro]].

In 1875, Klein married Anne Hegel, granddaughter of the philosopher [[Georg Wilhelm Friedrich Hegel]].<ref>Chislenko, Eugene; Tschinkel, Yuri. [https://www.ams.org/notices/200708/tx070800960p.pdf "The Felix Klein Protocols"], ''[[Notices of the American Mathematical Society]]'', August 2007, Volume 54, Number 8, pp. 960–970.</ref>

After spending five years at the Technische Hochschule, Klein was appointed to a chair of [[geometry]] at [[Leipzig University]]. His colleagues included [[Walther von Dyck]], Rohn, [[Eduard Study]] and [[Friedrich Engel (mathematician)|Friedrich Engel]]. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. In 1882, his health collapsed and he battled with depression for the next two years.<ref>{{Cite book|url=https://books.google.com/books?id=-pTwBwAAQBAJ&q=mental|title=Hilbert|last=Reid|first=Constance|publisher=Springer-Verlag|year=1996|isbn=9781461207399|location=New York|pages=19|language=en}}</ref> Nevertheless, his research continued; his seminal work on hyperelliptic sigma functions, published between 1886 and 1888, dates from around this period.

[[File:Max Liebermann Felix Klein 1912.png|thumb|upright|Klein (1912). Painting of [[Max Liebermann]].]]
Klein accepted a professorship at the [[University of Göttingen]] in 1886. From then on, until his 1913 retirement, he sought to re-establish Göttingen as the world's prime center for mathematics research. However, he never managed to transfer from Leipzig to Göttingen his own leading role as developer of [[geometry]]. He taught a variety of courses at Göttingen, mainly concerning the interface between mathematics and physics, in particular, [[mechanics]] and [[potential theory]].

The research facility Klein established at Göttingen served as model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. In 1895, Klein recruited [[David Hilbert]] from the [[University of Königsberg]]. This appointment proved of great importance; Hilbert continued to enhance Göttingen's primacy in mathematics until his own retirement in 1932. Klein and Hilbert jointly invited [[Emmy Noether]] to Göttingen in 1915 where she introduced Einstein to group theory and the relationship between symmetries and conservation principles.<ref>{{cite book | last=Phillips | first=Lee | title=Einstein's Tutor: The Story of Emmy Noether and the Invention of Modern Physics | year=2024 | publisher=Public Affairs | ISBN=978-1541702950}}</ref>

Under Klein's editorship, ''[[Mathematische Annalen]]'' became one of the best mathematical journals in the world. Founded by Clebsch, it grew under Klein's management, to rival,  and eventually surpass ''[[Crelle's Journal]]'', based at the [[University of Berlin]]. Klein established a small team of editors who met regularly, making decisions in a democratic spirit. The journal first specialized in [[complex analysis]], [[algebraic geometry]], and [[invariant theory]]. It also provided an 
important outlet for [[real analysis]] and the new [[group theory]].

In 1893,  Klein was a major speaker at the International Mathematical Congress held in Chicago as part of the [[World's Columbian Exposition]].<ref>{{cite book|editor=Case, Bettye Anne|editor-link=Bettye Anne Case|title=A Century of Mathematical Meetings|chapter=''Come to the Fair: The Chicago Mathematical Congress of 1893'' by David E. Rowe and Karen Hunger Parshall|year=1996|publisher=American Mathematical Society|page=64|isbn=9780821804650|chapter-url=https://books.google.com/books?id=UnkYqxyWGz8C&pg=PA64}}</ref> Due partly to Klein's efforts, Göttingen began admitting women in 1893. He supervised the first Ph.D. thesis in mathematics written at Göttingen by a woman, by [[Grace Chisholm Young]], an English student of [[Arthur Cayley]]'s, whom Klein admired. In 1897, Klein became a foreign member of the [[Royal Netherlands Academy of Arts and Sciences]].<ref>{{cite web|url=http://www.dwc.knaw.nl/biografie/pmknaw/?pagetype=authorDetail&aId=PE00001270 |title=Felix C. Klein (1849–1925) |publisher=Royal Netherlands Academy of Arts and Sciences |access-date=22 July 2015}}</ref>

Around 1900, Klein began to become interested in mathematical instruction in schools. In 1905, he was instrumental in formulating a plan recommending that [[analytic geometry]], the rudiments of differential and integral [[calculus]], and the [[Function (mathematics)|function]] concept be taught in secondary schools.<ref>{{cite book|editor1=Gary McCulloch|editor2=David Crook|title=The Routledge International Encyclopedia of Education|url=https://books.google.com/books?id=dCMuAgAAQBAJ&pg=PA373|year=2013|publisher=Routledge|isbn=978-1-317-85358-9|page=373}}</ref><ref>{{cite book|editor1=Alexander Karp|editor2=Gert Schubring|title=Handbook on the History of Mathematics Education|url=https://books.google.com/books?id=MYy9BAAAQBAJ&pg=PA499|year=2014|publisher=Springer Science & Business Media|isbn=978-1-4614-9155-2|pages=499–500}}</ref> This recommendation was gradually implemented in many countries around the world. In 1908, Klein was elected president of the [[International Commission on Mathematical Instruction]] at the Rome [[International Congress of Mathematicians]].<ref>{{cite book|editor1=Alexander Karp|editor2=Gert Schubring|title=Handbook on the History of Mathematics Education|url=https://books.google.com/books?id=MYy9BAAAQBAJ&q=%22Felix%20Klein%20as%20its%20first%20president%22&pg=PA503|year=2014|publisher=Springer Science & Business Media|isbn=978-1-4614-9155-2|page=503}}</ref> Under his guidance, the German part of the Commission published many volumes on the teaching of mathematics at all levels in Germany. In 1892 the [[Manchester Literary and Philosophical Society]] awarded Klein Honorary membership of the Society.<ref>Memoirs and proceedings of the Manchester Literary & Philosophical Society FOURTH SERIES  Eighth  VOLUME   1894</ref>
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 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]].

He died in Göttingen in 1925.

==Work==
[[File:Mobius to Klein.gif|thumb|Construction of a Klein Bottle from two Möbius strips]]
Klein's dissertation, on [[line geometry]] and its applications to [[mechanics]], 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 later investigated [[W-curve]]s, curves invariant under a group of [[projective transformation]]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]].<ref>{{MacTutor Biography|id=Klein}}</ref>

Klein devised the "[[Klein 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 the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional [[Möbius strip]], with one method of construction being the attachment of the edges of two [[Möbius strip]]s.<ref>{{Citation|last=Numberphile|title=Klein Bottles – Numberphile|date=22 June 2015|url=https://www.youtube.com/watch?v=AAsICMPwGPY |archive-url=https://ghostarchive.org/varchive/youtube/20211211/AAsICMPwGPY| archive-date=11 December 2021 |url-status=live|access-date=26 April 2017}}{{cbignore}}</ref>

During the 1890s, Klein began studying [[mathematical physics]] more intensively, writing on the [[gyroscope]] with [[Arnold Sommerfeld]].<ref>[[:de:Werner Burau]] and [[:de:Bruno Schoeneberg]] "Klein, Christian Felix." ''[[Dictionary of Scientific Biography|Complete Dictionary of Scientific Biography]]''. 2008. Retrieved 4 December 2014 from Encyclopedia.com: http://www.encyclopedia.com/doc/1G2-2830902326.html</ref> During 1894, he initiated the idea of an encyclopedia of mathematics including its applications, which became the [[Klein's Encyclopedia of Mathematical Sciences|''Encyklopädie der mathematischen Wissenschaften'']]. This enterprise, which endured until 1935, provided an important standard reference of enduring value.<ref>[[Ivor Grattan-Guinness]] (2009) ''Routes of Learning: Highways, Pathways, Byways in the History of Mathematics'', pp 44, 45, 90,  [[Johns Hopkins University Press]], {{isbn|0-8018-9248-1}}</ref>

===Erlangen program===
{{main|Erlangen program}}
[[File:Non-euclidean-geometry-model.svg|thumb|upright=1.1|Non-Euclidean geometry models proposed by Klein (left) and Poincaré (right)]]
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]]s determined by a [[Cayley–Klein metric]]. This insight had the corollary that [[non-Euclidean geometry]] was consistent if and only if [[Euclidean geometry]] was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. [[Arthur Cayley]] never accepted Klein's argument, believing it to be circular.

Klein's synthesis of [[geometry]] as the study of the properties of a space that is invariant under a given [[transformation group|group of transformations]], known as the ''[[Erlangen program]]'' (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of [[transformation group|transformation]]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]] as his major contribution to mathematics, specifically his work on:
*The link between certain ideas of [[Bernhard Riemann|Riemann]] and [[invariant theory]],
*[[Number theory]] and [[abstract algebra]];
*[[Group theory]];
*[[Geometry]] in more than 3 dimensions and [[differential equations]], especially equations he invented, satisfied by [[elliptic modular function]]s and [[automorphic function]]s.

Klein showed that the [[modular group]] moves the fundamental region of the [[complex plane]] so as to [[tessellation|tessellate]] the plane. In 1879, he examined the action of [[PSL(2,7)]], considered as an image of the [[modular group]], and obtained an explicit representation of a [[Riemann surface]] now termed the [[Klein quartic]]. He showed that it  was a complex curve in [[projective space]], that its equation was ''x''{{sup|3}}''y''&nbsp;+&nbsp;''y''{{sup|3}}''z''&nbsp;+&nbsp;''z''{{sup|3}}''x''&nbsp;=&nbsp;0, and that its group of [[symmetry group|symmetries]] was [[PSL(2,7)]] of [[order (group theory)|order]] 168. His ''Ueber Riemann's Theorie der algebraischen Funktionen und ihre Integrale'' (1882) treats complex analysis in a geometric way, connecting [[potential theory]] and [[conformal mapping]]s. 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 methods of [[Charles Hermite]] and [[Leopold Kronecker]], he produced similar results to those of Brioschi and later completely solved the problem by means of the [[icosahedral group]]. This work enabled him to write a series of papers on [[elliptic modular function]]s.

In his 1884 book on the [[icosahedron]], Klein established a theory of [[automorphic function]]s, associating algebra and geometry. [[Henri Poincaré|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]] that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. He came up with his proof during an [[asthma attack]] at 2:30 A.M. on 23 March 1882.<ref>{{Cite journal |last=Abikoff |first=William |date=1981 |title=The Uniformization Theorem |url=https://www.jstor.org/stable/2320507 |journal=The American Mathematical Monthly |volume=88 |issue=8 |pages=574–592 |doi=10.2307/2320507 |jstor=2320507 |issn=0002-9890}}</ref>

Klein summarized his work on [[automorphic function|automorphic]] and [[elliptic modular function]]s in a four volume treatise, written with [[Robert Fricke]] over a period of about 20 years.

==Selected works==
* 1882: ''[https://gutenberg.org/ebooks/20313 Über Riemann's Theorie der Algebraischen Functionen und ihre Integrale]'' {{JFM|14.0358.01}}
* 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]]
* 1886: ''Über hyperelliptische Sigmafunktionen''. Erster Aufsatz, pp.&nbsp;323–356, [[Mathematische Annalen]] Bd. 27, 
* 1887: [https://www.ams.org/journals/bull/1896-02-08/S0002-9904-1896-00340-2/S0002-9904-1896-00340-2.pdf "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.
* 1888: ''Über hyperelliptische Sigmafunktionen''. Zweiter Aufsatz, pp.&nbsp;357–387, Math. Annalen, Bd. 32,
* 1890: "[http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN516972847 Nicht-Euklidische Geometrie]"
* 1890: (with [[Robert Fricke]]) ''Vorlesungen über die Theorie der elliptischen Modulfunktionen'' (2 volumes)<ref>{{cite journal|url=https://www.ams.org/journals/bull/1892-01-05/S0002-9904-1892-00049-3/S0002-9904-1892-00049-3.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1892-01-05/S0002-9904-1892-00049-3/S0002-9904-1892-00049-3.pdf |archive-date=9 October 2022 |url-status=live|author=Cole, F. N.|author-link=Frank Nelson Cole|title=''Vorlesungen über die Theorie der elliptischen Modulfunktionen'' von Felix Klein, Erste Band|journal=Bull. Amer. Math. Soc.|year=1892|volume=1|issue=5|pages=105–120|doi=10.1090/S0002-9904-1892-00049-3 |doi-access=free}}</ref> and 1892)
* 1894: [https://catalog.hathitrust.org/Record/000438127 ''Über die hypergeometrische Funktion'']
* 1894: ''Über lineare Differentialgleichungen der 2. Ordnung''
* 1894: ''Evanston Colloquium'' (1893) reported and published by Ziwet (New York, 1894)<ref>{{cite journal|author=White, Henry S.|author-link=Henry Seely White|title=Review: ''The Evanston Colloquium: Lectures on Mathematics'' by Felix Klein|journal=Bull. Amer. Math. Soc.|year=1894|volume=3|issue=5|pages=119–122|url=https://www.ams.org/journals/bull/1894-03-05/S0002-9904-1894-00190-6/S0002-9904-1894-00190-6.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1894-03-05/S0002-9904-1894-00190-6/S0002-9904-1894-00190-6.pdf |archive-date=9 October 2022 |url-status=live|doi=10.1090/s0002-9904-1894-00190-6|doi-access=free}}</ref>
*{{Citation | last = Klein | first = Felix | title = Lectures on Mathematics | url = https://archive.org/details/lecturesonmathem00klei/page/n1/mode/2up | publisher = Macmillan and Co. | location = New York, London | year = 1894}}
* 1895: ''Vorträge über ausgewählte Fragen der Elementargeometrie''<ref>{{cite journal|url=https://www.ams.org/journals/bull/1896-02-06/S0002-9904-1896-00328-1/S0002-9904-1896-00328-1.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1896-02-06/S0002-9904-1896-00328-1/S0002-9904-1896-00328-1.pdf |archive-date=9 October 2022 |url-status=live|author=Scott, Charlotte Angas|author-link=Charlotte Scott|title=Review: ''Vorträge über ausgewählte Fragen der Elementargeometrie'' von Felix Klein|journal=Bull. Amer. Math. Soc.|year=1896|volume=2|issue=6|pages=157–164|doi=10.1090/s0002-9904-1896-00328-1|doi-access=free}}</ref>
** 1897: English translation by W. W. Beman and [[David Eugene Smith|D. E. Smith]] ''[https://archive.org/details/B-001-004-370 Famous Problems of Elementary Geometry]'' via Internet Archive
* 1897: (with [[Arnold Sommerfeld]]) ''Theorie des Kreisels'' (later volumes: 1898, 1903, 1910)
* {{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix | title=Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen | url=https://archive.org/details/vorlesungenber01fricuoft | publisher=Leipzig: B. G. Teubner | language=de| year=1897 | jfm=28.0334.01}}<ref name="HutchinsonRef">{{cite journal |author=Hutchinson, J. I. |author-link=John Irwin Hutchinson |year=1903 |title=Review: ''Vorlesungen über die Theorie der automorphen Functionen'' von Robert Fricke & Felix Klein, Erste Band & Zweiter Band |url=https://www.ams.org/journals/bull/1903-09-09/S0002-9904-1903-01020-9/S0002-9904-1903-01020-9.pdf |url-status=live |journal=Bulletin of the American Mathematical Society |volume=9 |issue=9 |pages=470–492 |doi=10.1090/S0002-9904-1903-01020-9 |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1903-09-09/S0002-9904-1903-01020-9/S0002-9904-1903-01020-9.pdf |archive-date=9 October 2022}}</ref> Zweiter Band. 1901.<ref name=HutchinsonRef/>
* 1897: [https://archive.org/details/mathematicaltheo00kleiuoft/page/n9/mode/2up ''Mathematical Theory of the Top''] (Princeton address, New York)<ref>{{cite journal|author=Thompson, Henry Dallas|title=Review: ''Mathematical Theory of the Top'' by Felix Klein|journal=Bull. Amer. Math. Soc.|year=1899|volume=5|issue=10|pages=486–487|url=https://www.ams.org/journals/bull/1899-05-10/S0002-9904-1899-00643-8/S0002-9904-1899-00643-8.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1899-05-10/S0002-9904-1899-00643-8/S0002-9904-1899-00643-8.pdf |archive-date=9 October 2022 |url-status=live|doi=10.1090/s0002-9904-1899-00643-8|doi-access=free}}</ref>
* 1901: {{Citation | title=Gauss' wissenschaftliches Tagebuch, ''1796—1814. Mit Anmerkungen von Felix Klein'' | url=http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002259079}}<ref>{{cite journal|author=Bôcher, Maxime|author-link=Maxime Bôcher|title=Review: ''Gauss' wissenschaftlichen Tagebuch'', 1796—1814. Mit Anmerkungen von Felix Klein|journal=Bull. Amer. Math. Soc.|year=1902|volume=9|issue=2|pages=125–126|url=https://www.ams.org/journals/bull/1902-09-02/S0002-9904-1902-00959-2/S0002-9904-1902-00959-2.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1902-09-02/S0002-9904-1902-00959-2/S0002-9904-1902-00959-2.pdf |archive-date=9 October 2022 |url-status=live|doi=10.1090/s0002-9904-1902-00959-2|doi-access=free}}</ref>
*{{Citation | last1=Fricke | first1=Robert | last2=Klein | first2=Felix | title=Vorlesungen über die Theorie der automorphen Functionen. Zweiter Band: Die funktionentheoretischen Ausführungen und die Anwendungen. 1. Lieferung: Engere Theorie der automorphen Funktionen | url=https://archive.org/details/vorlesungenber02fricuoft | publisher=Leipzig: B. G. Teubner.  | language=de | year=1912 | jfm=32.0430.01}}
* 1908: ''Elementarmathematik vom höheren Standpunkte aus'' (Leipzig)
* 1921: ''Gesammelte mathematische Abhandlungen'' (R. Fricke and A. Ostrowski, eds.). Berlin, Springer. 3 volumes. (online copy  at [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN237839962  GDZ])
* 1926: ''[[Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert]]'' (2 Bände), Julius Springer Verlag, Berlin<ref>{{cite journal |author=Smith, David Eugene |author-link=David Eugene Smith |year=1928 |title=Review: ''Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert'' von Felix Klein. Erste Band |url=https://www.ams.org/journals/bull/1928-34-04/S0002-9904-1928-04589-5/S0002-9904-1928-04589-5.pdf |url-status=live |journal=Bulletin of the American Mathematical Society |volume=34 |issue=4 |pages=521–522 |doi=10.1090/S0002-9904-1928-04589-5 |archive-url=https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/1928-34-04/S0002-9904-1928-04589-5/S0002-9904-1928-04589-5.pdf |archive-date=9 October 2022}}</ref> & 1927. S. [https://web.archive.org/web/20151225095843/http://gdz.sub.uni-goettingen.de/dms/load/img/?IDDOC=285982 Felix Klein ''Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert'']
* 1928: ''Vorlesungen über nichteuklidische Geometrie'', Grundlehren der mathematischen Wissenschaften, Springer Verlag<ref>{{cite journal|author=Allen, Edward Switzer|title=Three books on non-euclidean geometry|journal=Bull. Amer. Math. Soc.|volume=35|year=1929|pages=271–276|doi=10.1090/S0002-9904-1929-04726-8|doi-access=free}}</ref>
* 1933: ''Vorlesungen über die hypergeometrische Funktion'', Grundlehren der mathematischen Wissenschaften, Springer Verlag

==See also==
* [[Dianalytic manifold]]
* [[j-invariant]]
* [[Line complex]]
* [[Grünbaum–Rigby configuration]]
* [[Homomorphism]]
* [[Ping-pong lemma]]
* [[Prime form]]
* [[Leonardo polyhedron]]
* [[W-curve]]
* [[Uniformization theorem]]
* [[Felix Klein Protocols]]
* [[List of things named after Felix Klein]]

==References==
{{Reflist|30em}}

==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.
* {{cite book |title=The Legacy of Felix Klein |series=ICME-13 Monographs |year=2019 |url=https://link.springer.com/book/10.1007/978-3-319-99386-7 |publisher=Springer |doi=10.1007/978-3-319-99386-7 |isbn=978-3-319-99386-7}}
* [[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) [http://amshistorica.unibo.it/diglib.php?inv=7&int_ptnum=30&term_ptnum=401&format=jpg L'oeuvre mathematique de Klein] in ''Scientia''.

==External links==
{{Commons category|Felix Klein}}
{{Wikiquote|Felix Klein}}
* {{Gutenberg author | id=9227| name=Felix Klein}}
* {{Internet Archive author |sname=Felix Klein}}
* {{MacTutor Biography|id=Klein}}
* {{MathGenealogy |id=7401}}
* {{ScienceWorldBiography | urlname=KleinFelix | title=Klein, Felix (1849–1925)}}
* [http://page.mi.fu-berlin.de/moritz/klein/ Felix Klein, Klein Protokolle]
* [http://www.britannica.com/EBchecked/topic/319960/Felix-Klein Felix Klein (Encyclopædia Britannica)]
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_line_complexes.pdf F. Klein, "On the theory of line complexes of first and second order"]
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_line_geometry.pdf F. Klein, "On line geometry and metric geometry"]
*[http://neo-classical-physics.info/uploads/3/0/6/5/3065888/klein_-_canonical_forms.pdf F. Klein, "On the transformation of the general second-degree equation in line coordinates into canonical coordinates"]

{{Copley Medallists 1901-1950}}
{{De Morgan Medallists}}
{{Authority control}}

{{DEFAULTSORT:Klein, Felix}}
[[Category:1849 births]]
[[Category:1925 deaths]]
[[Category:Scientists from Düsseldorf]]
[[Category:19th-century German mathematicians]]
[[Category:20th-century German mathematicians]]
[[Category:Differential geometers]]
[[Category:Hyperbolic geometers]]
[[Category:German military personnel of the Franco-Prussian War]]
[[Category:Group theorists]]
[[Category:Members of the Prussian House of Lords]]
[[Category:People from the Rhine Province]]
[[Category:Recipients of the Copley Medal]]
[[Category:University of Bonn alumni]]
[[Category:Humboldt University of Berlin alumni]]
[[Category:Academic staff of the University of Göttingen]]
[[Category:Academic staff of the University of Erlangen-Nuremberg]]
[[Category:Academic staff of the Technical University of Munich]]
[[Category:Academic staff of Leipzig University]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Foreign members of the Royal Society]]
[[Category:Members of the Royal Netherlands Academy of Arts and Sciences]]
[[Category:Recipients of the Pour le Mérite (civil class)]]
[[Category:De Morgan Medallists]]
[[Category:Prussian Army personnel]]
[[Category:Scientists from North Rhine-Westphalia]]
[[Category:Presidents of the German Mathematical Society]]