Felix Klein

German mathematician

• Born: April 25, 1849; Düsseldorf, Germany
• Died: June 22, 1925; Göttingen, Germany

Nineteenth-century German mathematician Felix Klein discovered many new non-Euclidean geometric systems. He established their consistency with traditional Euclidean geometry through applying algebra to the study of symmetry.

Also known as: Christian Felix Klein

Primary field: Mathematics

Specialty: Geometry

Early Life

Christian Felix Klein was born in Düsseldorf, Germany on April 25, 1849. Klein studied from 1857 to 1867 at the Düsseldorf Gymnasium and went from there to the University of Bonn at age seventeen. He intended to study physics and became an assistant to the eminent physicist, Julius Plucker. Plucker, however, was also an important geometer whowas near the end of his career. Klein assisted Plucker in working on line geometry, a modern development of the Euclidean geometry that had been the guiding geometrical system since antiquity.

When Plucker passed away on May 22, 1868, Klein continued his work and saw that it was published. Through Plucker’s work, he became acquainted with Alfred Clebsch, an influential professor at Göttingen. Clebsch founded a mathematical journal in Göttingen, Mathematische Annalen. He recognized the potential for greatness in Klein and helped to advance his career.Klein earned his doctorate in December 1868 at the age of nineteen. His dissertation was much influenced by Plucker and contained several references to his mentor’s work. Klein’s doctoral thesis gave algebraic formulae for geometrical transformations in which straight lines refer to coordinate tetrahedrons. Klein’s thesis opened new avenues for study and helped establish his career path. In his dissertation, Klein also called for instruction in geometry to be amended to promote the new, non-Euclidean geometry that had been discovered in 1823. Without devaluing applied mathematics, Klein saw the value of purely abstract research.

Life’s Work

In 1870, Klein began work with the Norwegian mathematician Sophus Lie. Lie’s work in groups and algebra influenced a paper by Klein, presented to the Göttingen Academy of Science. Klein forwarded problems concerning parallels in projective geometry. At the time, many scientists believed the only knowledge of value was obtained from pure physical measurement and manipulation. Others felt abstract and intuitive mathematics, physics, and other sciences had high intrinsic value. Klein’s synthesis of intuition and strict research earned him high regard.

At the age of twenty-three, Klein was called to a departmental chair at the University of Erlangen. In 1872, he delivered a paper called the Erlangen Program. The work was celebrated for its far-reaching influence on mathematics and geometry. He pulled together studies from algebra and the new field of non-Euclidean geometry and created a system for organizing the field in groups. The groups were based on kinds of repetitive movements. Klein proved that projective geometry could vastly simplify the growing field by providing its base theory. Every symmetry could be defined and grouped by its transformation. Applying algebra to geometry added powerful tools to the understanding of non-Euclidean space.

Klein married Anne Hegel in 1875, granddaughter of the famous German philosopher Wilhelm Friedrich Hegel. The same year, Klein was called to the University at Munich, where he taught a number of students who would later become famous in their own right, including physicist Max Planck.

From 1880 to 1886, Klein continued his work as a scientist and professor at Leipzig. He developed what he called “the Klein surface,” an object since then referred to mistakenly as “the Klein Bottle.” The object features a nonorientable surface (no consistent right or left) and no boundaries. Geometrically interesting, the Klein surface is often pictured and sometimes built, but it can only be fully realized in four dimensions.

Klein also worked on hyperelliptic sigma functions, publishing papers on this work in 1886 and 1888. During these years, Klein made what he considered to be his greatest contribution to mathematics: his works on function theory, group theory, number theory, and abstract algebra. However, he also suffered from depression and poor health during this period in his life. Klein engaged in competitive mathematical collaboration with the French mathematician, Jules Henri Poincaré. Working on automorphic functions, Poincaré “won” the competition informally by first discovering the foundations of what he named “Klein groups.”

In 1885, Klein was admitted to the Royal Society. The next year, he was called to chair at Göttingen, where he remained until retirement. His interest in education deepened. He aimed to transform Göttingen into one of the world’s first-order schools of mathematics and applied physics. He opened a library for mathematics, introduced weekly group discussion, and designated reading rooms. He was well connected among politicians and industrialists, garnering funding from such companies as Krupp, Bayer, and others. Klein attracted many excellent students and professors. Under his auspices, the Göttingen journal Mathematische Annalen became one of the world’s best. In 1893, Klein’s support urged Göttingen to admit its first female students. In 1890 and 1892 Klein and his erstwhile student, mathematician Robert Fricke, published a two-volume collaborative work on elliptic modular functions.

Klein became an influential academic, garnering titles awards within the international scientific community. He won the De Morgan medal of the London Mathematical Society in 1893. In the same year, he promoted mathematical study at the International Mathematical Congress, part of the Chicago World’s Fair. Following this, he lectured for two weeks at Northwestern University.

In 1897, the first volume of the second part of his collaboration with Fricke was published, which covered automorphic function theory. Fricke was responsible for much of the volume’s content, but Klein’s advisory relationship had a powerful influence. The second and final volume came out in 1912.

Beginning in 1900, Klein published books for math teachers at the gymnasium, many of them his students at the university. He intended to modernize the transition of mathematical education between gymnasium and university. Applying modern mathematics to ancient problems, he sought to enable gymnasium teachers to introduce calculus and functions at the secondary level. In 1908 Klein was elected to chair the International Commission on Mathematical Instruction.

Klein died June 22, 1925 in Göttingen. Posthumously, Klein’s race was called into question by Nazi officials. In 1933, the Prussian Ministry of the Interior sent a missive to the Bavarian Ministry of Culture, blaming Jewish mathematicians for taking over at Göttingen, led by Klein. After Nazi leader Adolf Hitler gained power in Germany, many of the department’s faculty members were either fired or left.

Impact

Klein felt that his greatest contribution to mathematics was his work on functions. He and Poincaré inspired each other, and each sought to prove the uniformization theorem first.

Euclidean geometry assumes space is flat. For example, triangles are studied on two-dimensional surfaces. Klein adopted Georg Riemann’s notion of non-Euclidean space, which can have three or more dimensions. For instance, a hyperbolic plane is only “flat” in that it has two dimensions. But its spread increases exponentially as you move from a given point. Hyperbolic space has negative curvature; elliptic space is similar, but has positive curvature. Klein’s work on the uniformization theorem would state that any connected two surfaces can demonstrate symmetry in the following way: they can be algebraically manipulated or projected such that the surface in question matches exactly a piece of one of the other kinds of plane; this theorem makes it possible to measure some extremely complicated surfaces.

The impact of Klein’s work in elliptic and automorphic functions in this area has had a far-reaching effect, especially since computers have evolved and vastly more complex surfaces have come under study. Through his teaching, Klein inspired many students, and through his combination of spatial mathematical intuition, rigorous research, and his larger-than-life personality, he left a lasting influence on the fields of geometry, group theory, and set theory.

Bibliography

Ash, Avner. Elliptic Tales: Curves, Counting, and Number Theory. Princeton: Princeton UP, 2012. Print. Elliptic curves and number theory are two aspects of modern mathematics that build on Klein’s discoveries. Understandable for the popular intellectual and the university math student.

Mumford, David, Caroline Series, and David Wright. Indra’s Pearls: The Vision of Felix Klein. Cambridge: Cambridge UP, 2002. Print. Images made possible by Klein’s groups are explored in mathematical terms for an algebra student or educated lay reader. Computer-generated color illustrations help convey Kleinian concepts of symmetry.

Trudeau, Richard J. The Non-Euclidean Revolution. Boston: Birkhäuser, 2008. Print. Reviews the discovery of non-Euclidean geometry, which many consider to be as important as the Copernican and Darwinian revolutions. Discusses the work of Klein and his peers and speculates on the search for truth through science.