## Felix KleinGerman mathematician.
Country:
Germany |

**Content:**

## Biography of Felix Klein

Felix Klein was a German mathematician who played an active role in the reform of school education in the early 20th century. He was also the author and initiator of several studies on the state of mathematics education in different countries. Klein was born in Düsseldorf, Germany, into a civil servant family. He received his education at the University of Bonn (1865-1868), where he was a student of Julius Plücker. In 1868, Plücker passed away, and Klein, who was his assistant at the time, embarked on a trip across Germany, where he met prominent mathematicians such as Clebsch. He was particularly influenced by Sophus Lie. In 1870, during the Franco-Prussian War, Klein traveled to Paris with Lie, where he met Darboux and Jordan. He returned to Germany after the war, narrowly escaping a typhoid epidemic that ravaged the country.

## Professorship and Publications

In 1872, Klein became a professor at the University of Erlangen, recommended by Clebsch. He published his famous "Erlangen Program" and immediately gained recognition throughout Europe. In 1875, he became a professor at the Technical University of Munich. He married Anna Hegel, the granddaughter of the renowned philosopher, in the same year. In 1876, Klein became the co-editor of the journal "Mathematische Annalen" alongside Adolf Mayer. He moved to the University of Leipzig in 1880.

From 1882 to 1884, Klein suffered a serious illness due to overwork. He redirected his immense energy towards pedagogical and public work. In 1888, he became a professor at the University of Göttingen, where he taught captivating and profound optional courses on various subjects, ranging from number theory to technical mechanics. Students from all over the world attended his lectures.

## Contributions to Mathematics

In the early 20th century, Klein actively participated in the reform of school education and authored several studies on the state of mathematics education in different countries. He also contributed to the establishment of scientific research institutes for applied research in various technical fields at the University of Göttingen. Klein played a role in the publication of Gauss's complete works and the first Mathematical Encyclopedia. He represented the University of Göttingen in the parliament. It is noteworthy that during World War I, Klein did not participate in the numerous chauvinistic actions of that time.

In 1924, Klein's 75th birthday was widely celebrated. The following year, the same newspapers published his obituary. By the mid-19th century, geometry had divided into numerous poorly coordinated branches, such as Euclidean, spherical, hyperbolic, projective, affine, Riemannian, multidimensional, complex, and more. Pseudo-Euclidean geometry and topology were added to the mix at the turn of the century.

Klein came up with the idea of algebraic classification of different branches of geometry based on the classes of transformations that are irrelevant to each geometry. More precisely, one branch of geometry differs from another in terms of the groups of transformations of space that correspond to it, and the objects under study are the invariants of such transformations.

For example, classical Euclidean geometry studies the properties of figures and solids that are preserved under non-deforming motions. It corresponds to a group containing rotations, translations, and their combinations. Projective geometry can study conic sections but has no concern with circles or angles because circles and angles are not preserved under projective transformations. Topology examines the invariants of arbitrary continuous transformations (incidentally, Klein noted this even before topology was born). By studying the algebraic properties of transformation groups, we can discover new profound properties of the corresponding geometry and provide simpler proofs for old ones. For example, the median is an affine invariant; if the medians of an equilateral triangle intersect at one point, then it will be true for any other triangle because any triangle can be transformed into an equilateral one and back using affine transformations.

Klein presented all these ideas in his 1872 lecture titled "Vergleichende Betrachtungen über neuere geometrische Forschungen" ("Comparative Considerations on Recent Geometric Research") [1], which became known as the "Erlangen Program." It attracted the attention of mathematicians throughout Europe because it not only provided a new understanding of the subject of geometry but also outlined clear prospects for further research. It marked a new level of discovery, similar to Descartes' algebraization of geometry, which allowed for results that were extremely difficult or impossible to achieve with the old tools. The influence of the "Erlangen Program" on the further development of geometry was remarkably significant.

In the following three years, Klein published more than 20 works on non-Euclidean geometry, Lie groups, polyhedra theory, and elliptic functions. One of his most important achievements was the first proof of the consistency of Lobachevsky's geometry, for which he constructed its interpretation in Euclidean space (see Klein's model).

Klein also published several works on the solution of equations of the 5th, 6th, and 7th degrees, differential equation integration, Abel functions, and non-Euclidean geometry. His works were mainly published in "Mathematische Annalen," of which he had been the co-editor since 1875 alongside Adolf Mayer. Later, Klein explored automorphic functions and the theory of the spinning top.

Klein's lectures were highly popular, and many of them were repeatedly reprinted and translated into multiple languages. He also published several monographs on analysis, consolidating the achieved results up to that moment.

During Klein's lifetime, a three-volume edition of his Collected Works was published.

**Mathematicians**

Honore Fabri | Leopold Kronecker | Karl Gauss |

Witold Hurewicz | Gosta Mittag-Leffler | Evklid |