### Felix Christian Klein was a well-known 19th-century German mathematician who made significant contributions to group theory, complex analysis, non-Euclidean geometry, and the Erlangen Program. He was born in Düsseldorf to Prussian parents and attended the University of Bonn for his education. Before being appointed a Privatdozent at the University of Göttingen, Klein collaborated closely with physicist Julius Plücker and mathematician Alfred Clebsch. He authored two key works on Euclidean and non-Euclidean geometry here at the age of 22. He became a full professor at the University of Erlangen at the age of 23, where he founded the Erlanger program. Later, he was a mathematics professor at Munich’s Technische Hochschule and Leipzig University, where he published a number of notable works. However, he was never in good health, and in his early thirties, he had a psychological breakdown, which severely impeded his studies. After that, he returned to Göttingen University and began teaching a wide range of subjects. But, shortly after, his focus shifted to educational reform, both at the university and at the school level, and he devoted much of his efforts to it in his latter years.

## Childhood and Adolescence

Felix Christian Klein was born on April 25, 1849, in Düsseldorf, then the capital of the Rhine Province of the Kingdom of Prussia, and today the capital of the German state of North Rhine-Westphalia. Casper Klein and Sophie

Elise Klein née Kayser, his parents, were both Prussians.

He began his elementary education at home with his mother, and at the age of six, he was enrolled in a private school, where he spent two and a half years. He was sent to the Düsseldorf Gymnasium in 1857, where he graduated in 1865.

Although classical education has typically been prioritized in gymnasiums, the one he attended placed equal emphasis on mathematics and science. He developed in-depth understanding of mathematics here, under the tutelage of several trained tutors, and did fairly well on the final examination.

He enrolled at the University of Bonn in the winter semester of 1865-1866 after graduating from the Gymnasium with the intention of becoming a physicist. He studied mathematics and natural sciences at this university, where he studied physics, chemistry, geology, botany, and zoology in the ‘Seminar für die gesamten Naturwissenschaften.’

Despite his accomplishments in botany and zoology, he excelled in physics. Julius Plücker, professor of mathematics and experimental physics, praised him from the start for his strengths in theoretical and experimental physics.

Felix Klein began working as an assistant in Plücker’s laboratory in 1866. However, the professor’s attention had shifted to geometry by that time, and Klein began studying mathematics alongside him. Plücker’s analytical-geometric approach quickly affected him.

Klein began concentrating on mathematics in the winter semester of 1867-68, despite continuing to study natural sciences. He studied analytic geometry, number theory, mechanics, statics, differential equations, and calculus with Rudolf Lipschitz during this time, in addition to attending Plücker’s lectures.

Klein got his doctorate from the University of Bonn in December 1868, after working with Plücker on his thesis, ‘Über die Transformation der allgemeinen Gleichung des zweiten Grades zwischen Linien-Koordinaten auf eine kanonische’. He used the Weierstrass theory of elementary divisors to classify second degree line complexes in this thesis.

## In the Gttingen

Julius Plücker died on May 22, 1868, just before Klein obtained his doctorate, leaving his dissertation on the foundations of line geometry, ‘Neue Géometrie des Raumes,’ unfinished. Alfred Clebsch, Professor of Mathematics at the University of Göttingen, asked Klein, his closest assistant, to finish the job.

Felix Christian Klein arrived at the University of Göttingen in January 1869 and studied there with Alfred Clebsch until August 1869. He then proceeded to Berlin, where he studied until April 1870, attending Kronecker’s lectures and the Mathematisches Seminar. He also met Norwegian mathematician Sophus Lie here, with whom he forged a strong friendship.

Klein and Lie found the fundamental properties of the asymptotic lines on the Kummer surface while working together. They went on to work on W-curves, which are curves that are invariant under a set of projective transformations.

Klein and Lie travelled for Paris in April 1870 with the intention of studying new advancements in French mathematics. Klein was compelled to abandon the plan when the Franco-Prussian War broke out in July 1870.

He volunteered for the war and served as a medical orderly in the Prussian Army; however, after contracting typhus, he was discharged from the army. He began studying for his habilitation while healing from his infirmities, and he received his degree in January 1871.

## Early on in your career

Felix Christian Klein began his career as a Privatdozent at the University of Göttingen in early 1871, shortly after getting his degree. He kept working on his study throughout, releasing two significant articles on Euclidean and non-Euclidean geometry.

Klein was named professor of mathematics at the University of Erlangen, Bavaria, in October 1872, on Clebsch’s proposal. He was only 23 years old at the time, but he was already well-known for his geometry works.

Traditionally, when a professor assumed his position, he or she was required to write a scientific study. He produced ‘Vergleichende Betrachtungen über neuere geometrische Forschungen,’ which was a continuation of his earlier work on Euclidean geometry, as was customary at the time. The ‘Erlangen Program’ was founded as a result of the study.

Klein had few students at Erlangen, much to his dismay. This is primarily due to the fact that mathematics was still being established in Bavaria, and so few students were interested in it. The period, however, was not entirely uneventful.

Klein joined Erlangen a month after Clebsch, who was not only the cofounder but also one of the editors of the mathematical journal ‘Mathematische Annalen.’ Klein joined the editorial board of the journal in 1874, eventually taking over the Clebsch School’s leadership.

He came to Munich in 1875, when he was offered a position at the Technische Hochschule. He had a large number of good pupils here, the most of whom were studying mathematics as a service subject. He had to watch after a few students who were math specialists in addition to them.

Klein stayed at Technische Hochschule for five years, ensuring that his pupils had a strong mathematical basis. During this time, his talent to teach was fully developed.

Klein became Professor of Geometry at Leipzig University in 1880. Here, he began to develop his mathematics studies in a systematic manner. He also had a separate mathematics building constructed. It contained distinct chambers for the Mathematisches Seminar in addition to lecture halls and a dedicated library.

He began working on elliptic functions in earnest in the early 1880s. He was competing with Henri Poincare, who was also working on the same issue, to see who could outdo the other.

‘Ueber Riemann’s Theorie der Algebraischen Functionen and ihre Integrale’ was his first work on the subject, published in 1882. However, in the autumn of 1882, he suffered a psychological collapse that necessitated extensive recuperation. The years 1883-1884 were spent in a deep depression, unable to find job.

His well-known work, ‘Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade,’ was published in 1884. (Lectures on the Ikosahedron; and the Solution of Equations of the Fifth Degree). Poincare, on the other hand, was much ahead of him at the time.

Klein continued to work on the theory of functions until the 1890s, despite the fact that he understood he would never catch up to Poincare. However, he quickly shifted his focus to educational reform.

## Returning to Gttingen

Felix Christian Klein accepted the chair of mathematics at Göttingen University in 1886 and remained there until his retirement in 1913. He taught a wide range of courses at this university, with a focus on the intersection of mathematics and physics, such as mechanics and potential theory. He also had a lot of administrative responsibilities at the same time.

Klein founded a mathematical research center in Göttingen, which became a model for similar institutions around the world over time. He also started monthly discussion gatherings and created a mathematical reading room with a mathematical library, following the Leipzig model.

Despite the fact that he had been on the editorial board of ‘Mathematische Annalen’ since 1874, he began to take a more active role in it today. He formed a small group of editors that met on a regular basis and made all decisions democratically.

Under his direction, the publication began to compete with and eventually surpass the University of Berlin’s ‘Journal für die reine und angewandte Mathematik,’ also known as ‘Crelle’s Journal.’ In actuality, the two publications reflected two schools of thought, one from Berlin and the other from Clebsch.

Klein visited America in 1893 to attend the International Mathematical Congress, which was held in Chicago as part of the World’s Columbian Exposition. He was one of the main speakers at this event.

Göttingen began to allow women in 1893, thanks to Klein’s efforts. One of his students was Grace Chisholm Young, the first woman to get a doctorate in Germany. Her doctoral thesis was written under his supervision, and she received her degree in 1895.

Klein was also appointed to oversee the ‘Encyklopädie der mathematischen Wissenschaften with Einschluss iher Anwendungen (“Encyclopedia of Pure and Applied Mathematics”), which was published in 1895. In the same year, he persuaded David Hilbert to join his Göttingen research team, assigning him to the other mathematics chair.

Klein and Hilbert collaborated extensively from 1896 onwards to make Göttingen into a magnet for mathematicians from all over the world. The Chair for Applied Mathematics was one of the most important new jobs created (1904).

He set out to redefine mathematics so that it would be consistent with the curricula of technical schools, in order to move it to a more practical level. In 1898, he formed the Göttinger Vereinigung für angewandte Physik with the help of important German industrialists.

He began striving to improve mathematics teaching skills at the school level in 1900, organizing summer schools for mathematics teachers to receive in-service training. He also suggested that rudimentary differential and integral calculus, as well as the concept of functions, be taught in secondary schools.

He wrote a lot of books on elementary mathematics in addition to technical papers. ‘Elementarmathematik vom höheren Standpunkte aus (Elementary Mathematics from a Higher Standpoint),’ released in three volumes in 1908, was one of the most popular and is still used as a model today.

He attended the Internationale Mathematische Unterichtskommission (IMUK) in Rome in 1908, where he was chosen the organization’s first President. It became an agent for curricular reform under his leadership, and its German chapter began publishing several volumes on the teaching of mathematics at all levels.

Klein worked extremely hard to achieve the four-year deadline set by IMUK, falling ill sometime in 1911 and being compelled to reside at a sanatorium at Hahnenklee in the Harz Mountains from 1911 to 1912. He did not, however, cease working, and he managed to meet with both his German assistants and foreign colleagues.

Klein’s health had worsened to the point where he had to leave from his position at the University of Göttingen in 1913. During the First World War, he continued to teach at home while also working for the IMUK until it was officially dissolved in 1920.

## Major Projects of Felix Christian Klein

Felix Christian Klein is most known for developing the ‘Erlanger Program.’ Despite the fact that non-Euclidean geometries have emerged at that time, there was no way of determining their hierarchy or linkages. The Erlanger Program, which began in 1872 and characterized geometries based on group theory and projective geometry, was a significant step in that direction.

His work on functional theory was another significant addition to mathematics. In 1882, he published a study that connected conformal mappings with potential theory to treat functional theory geometrically. He was able to incorporate physical notions into his work by using fluid dynamics.

Klein used transcendental methods to solve general equations of the fifth degree. He solved problems with the icosahedron group by incorporating the ideas of Charles Hermite and Leopold Kronecker into his methodology. As a result of his findings, he began working on elliptic modular functions.

Klein developed the theory of automorphic functions by connecting geometric and algebraic conclusions, which he published in his work on the icosahedron in 1884.

## Achievements & Awards

Felix Christian Klein was awarded the De Morgan Medal for distinguished contributions to mathematics in 1893.

He received the Copley Medal in 1912 for “his mathematical researches.”

For his analytical studies in mathematics, he received the inaugural Ackermann–Teubner Memorial Award in 1914.

In 1885, he was elected a member of the Royal Society.

## Personal History and Legacy

Klein married Anna Maria Carolina Hegel in 1875, the granddaughter of Georg Wilhelm Friedrich Hegel, a well-known philosopher. The marriage had three children: Otto Klein, a boy, and Luise Süchtig née Klein and Elisabeth Steiger nee Klein, two daughters. He had another daughter, according to some biographers, whose name is unknown.

Klein died in Göttingen on June 22, 1925, at the age of 76.

Klein’s influence lives on in concepts like the Klein bottle and the Beltrami–Klein model.

## Estimated Net Worth

Felix Christian Klein net worth is unknown.