Felix Klein

What The Transformations Leave Alone · Mathematician · 1849–1925

The mathematician who defined a geometry as the study of what a family of transformations leaves unchanged, and who built Göttingen into the mathematical centre of the world.

Who was Klein?

Felix Klein was born at Düsseldorf in 1849 and took his doctorate at twenty under Julius Plücker. In 1872, on being appointed to a chair at Erlangen at the age of twenty-three, he set out an inaugural programme that became the most quoted sentence in nineteenth-century mathematics. The problem it solved was real. Through the nineteenth century geometry had been fragmenting: projective geometry, affine geometry, the geometries in which Euclid's parallel assumption fails, and more. Each had its own results and there was no way to say how they were related, or even whether they were rivals. Klein's answer was that a geometry is not a body of facts about figures at all. It is the study of the properties that a particular family of transformations leaves unchanged. Fix the family and you have fixed the geometry; allow more transformations and fewer properties survive, so the geometries arrange themselves in an order. Ordinary school geometry is what survives sliding, turning and reflecting; allow enlargements and lengths stop mattering while shapes persist. His own research included the one-sided surface named after him, the theory of functions invariant under substitutions, and a celebrated book connecting the icosahedron to the equation of the fifth degree. In the early 1880s he competed intensely with Poincaré over automorphic functions, worked himself into a breakdown, and never fully recovered his powers as a researcher. What he did instead was build. From 1886 he was at Göttingen, and he turned it into the place mathematicians came to from everywhere: he brought Hilbert, he founded a mathematical reading room and a tradition of applied work, he cultivated engineering and industry, and he supported the admission of women — supervising Grace Chisholm Young, the first woman to take a German doctorate in mathematics, in 1895. He spent his later years on the reform of school mathematics teaching, a subject most of his colleagues thought beneath a research mathematician. The Erlangen programme was little noticed for two decades. It became famous partly because Klein kept reissuing it.

Major achievements

• Set out the Erlangen programme in 1872, defining a geometry by its transformations
• Gave the one-sided surface that carries his name
• Connected the icosahedron to the solution of the equation of the fifth degree
• Built Göttingen into the leading mathematical institute in the world
• Brought Hilbert to Göttingen and supervised the first German female doctorate in mathematics

Life in brief

• 1849 — Born at Düsseldorf: Doctorate at twenty under Plücker.
• 1872 — The Erlangen programme: A geometry defined by what its transformations leave unchanged.
• 1880s — Poincaré and the breakdown: Competition over automorphic functions, then collapse from overwork.
• 1886 — Göttingen: The beginning of nearly forty years building the institute.
• 1895 — Hilbert arrives: And Grace Chisholm Young takes her doctorate.
• 1908 — School reform: Chairing the international commission on mathematics teaching.
• 1925 — Died at Göttingen: Having outlived the institute's greatest years by a decade.

Explore the life of Klein — five chapters

1. A Chair at Twenty-Three — Düsseldorf, Plücker, and a rapid start
2. What the Transformations Leave Alone — The Erlangen programme, 1872
3. The Race with Poincaré — And what it cost
4. Building Göttingen — Hilbert, engineers, and the first women
5. Elementary Mathematics from an Advanced Standpoint — The campaign nobody expected

Read the five-chapter life of Felix Klein →

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