On April 25, 1849, German mathematician and mathematics educator Felix Klein was born. Klein is known for his work in group theory, complex analysis, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day. Klein also devised the Klein-bottle, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down.
A mathematician named Klein
Thought the Möbius band was divine.
Said he: “If you glue
The edges of two,
You’ll get a weird bottle like mine.”
Felix Klein studied mathematics and physics at the University of Bonn and intended to become a physicist after. He was appointed assistant of Julius Plücker, who held Bonn’s chair of mathematics and experimental physics. Klein received his doctorate, supervised by Plücker, from the University of Bonn in 1868. Unfortunately, Plücker passed away in the same year. He left behind an unfinished book on the foundations of line geometry, which was to be completed by Felix Klein.
The University of Erlangen appointed Klein professor when he was only 23 years old. He switched to Munich’s Technische Hochschule in 1875 and later to Leipzig, where he held the chair of geometry. During his years there, Klein published his seminal work on hyperelliptic sigma functions. In 1886, Felix Klein accepted a chair at the University of Göttingen and taught for instance mechanics and potential theory.
In 1895, Felix Klein hired David Hilbert away from Königsberg, which proved very fateful, because Hilbert continued the great work in Göttingen until his own retirement in 1932.
Klein’s first important mathematical discoveries were made in his early career. Along with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They further investigated W-curves, curves invariant under a group of projective transformations.
Klein devised the bottle named after him, the Klein Bottle, 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 Euclidean space of dimensions 4 and higher.
Like the Möbius strip, the Klein bottle is a two-dimensional manifold which is not orientable. However, its a closed manifold, meaning it is a compact manifold without boundary. The Klein Bottle can be constructed in a mathematical sense, because it cannot be done without allowing the surface to intersect itself by joining the edges of two Möbius strips together.
At yovisto, you can learn more about non-orientable surfaces, especially the Möbius Band, in a lecture by N.J. Wildberger.
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